Number Theory

A Conjecture Regarding the Primes and π

Authors: Shreyansh Jaiswal
Comments: 5 Pages.

In this article we introduce a new function:f (ϕ) We also conjecture that: f(π) = π We also rigorously show that 3.85 > f(π) > 3.1240.
Category: Number Theory

Multiple of a 6th Power is a Sum of 2 Cubes

Authors: Oliver Couto
Comments: 3 Pages. (Note by viXra Admin: An abstract is required; also please cite and list scientific references)

Multiple of a sixth power is a sum of two cubes.
Category: Number Theory

How to Know All Prime Numbers and All Numbers Divisible with a Prime Number Only: Seven

Authors: Filiberto Marra
Comments: 17 Pages.

This study on prime numbers presents a method that allows us to know divisible numbers without performing complex calculation system using examination of composition of numbers. Example 143=11x13, 493=17x29. I can know all prime numbers with this examination of composed number.
Category: Number Theory

Gaussian Integral

Authors: Edgar Valdebenito
Comments: 4 Pages.

The Gaussian integral is defined as the integral of the function exp(-x^2) over the entire real line. Mathematically, it is expressed as:integral from -infinity to infinity(exp(-x^2))dx=sqrt(pi). In this note we give some formulas related to the Gaussian integral.
Category: Number Theory

The Proof of the Riemann Conjecture

Authors: Liao Teng
Comments: 34 Pages.

In order to strictly prove the hypothesis and conjectures in Riemann's 1859 paper on the Number of Prime Numbers Not greater than x from a pure mathematical point of view, and in order to strictly prove the Generalized hypothesis and the Generalized conjectures, this paper uses Euler's formula to study the relationship between symmetric and conjugated zeros of Riemann's ζ(s) function and Riemann's ξ(s) function, and proves that Riemann's hypothesis and Riemann's conjecture are completely correct.
Category: Number Theory

Power-spectralNumbers

Authors: Walter A. Kehowski
Comments: 61 Pages.

Given $n=p_1^{e_1}cdots p_k^{e_k}$, there is a canonical isomorphism $mathbb{Z}_ncongmathbb{Z}_{p_1^{e_1}} oplus cdots oplus mathbb{Z}_{p_k^{e_k}}$. The spectral basis of $mathbb{Z}_n$ is an explicit realization of this isomorphism within $mathbb{Z}_n$ itself. This paper is a study of those numbers whose spectral basis consists of primes and powers. For example, if $M_p$ is a Mersenne prime with exponent $p$, then $2M_p$ has spectral basis ${M_p,2^{p}}$, while $2^pM_p$ has spectral basis ${M_p^2,2^{p}}$. Isospectral and isotropic numbers are introduced and many other numbers with interesting spectral bases are presented.
Category: Number Theory

Proof of the Collatz Conjecture Based on Directed Graph

Authors: Wiroj Homsup, Nathawut Homsup
Comments: 6 Pages.

The Collatz conjecture considers recursively sequences of positive integers where n is succeeded by n/2 , if n is even or (3n+1)/2 , if n is odd. The conjecture states that for all starting values n the sequence eventually reaches a trivial cycle 1, 2, 1, 2u2026u2026.The Collatz sequences can be represented as a directed graph. If the Collatz conjecture is false, then either there is a nontrivial cycle, or one sequence goes to infinity. In this paper, we construct a Collatz directed graph by connecting infinite number of basic directed graphs. Each basic directed graph relates to each natural number. We prove that the Collatz directed graph covers all positive integers and there is only a trivial cycle and no sequence goes to infinity.
Category: Number Theory

Primality Criterion for N=4*3^n-1

Authors: Predrag Terzic
Comments: 4 Pages.

Polynomial time primality test for numbers of the form 4*3^n−1 is introduced.
Category: Number Theory

"Infinitely Often"="infinity" [:] a Statistics Approach to Small Gaps Between Primes

Authors: Yung Zhao
Comments: 2 Pages.

We construct a sequence of consecutive primes. From the perspective of statistics, we analyze and handle them by the combination of the fundamental property of primes with James Maynard's result. It reveals that there are infinitely many pairs of primes which differ by two.
Category: Number Theory

Complete Proof of the Collatz Conjecture

Authors: Ugur Pervane
Comments: 5 Pages.

The Collatz conjecture has remained unsolved for a long time. In this paper, a proof of this conjecture will be provided. We know that almost all numbers will reach one by following the steps of the Collatz algorithm. Terence Tao has previously proven this. This paper uses Tao's proof to demonstrate that all numbers will eventually reach one. If almost all numbers reach the number one, then the probability that a randomly selected number from an infinite set will reach one is one. Conversely, the probability of not reaching one is zero. The probability of selecting elements of a sequence associated with a number n that violates the conjecture from an infinite set is a non-zero number, such as c, but this contradicts the proof that almost all numbers will reach one. Therefore, there is no such number n that violates the conjecture, and the conjecture holds true for all numbers. To prove that the probability of selecting elements of a sequence associated with a number n that violates the conjecture from an infinite set is a non-zero number, such as c, we look at the sequences associated with the number one. If the probability of selecting elements of each sub-branch of these sequences from an infinite set is a non-zero number, then we reach the proof we are looking for.
Category: Number Theory

Pseudo Derivative Calculus

Authors: Ahcene Ait Saadi
Comments: 5 Pages. (Name added to Article by viXra Admin as required - Please conform)

Pseudo derivatives. This is a kind of derivation that I apply to polynomials with several variables of degree n. Such that the set of solutions of the system of pseudo derivatives of a polynomial P, belong to the set of solutions of the polynomial P. With this very powerful mathematical tool, I manage to solve: -an infinity of Diophantine equations - -The equations of higher degrees -systems of linear and nonlinear equations with several variablesFinally, this calculation allows me to question the notion of 0/0.
Category: Number Theory

Proof of the Conjectures of Legendre, Andrica, Oppermann and Their Generalizations

Authors: Philippe Sainty
Comments: 21 Pages.

Proofs of the conjectures of Legendre, Andrica, Oppermann, Brocard and their generalizations using Lgk function sequences: x--> sqrt(pk/pk+1).sqrt(x).ln(x) , pk , kth prime number , x between pk and pk+1; we use reasoning by recurrence and absurdity; we obtain gaps framings between two consecutive primes, as well as the number of primes between N² and (N+1)². Generalizations with Lqk : x-->racineqième(pk/pk+1)*racineqième(x)*ln(x).
Category: Number Theory

Symmetries in Goldbach's Conjecture

Authors: Timothy Jones
Comments: 7 Pages.

We define a Goldbach table as a table consisting of two rows. The lower row counts from 0 to any n and and the top row counts down from 2n to n. All columns will have all numbers that add to 2n. Using a sieve, all composites are crossed out and only columns with primes are left. We then define a novel prime decimal system: it gives for every n remainders when n is divided by all primes less than the n. This suggests linear functions, the divisions used can give another perspective on all the column pairs. The inverses of these functions when put into tabular form give symmetries that suggest Goldbach's conjecture is correct.
Category: Number Theory

The Magic Squares of Khajuraho, Durer and the Golden Proportion

Authors: Andrey V. Voron
Comments: 4 Pages.

Based on the theoretical analysis of 4×4 pandiagonal squares, their "structure" features are shown: the invariants of the structure of 4×4 pandiagonal squares are pairs of numbers equal in sum to one of the two Fibonacci numbers — 13 or 21. It is revealed that any variant of the set of six digits of the Durer square and similar 4×4 pandiagonal squares, forming a continuous symmetric configuration, is equal in total to the integer 51. A geometric figure "cube in a cube" is constructed, which has the properties of the "golden symmetry" of 4×4 pandiagonal squares. All the numbers of the diagonals of the cube have the properties of "golden symmetry" (two numbers form in one case the total number 13, in the other — 21), and all planes having 4 angles (numbers) both the inner and outer squares of the geometric figure form a total of the Fibonacci number — 34.
Category: Number Theory

Proof of the Riemann Hypothesis Using the Decomposition ζ(z)= X(z) — Y(z) and Analysis of the Distribution of the Zeros of ζ(z) Based on X(z) and Y(z)

Authors: Pedro Caceres
Comments: 45 Pages.

Prime numbers are the atoms of mathematics and mathematics is needed to make sense of the real world. Finding the Prime number structure and eventually being able to crack their code is the ultimate goal in what is called Number Theory. From the evolution of species to cryptography, Nature finds help in Prime numbers. One of the most important advances in the study of Prime numbers was the paper by Bernhard Riemann in November 1859 called "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse" (On the number of primes less than a given quantity).In that paper, Riemann gave a formula for the number of primes less than x in terms the integral of 1/log(x) and the roots (zeros) of the zeta function defined by:[RZF] ζ(z)=∑(n=1,∞) 1/n^z Where ζ(z) is a function of a complex variable z that analytically continues the Dirichlet series. Riemann also formulated a conjecture about the location of the zeros of RZF, which fall into two classes: the "trivial zeros" -2, -4, -6, etc., and those whose real part lies between 0 and 1. Riemann's conjecture Riemann hypothesis [RH] was formulated as this:[RH]The real part of every nontrivial zero z* of the RZF is 1/2.Proving the RH is, as of today, one of the most important problems in mathematics. In this paper we will provide proof of the RH. The proof of the RH will be built following these five parts:PART 1:Description of the Riemann Zeta Function RZF ζ(z) - Introducing s limit and an approximationPART 2: The C-transformation. An artifact to decompose ζ(z) PART 3: Application of the C-transformation to f(z)=1/x^z in Re(z)≥0 to obtain ζ(z)=X(z)-Y(z) - Decomposition of ζ(z)=X(z)-Y(z) - Analysis of X(z),|X(z)|,|X(z)|^2 - Analysis of Y(z),|Y(z)|,|Y(z)|^2PART 4: Proof of the Riemann Hypothesis - Analysis of the values of z such that X(z)=Y(z), that equates to ζ(z)=0 - Proof that |X(z)|=|Y(z)| only if Re(z)=1/2 - Conclude that ζ(z)=0 only if Re(z)=1/2 for Re(z)≥0PART 5: On the distribution of the non-trivial zeros of Zeta in the critical line α= 1/2. - Algorithm N1, Algorithm H1, Algorithm H2
Category: Number Theory

A Family of Elliptic Curves with the Rank of at Least Three Arising from Quartic Curves

Authors: Seiji Tomita
Comments: 3 Pages.

In this paper, we will construct a new family of elliptic curves with the rank of at least three arising from quartic curves.
Category: Number Theory

A New Method to Find Trivial Zeros of Riemann Hypothesis

Authors: Zhiyang Zhang
Comments: 15 Pages.

The counterexample of the Riemann hypothesis causes a significant change in the image of the Riemann Zeta function, which can be distinguished using mathematical judgment equations. The counterexamples can be found through this equation.
Category: Number Theory

New Approach to Affirm the Riemann Hypothesis

Authors: Khazri Bouzidi Fethi
Comments: 4 Pages. (Note by viXra Admin: Article title should be above the author's name)

We will use another relationship between the estimation of gauss for the prime numbers and the zeta function to arrive at affirming the reimann hypothesis.
Category: Number Theory

Discussion on Modular Exponentiation

Authors: V. Barbera
Comments: 5 Pages.

This paper presents an extension of the left-to-right binary method to perform modularexponentiation 2^c (mod m) by representing the exponent not in binary notation but in base 2^b.
Category: Number Theory

A Geometric Approach to the Riemann Hypothesis: Analyzing Non-Trivial Zeros in Polar Coordinates

Authors: Bryce Petofi Towne
Comments: 31 Pages.

The Riemann Hypothesis asserts that all non-trivial zeros of the Riemann zeta function lie on the critical line ( Re(s) = frac{1}{2} ) in the complex plane. This paper explores an alternative geometric approach by analyzing the zeta function and its non-trivial zeros in polar coordinates. Transforming the problem into this framework reveals a natural symmetry about the polar axis, which corresponds to the critical line in Cartesian coordinates.We demonstrate that the (Xi(s)) function, a redefined version of the zeta function, retains the symmetry ( Xi(s) = Xi(1 - s) ) in polar coordinates, supporting the hypothesis that non-trivial zeros must lie on the critical line.This geometric perspective suggests a potential simplification in verifying the Riemann Hypothesis and offers new insights into the distribution of non-trivial zeros.
Category: Number Theory

Proof of Goldbach's Conjecture and Twin Prime Number Conjecture

Authors: Wenbin Hu
Comments: 8 Pages.

Goldbach's conjecture has been around for more than 300 years and the twin prime conjecture for more than 160 years and both are still unsolved ， both conjectures are important number theory conjectures for studying prime numbers,this article proposes a method of sequence shift to prove Goldbach's conjecture and the conjecture of twin primes, which may be a good method and seems quite easy to understand.
Category: Number Theory

An Advanced Quantum-Resistant Algorithm: Design, Implementation, and Analysis

Authors: Daniil Krizhanovskyi
Comments: 9 Pages.

The advent of quantum computing represents a paradigm shift with profound implications for the field of cryptography. Quantum algorithms, particularly Shor's algorithm, threaten to undermine the security foundations of traditional cryptographic schemes such as RSA, ECC, and DSA, which rely on the computational difficulty of integer factorization and discrete logarithms. As these algorithms become obsolete in the face of quantum capabilities, there is an urgent need for cryptographic systems that can withstand quantum-based attacks. In response to this looming threat, this paper introduces the Quantum Cryptographic Toolkit (QCT), a robust and versatile framework designed to facilitate the development, testing, and deployment of quantum-resistant cryptographic algorithms. The QCT integrates a diverse set of post-quantum cryptographic algorithms, including lattice-based methods like NewHope, code-based approaches exemplified by the McEliece cryptosystem, and isogeny-based cryptography, such as SIKE. Each of these algorithms is implemented with a focus on maintaining security even in the face of quantum computing advancements, addressing both theoretical and practical challenges. The toolkit is structured to be modular and extensible, allowing researchers and developers to seamlessly incorporate additional algorithms and cryptographic primitives as the field evolves. This paper details the design principles underlying the QCT, emphasizing the importance of modularity, extensibility, and performance optimization. We discuss the implementation strategies employed to ensure the toolkit's effectiveness across a range of cryptographic scenarios, from key exchange protocols to encryption and digital signatures. A comprehensive security analysis is provided, highlighting the resistance of each algorithm to quantum attacks, and comparing their performance to other post-quantum cryptographic solutions. In addition to the security analysis, we include extensive performance benchmarks that evaluate the computational efficiency, memory usage, and scalability of the algorithms within the QCT. These benchmarks demonstrate the practical viability of the toolkit for real-world applications, offering insights into the trade-offs between security and performance that are inherent in post-quantum cryptography. The results indicate that the QCT not only meets the stringent security requirements
Category: Number Theory

Proof of Existence of Infinite Twin Primes

Authors: Nikhil Datar
Comments: 10 Pages. (Note by viXra Admin: Please cite and list scientific references)

This submission gives a proof of existence of infinite twin primes. The basic principle used here is of Fundamental Principle of Counting and Excel has been as a major tool for explanation
Category: Number Theory

Sieving Primes from Composite Pairs Within Sets of 6n±1 Numbers

Authors: Adrian M. Stokes
Comments: 5 Pages.

The prime numbers ≥ 5 within a finite sequence of natural numbers can be found by calculating all of the values given by 6n±1 that fall within the sequence and subtracting the composites given by (6n_1±1)(6n_2±1), where n is a natural number. A test model for finding primes based on this method uses three reference sub-set multiplication tables to calculate composites and then matches these to the corresponding values in the sets {6n-1} and {6n+1}. The unmatched numbers are primes. Although this model provides a useful proof of concept, it is impractical at scale. A new method that replaces the sub-set tables with an equation to calculate 6n±1 composite pairs forms the basis of an improved model using sieve methodology.
Category: Number Theory

A Simple Probabilistic Heuristic Supporting the Collatz Conjecture

Authors: Shreyansh Jaiswal
Comments: 11 Pages.

The Collatz Conjecture, also known as the 3x+1 problem, posits that for any positive integer n, the sequence defined by the Collatz functionwill eventually reach the number 1. This conjecture has been extensively tested for a vast range of values, consistently supporting its validity. Inthis paper, we explore a probabilistic perspective to provide additional support for the conjecture. We focus on the probability that the Collatzsequence T(n), for any starting value n, reaches a power of 2—an essential step in the sequence’s progression toward 1. Our approach suggeststhat as n tends to infinity, the likelihood of the Collatz conjecture being satisfied becomes very high. This probabilistic argument aligns withthe extensive empirical evidence supporting the conjecture and offers a novel perspective on its validity. While not a formal proof, our findingscontribute to the broader understanding of the Collatz Conjecture and reinforce the conjecture’s plausibility through probabilistic reasoning.
Category: Number Theory

Geometric Symmetry of Non-Trivial Zeros of the Riemann Zeta Function in Polar Coordinates

Authors: Bryce Petofi Towne
Comments: 14 Pages.

This paper investigates the symmetry of non-trivial zeros of the Riemann zeta function (zeta(s)) through geometric analysis in polar coordinates. By transforming the complex number (s = sigma + it) into polar form, we demonstrate that the symmetry about the critical line (sigma = frac{1}{2}) necessitates (sigma = frac{1}{2}) for all non-trivial zeros. Numerical simulations further confirm the accuracy and consistency of this geometric approach. And we introduce a formula for the distribution pattern of all non-trivial zeros:[zetaleft(sqrt{frac{1}{4} + t^2} , e^{i arctan(2t)}ight) = 0]where: [r = sqrt{frac{1}{4} + t^2} quad text{and} quad theta = arctan(2t)]
Category: Number Theory

An Induction Proof For Goldbach's Conjecture

Authors: Timothy Jones
Comments: 4 Pages.

We use a series of tables with an induction argument to show Goldbach's conjecture is true.
Category: Number Theory

Fransén-Robinson Constant

Authors: Edgar Valdebenito
Comments: 5 Pages.

We give some formulas related to the Fransén-Robinson constant F=2.80777024...
Category: Number Theory

The Nicolas Criterion for a Proof of the Riemann Hypothesis

Authors: Dmitri Martila, Stefan Groote
Comments: 3 Pages.

A criterion given by Jean-Louis Nicolas is used to offer a proof for the Riemann Hypothesis in a straightforward way.
Category: Number Theory

Best Estimâtes of Density Sum Error the Prime Number Theorem for a Sequence of X Well Chosen

Authors: Khazri Bouzidi Fethi
Comments: 3 Pages.

��(��)The TNP prime number counting function��(��)Gaussian approximation for prime numbersDensity sum error for x superior 600 So 0 ≤ �� — �� ≤ 0.04.
Category: Number Theory

Geometric Analysis of Non-Trivial Zeros of the Riemann Zeta Function and Proof of σ as a Constant

Authors: Bryce Petofi Towne
Comments: 16 Pages.

This paper investigates the non-trivial zeros of the Riemann zeta function using polar coordinates. By transforming the complex plane into a polar coordinate system, we provide a geometric perspective on the distribution of non-trivial zeros. We focus on the key formula:[zetaleft(sqrt{frac{1}{4} + t^2} , e^{i arctan(2t)}ight) = 0]This formula reveals the distribution pattern of non-trivial zeros and supports the hypothesis that (sigma) must be (frac{1}{2}) and is a constant.
Category: Number Theory

Sixth Degree Diophantine Polynomial Equation

Authors: Oliver Couto
Comments: 5 Pages.

Historicaly, we note that finding a prametrization of degree six has not been easy. In the below paper the author has followed in the footsteps of below mentioned paper, ref. no. (1). In ref. no.(1), Ajai Choudhry on page 356 has parametrized equation: m(abc)(ab+bc+ca)(a+b+c)=n(pqr)(pq+pr+qr)(p+q+r), where, (m,n)=(3,32) using one parameter. In the below paper the author has parametrized the said equation for (m,n)=(1,1), but with two parameters. The trick used is to split equation (1) into three parts, so as to balance the coefficents (m,n) & such that, (m,n)=[(uvw),(xyz)]. And [(uvw),(xyz)] =(72,72), & hence they cancel each other out.
Category: Number Theory

A Set of Formulas for Prime Numbers

Authors: Rédoane Daoudi
Comments: 7 Pages.

Here I present several formulas and conjectures on prime numbers. I’m interested in studying prime numbers, Euler’s totient function and sum of the divisors of natural numbers.
Category: Number Theory

A Direct Proof that Goldbach's Conjecture is True

Authors: Timothy Jones
Comments: 4 Pages.

We introduce a Goldbach table. It consists of two rows. A bottom row counts from zero to a given n and the top counts from the right from n to 2n. The columns generated give all the whole numbers that add to 2n. We confirm that using a sieve, we do always seem to get top and bottom primes that show Goldbach's conjecture is true for the particular 2n depicted by this table. Next we cumulatively depict these tables and we see some interesting patterns. We can infer that all prime pairs will occur in one of these tables. We also see diagonal prime lines that seem to start and stop in symmetrical ways. These patterns suggest that for any even number we can choose a column and then find a prime pair.
Category: Number Theory

A Novel Framework for Improving the Prime Number Theorem Regarding Estimating the Nth Prime Number

Authors: Shreyansh Jaiswal
Comments: 20 Pages. Distributed under CC BY-SA 4.0

The Prime Number Theorem (PNT) offers a foundational approximation for the distribution of prime numbers and aids in estimating the nth prime number p(n) through p(n) ∼ n log n. This paper proposes enhancements to this approximation by introducing a correction factor C(k), refining the estimate to p(n) ≈ C(k)·n·log n. The derivation of C(k) is explored, alongside its asymptotic behavior and empiricalanalysis. A generalized formula for p(n) is also derived, eliminating variables other than n and e (Euler’s number). Empirical comparisons with traditional methods demonstrate the accuracy and computational efficiency of these new approaches. Ideal conditions for optimal performance of C(k) are examined.Graphical representations and statistical analyses support the validity of the proposed refinements. The paper concludes with a discussion on the implications of these findings and potential areas for future research.
Category: Number Theory

On Diophantine Equation Ax^4 + By^4 + Cz^4 + Dw^4 + Eu^4 = 0

Authors: Seiji Tomita
Comments: 6 Pages.

In this paper, we prove that there are infinitely many integer solutions of ax^4 + by^4 + cz^4 + dw^4 + eu^4 = 0 where a+b+c+d+e=0.
Category: Number Theory

An Original Method to Find Probable Prime Numbers

Authors: David Hill, Silvio Gabbianelli
Comments: 36 Pages. Distributed under Creative Commons Attribution-NoDerivatives 4.0 License

This paper presents an original method devised by David Hill for identifying probable prime numbers through a series of systematic steps involving division and rounding. The method begins with selecting any natural number, repeatedly dividing it by 2 until the result ends in a decimal of .5. Based on the parity of the original number, the resulting decimal is then rounded to the nearest even or odd number. This rounded number is either added to or subtracted from the original input number, often resulting in a prime number. While the method does not guarantee a prime in every case, it demonstrates a high success rate, particularly within the range of 2 to 100. An exception is made for generating the prime number 2 from the input number 1. To validate this method, two Python programs were utilized. One program tested integer numbers within a given range one by one, and the other produced a list of probable prime numbers found. Analysis of the results revealed that the method consistently found a higher number of primes than initially estimated. For example, starting in the range of 2 to 100 integers, the method found 34 primes compared to the estimated 25. This pattern held true across larger ranges, with the method finding up to 46.06% more primes out of the estimated in the range of 2 to 10,000,000.Additionally, the method identified the greatest prime numbers that extended significantly beyond the initial range limits. The trend line for the percentage increase in found primes suggested that the method becomes increasingly effective at identifying additional primes as the range expands. These findings suggest that the method has the potential to uncover a greater number of prime numbers than traditional estimation methods predict, providing a new approach to prime number discovery. This could indicate a deeper connection between composite numbers and primes through systematic division and balancing of odds and evens. Further research is needed to determine the method's effectiveness across larger and more varied ranges, but the initial results are promising.[Note: Silvio Gabbianelli created Python programs to test the method. This paper, written by him for me to present. The tests, conducted up to 10,000,000, yielded promising results.]
Category: Number Theory

On Degrees of Carry and Scholz's Conjecture

Authors: Theophilus Agama
Comments: 10 Pages.

Exploiting the notion of carries, we obtain improved upper bounds for the length of the shortest addition chains $iota(2^n-1)$ producing $2^n-1$. Most notably, we show that if $2^n-1$ has carries of degree at most $$kappa(2^n-1)=frac{1}{2}(iota(n)-lfloor frac{log n}{log 2}floor+sum limits_{j=1}^{lfloor frac{log n}{log 2}floor}{frac{n}{2^j}})$$ then the inequality $$iota(2^n-1)leq n+1+sum limits_{j=1}^{lfloor frac{log n}{log 2}floor}bigg({frac{n}{2^j}}-xi(n,j)bigg)+iota(n)$$ holds for all $nin mathbb{N}$ with $ngeq 4$, where $iota(cdot)$ denotes the length of the shortest addition chain producing $cdot$, ${cdot}$ denotes the fractional part of $cdot$ and where $xi(n,1):={frac{n}{2}}$ with $xi(n,2)={frac{1}{2}lfloor frac{n}{2}floor}$ and so on
Category: Number Theory

[Attempted] Proof of Fermat's Conjecture in Just a Few Lines

Authors: Atsu Dekpe
Comments: 9 Pages.

We present in this paper a formula for decomposing a power of an integer into a product of consecutive integers and its properties. We also discuss properties of some specific vectors (polynomials). By using these concepts, we provide simple proofs for both of Fermat's theorems. Furthermore, the proof of the great Fermat theorem is accessible to all students who have studied the notion of vector space.
Category: Number Theory

Invariant Polynomial

Authors: Ahcene Ait Saadi
Comments: 3 Pages. (Author name added to the article by viXra Admin as required; also, please cite and list scientific references)

For centuries, mathematicians have been studying polynomials, especially the zeros of polynomials. the theory of Galois states that we cannot find a general formula for solving equations greater than 4. In this article I study the invariant polynomials of degrees 6,10 and 12. When we make a variable change to these polynomials, they become two-square. Which allows us to solve equations of higher degree.
Category: Number Theory

Collatz Conjecture Proof with "Branches" of Tree

Authors: SeongJoo Han
Comments: 11 Pages.

The Collatz's Conjecture is still unsolved as ancestors said thatit is impossible. However, we did ?nd the 'In?nite Beautiful Branches' in"Collatzs's Tree". And the Branches of tree shows us the way to prove"Collatz is right" like blue sky in autumn.
Category: Number Theory

On the L-Functions from Generalized Riemann Hypothesis, Birch and Swinnerton-Dyer Conjecture, and the Prime Numbers from Polignac's and Twin Prime Conjectures

Authors: John Yuk Ching Ting
Comments: 54 Pages. Generalized Riemann hypothesis, Birch and Swinnerton-Dyer conjecture, Polignac's and Twin prime conjecture submitted Wednesday 8 November 2023.

Dirichlet eta function (proxy function for Riemann zeta function as generating function for all nontrivial zeros) and Sieve of Eratosthenes (generating algorithm for all prime numbers) are essentially infinite series. We apply infinitesimals to their outputs and analyze L-functions of elliptic curves for Birch and Swinnerton-Dyer conjecture. Riemann hypothesis asserts the complete set of all nontrivial zeros from Riemann zeta function is located on its critical line. It is proven to be true when usefully regarded as an Incompletely Predictable Problem. We ignore even prime number 2. The complete set with derived subsets of Odd Primes contain arbitrarily large number of elements and satisfy Prime number theorem for Arithmetic Progressions, Generic Squeeze theorem and Theorem of Divergent-to-Convergent series conversion for Prime numbers. Having these theorems satisfied by all Odd Primes, Polignac's and Twin prime conjectures are separately proven to be true when usefully regarded as Incompletely Predictable Problems.
Category: Number Theory

Other Relationship Between Prime Numbers and the Zeta Function

Authors: Khazri Bouzidi Fethi
Comments: 2 Pages. In French (Correction made by viXra Admin - Please conform!)

we will establish another relationship between the prime numbers and the classic zeta function of Riemman then we will prove that the sum of the inverses of the count function of the prime numbers is equal to a constant 1.48 for a well-chosen sequence of x.

on vas établir une autre relation entre les nombres premiers et la fonction zeta classique de Riemman ensuite on vas prouver que la somme des inverses de fonction de compte des nombres premiers est égal a la à une constante 1.48 pur une suite des x bien choisi.
Category: Number Theory

Estimating the Number of Primes Within a Limited Boundary

Authors: Junho Eom
Comments: 21 pages, 2 figures, 2 tables, 3 appendices

Within n^2, n boundaries were generated from the 1st to the nth, each containing n numbers. Primes less than n^2/2 were multiplied, intersected, and formed composites. At least one prime less than n or in the 1st boundary was used as a factor for the composites between n and n^2, or 2nd and nth boundaries, limiting the number of composites to (2n^2)/λ, where λ represented the wavelength of primes in the 1st boundary. Under these conditions, passively remaining numbers that were not connected to the wave of primes in the 1st boundary were all new primes between the 2nd and nth boundaries. Considering the cause-and-effect relationship among the primes less than n and the composites and new primes between 2nd and nth boundaries, the characteristics of composites could represent the characteristics of primes, and both were defined within a limited n^2 boundary. In this paper, these boundary characteristics were utilized to obtain the average number count per boundary, which led to obtaining the average number of primes per boundary. The average number of primes was multiplied by n boundaries with a coefficient of either β1 or β_√2, denoting the ratio of the number of primes. Using either β1 or β_√2, the number of primes was estimated between 10^6 and 10^28 and compared to the actual number of primes. Considering the relative error between β1 (Average 1.42%: maximum 2.92%, minimum 0.16%) or β_√2 (Ave. 0.37%: max. 0.96, min. 0.04%), it was concluded that the number of primes could be estimated with β_√2, allowing for a relative average error of 0.37%, in an equation of π(n^2)=π(n)∙n/β_√2, where 10^3 ≤ n ≤ 10^14, π(n) was the known number of primes within n, and β_√2 = ln(2√2∙n)/ln(n)+1.
Category: Number Theory

A Proof of the Collatz Conjecture

Authors: Xingyuan Zhang
Comments: 6 Pages.

In this paper, we had given a proof of the Collatz conjecture in elementary algebra. Since any given positive integer is conjectured to return to odd 1 in operations, we analyze continuous inverse operations starting with odd 1, it had proved that all of the inverse path numbers of a given non-triple is obtainable and any inverse operation path tends to infinity, we can get any odd and even, to do continuous forward operations for a positive integer obtained it will return to the odd 1 along the inverse operation paths.
Category: Number Theory

[attempted] Proof for Goldbach’s Conjecture Verification with Mathematical Induction Formula

Authors: Budee U. Zaman
Comments: 13 Pages.

This document forwards a freshly unearthed test of the Goldbach Conjecture, a longstanding enigma in the theory of numbers put forth byChristian Goldbach in 1742. In our point of view, we have been able to come up with a simple and yet stunning explanation on how numberswhich are divisible by 2 could be permanently expressed as the sum of two prime numbers. Through an extensive analysis, it will be seen that every other two numbers above 2 can always be expressed in that manner. Our evidence is based on fundamental theories of numbers and original methods that solve the problem without any difficulty. Consequently, understandingis not difficult at all. The pathway for further research in number theory has just been brought to light while at the same time indicatinghow vital determination and a variation of outlook are for any endeavour.
Category: Number Theory

Pascal Triangle: a Combinatorial Approach to Power Sums

Authors: Yasser A. Chavez
Comments: 15 Pages.

We explore three combinatorial sequences derived from Pascal’s triangle: Binomial Coefficients, the Narayana Numbers and a variant of the Binomial Coefficients. The goal is to express particular cases of the sum of powers of the first n natural numbers using combinatorial sequences. 1^p + 2^p + 3^p + 4^p + ... + n^p, where p , n ∈ ℕ The methodology we employ is based on the differences between terms. We multiply each term by n to equal the next exponent and then add each term. Finally, we identify patterns in the sequences at the intermediate or final stage.
Category: Number Theory

The Strict Proof That the Riemann Zeta Function Equation Has No Non-Trivial Zeros

Authors: Xiaochun Mei
Comments: 13 Pages.

A standard method is proposed to prove strictly that the Riemann Zeta function equation has no non-trivial zeros. The real part and imaginary part of the Riemann Zeta function equation are separated completely. By comparing the real part and the imaginary part of Zeta function equation individually, a set of equation is obtained. It is proved that this equation set only has the solutions of trivial zeros. So the Riemann Zeta function equation has no non-trivial zeros. The Riemann hypothesis does not hold.
Category: Number Theory

Analysis and Improvement of Twin Prime Density Estimation

Authors: Bruce R. Nye
Comments: 3 Pages.

The ’twin prime conjecture’ was first proposed over 100 years ago. The work of Hardy and Littlewood still remains the dominant authority with respect to identifying twin prime density. The Hardy-Littlewood conjec- ture is paired with a counting function alongside the twin prime constant (0.660016). This process estimates twin prime count to ’x’, as the error is infinitely sieved to zero. The proposed limit represents a nuanced more precise approach to estimating the number of twin primes up to n, making this formula a technical improvement over the Hardy-Littlewood formula. By incorporating additional logarithmic terms and scaling factors, this for- mula refines the asymptotic estimate, offering deeper accuracy and deeper insights into the distribution of twin primes. This refinement is significant for both theoretical studies and practical applications in number theory, as it provides a more detailed and more accurate framework.
Category: Number Theory

Conditions for Convergence of the Sequence 1/(��^��|sin��|^��)

Authors: Yudai Sakuma
Comments: 4 Pages.

It is known that if the sequence 1/(��^��|sin��|^��) converges then ��(��)≤1+��/�� , but the convergence of this sequence has not been solved. In this study, the conditions for convergence of 1/(��^��|sin��|^��) were clarified by focusing on �� such that the value of |sin��| becomes explosively small. As a result, it was confirmed that ��(��)<1+��/�� is a sufficient condition for convergence of 1/(��^��|sin��|^��) . This is the same result as in the previous study, but because the method of proof is different, we succeeded in identifying a range of values for lim��→∞1/(��^��|sin��|^��) when ��(��)=1+��/�� .
Category: Number Theory

A New Understanding on the Problem That the Quintic Equation Has No Radical Solutions

Authors: Xiaochun Mei
Comments: 30 Pages. In Chinese

It is proved in this paper that Abel’s and Galois's proofs that the quintic equations have no radical solutions are invalid. Due to Abel’s and Galois's work about two hundred years ago, it was generally accepted that general quintic equations had no radical solutions. However, Tang Jianer etc. recently proves that there are radical solutions for some quintic equations with special forms. The theories of Abel and Galois can not explain these results. On the other hand, Gauss etc. proved the fundamental theorem of algebra. The theorem declared that there were n solutions for the n degree equations, including the radical and non-radical solutions. The theories of Abel and Galois contradicted with the fundamental theorem of algebra. Due to the reasons above, the proofs of Abel and Galois should be re-examined and re-evaluated. The author carefully analyzed the Abel’s original paper and found some serious mistakes. In order to prove that the general solution of algebraic equation he proposed was effective for the cubic equation, Abel took the known solution of cubic equation as a premise to calculate the parameters of his equation. Therefore, Abel’s proof is a logical circular argument and invalid. Besides, Abel confused the variables with the coefficients (constants) of algebraic equations. An expansion with 14 terms was written as 7 terms, 7 terms were missing.We prefer to consider Galois’s theory as a hypothesis rather than a proof. Based on that permutation group had no true normal subgroup, Galois concluded that the quintic equations had no radical solutions, but these two problems had no inevitable logic connection actually. In order to prove the effectiveness of radical extension group of automorphism mapping for the cubic and quartic equations, in the Galois’s theory, some algebraic relations among the roots of equations were used to replace the root itself. This violated the original definition of automorphism mapping group, led to the confusion of concepts and arbitrariness. For the general cubic and quartic algebraic equations, the actual solving processes do not satisfy the tower structure of the Galois’s solvable group. The resolvents of cubic and quartic equations are proved to have no the symmetries of Galois’s soluble group actually. It is invalid to use the solvable group theory to judge whether the higher degreeequation has a radical solution. The conclusion of this paper is that there is only the Sn symmetry for the n degree algebraic equations. (Truncated by viXra Admin to < 400 words)
Category: Number Theory

Prime Number Distribution Proving the Twin Prime and Goldbach Conjectures

Authors: Budee U. Zaman
Comments: 7 Pages.

The paper investigates the dispersion of prime numbers as well as the twin prime and goldbach’s conjectures. The initial key feature that primenumbers are never even (apart from 2) will be presented as the basis on which a new rule concerning their distribution can be developed. In that wise, this will help us to come up with a demonstration of why there exist an infinite number of odd pairs such that their difference is equal to 2. We also show that the Goldbach conjecture is true. This means that it is possible to write any even number greater than two as the sum of two prime numbers. The results contribute fresh knowledge concerning old mathematics subjects, especially those concerning the origins of prime numbers.
Category: Number Theory

The Inconsistency Problem of Riemann Zeta Function Equation

Authors: Xiaochun Mei
Comments: 16 Pages. In Chinese

Four basic problems are found in Riemann’s original paper proposed in 1859. The Riemann hypothesis becomes meaningless. 1. It is proved that on the real axis of complex plane, the Riemann Zeta function equation holds only at point Re(s)=1/2 (s=a+ib) . However, at this point, the Zeta function is infinite, rather than zero. At other points of real axis with a be not equal to zero and b=0 and , the two sides of function equation are contradictory. When one side is finite, another side may be infinite. 2. An integral item around the original point of coordinate system was neglected when Riemann deduced the integral form of Zeta function. The item was convergent when Re(s)>1 but divergent when Re(s)<1. The integral form of Zeta function does not change the divergence of its series form. Two reasons to cause inconsistency and infinite are analyzed. 3. A summation formula was used in the deduction of the integral form of Zeta function. The applicative condition of this formula is x>0. At point x=0 , the formula is meaningless. However, the lower limit of Zeta function’s integral is x=0, so the formula can not be used. 4. The formula of Jacobi function was used to prove the symmetry of Zeta function equation. The applicable condition of this formula is also x>0. Because the lower limit of integral in the deduction was , this formula can not be used too. The zero calculation of Riemann Zeta function is discussed at last. It is pointed out that because approximate methods are used, they are not the real zeros of strict Riemann Zeta function.
Category: Number Theory

Pi's Irrationality Using Maclaurin Polynomials

Authors: Timothy Jones
Comments: 4 Pages.

Using the Maclaurin series and polynomials for sin(x) with some simple algebraic manipulations, we give a simple proof that pi is irrational.
Category: Number Theory

Is it Really that Difficult to Prove the Goldbach Conjecture?

Authors: Mary Anne Ji You, Óscar E. Chamizo Sánchez
Comments: 6 Pages.

The Goldbach conjecture, that is to say, every even number greater than 4 can be represented by the sum of two primes, is a simple and intractable statement that has been torturing mathematicians for more than 250 years. We wondered if the divide et impera method, so useful in programming and algorithmics, could provide some service here. The goal is simplify and separate the whole problem into three independent and fairly manegeable subproblems. An approach that, as far as I know, has not been tested before.
Category: Number Theory

Transcendence of Deformations of Polylogarithm Functions in Non-Zero Characteristic

Authors: David Adam
Comments: 15 Pages. In French

Let p be a prime number. In 2005, with the aim of reproving by Wade's method the transcendence for n ∈ Zp N of Γ∞(n), where Γ∞ is the Carlitz-Goss factorial, Yao introduces an uncountable class of deformations of Kochubei polylogarithm functions of which he shows the transcendence of their values u200bu200bin 1 (which is sufficient for his objective) and more generallyin 1/T^k (k ∈ N∗). In this present article, we answer Yao's question to extend this result into a non-zero algebraic. We prove this fact not only in the context of infinite place but also in that of finite places. We show an analogous result for the deformations of the polylogarithm function of Carlitz.
Category: Number Theory

Riemann Hypothesis Via Nicolas Criterion

Authors: Dmitri Martila
Comments: 3 Pages.

The Robin's Theorem with Nicolas criterion were used to prove the Riemann Hypothesis in a straightforward way.
Category: Number Theory

Empirical and Theoretical Validation of Beal's Conjecture

Authors: Óscar Reguera García
Comments: 8 Pages.

Beal's Conjecture posits that for any solution to the equation A^x + B^y = C^z with A, B, C beingpositive integers without common prime factors and x, y, z being integers greater than 2, A, B, and C must share at least one common prime factor. This study conducts a comprehensive empirical andtheoretical validation of the conjecture, using a combined theoretical analysis with computationalsimulations. No counterexamples were found in the extended range of 2 to 10,000 for A, B, C, and 3to 10 for x, y, z.
Category: Number Theory

Notes and Problems in Number Theory (Volume II)

Authors: Taha Sochi
Comments: 178 Pages.

This is the second volume of my book "Notes and Problems in Number Theory".
Category: Number Theory

Notes and Problems in Number Theory (Volume I)

Authors: Taha Sochi
Comments: 237 Pages.

This book is the first volume of a collection of notes and solved problems about number theory. Like my previous books, maximum clarity was one of the main objectives and criteria in determining the style of writing, presenting and structuring the book as well as selecting its contents.
Category: Number Theory

Fractal Patterns in Prime Number Distribution - A Novel Approach to Number Theory

Authors: Athon Zeno, Aeon Zeno
Comments: 6 Pages. (Note by viXra Admin: Please cite and list scientific references)

This paper presents a new perspective on prime number distribution, proposing a fractal-like structure that manifests at multiple scales. We introduce a mathematical framework, utilizing modular arithmetic and the Chinese Remainder Theorem, to prove self-similarity in prime distribution. Our model offers potential insights into the Riemann Hypothesis and suggests new approaches to understanding prime number gaps. Computational evidence up to 10^9 demonstrates consistent fractal dimensions across scales, agreement with predicted scaling factors, and self-similar prime gap distributions, strongly supporting our theoretical framework.
Category: Number Theory

On the No-trivial Zeros of the Zeta Function C(S)

Authors: Aziz Arbai, Amina Bellekbir
Comments: 8 Pages.

We research and explicitly expose example of an infinity of zeros (C(r+ic)=0) of RH (The Riemann hypothesis) in the critical line (having for real part r= 1/2 and c=[+or-pi/4+2kpi]/ln(2)). So there is infinity of no- trivial zeros of Riemann’s zeta function which have the real part equal to 1/2, which shows (using simple mathematics baggage) Hardy and Littlewood Theorem and give as a hope that the Riemann’s Conjecture would be true....
Category: Number Theory

An Elementary Approach to X^2 +7 = 2^n

Authors: Seiji Tomita
Comments: 2 Pages.

In this paper, we prove that the positive integer solutions of the equation x^2 +7 = 2^n are x = 1, 3, 5, 11, 181, corresponding to n = 3, 4, 5, 7, 15.
Category: Number Theory

Simple Symmetry Proves All Three Riemann's Hypotheses

Authors: Dmitri Martila
Comments: 2 Pages.

Suppose the Riemann Zeta function is multiplied by two arbitrary functions, and the resulting functions' values are equated at symmetrical points concerning the critical line Re s = 1/2. In that case, the resulting system of fourequations has to give the positions of the Zeta function's zeros. However, since the functions are arbitrary, the positions of the zero places are arbitrary, making a zero coincide with non-zero. Hence, the Riemann Hypothesis that the only zeroes are those on the critical line is true. This simple text is proof of the Riemann hypothesis, Generalized Riemann hypothesis and Extended Riemann hypothesis with accordingfunctions.
Category: Number Theory

On Prime Cycles in Directed Graphs Built with Primality Test

Authors: Marcin Barylski
Comments: 4 Pages.

One of the most famous unsolved problems in mathematics is Collatz conjecture which is claiming that all positve numbers subjected to simple 3x + 1/2 formula will eventually result in 1, with only one known cycle (1, 4, 2, 1) present in the calculations. This work is devoted to finding cycles in other interesting sequences of integer numbers, constructed with the use of some aspect of primality test.
Category: Number Theory

Clarifying an Early Step in Hardy's Transcendence of Pi Proof

Authors: Timothy Jones
Comments: 3 Pages.

We clarify and strengthen Hardy's footnote proof of an essential step in his proof of the transcendence of pi. We show that ri is algebraic if and only if r is algebraic.
Category: Number Theory

General Solution Conditions

Authors: Hajime Mashima
Comments: 41 Pages.

Modulo [is] not divisible by xyz and possible expansions.
Category: Number Theory

On the Diophantine Equation X^6 + Y^6 = W^n + Z^n, N=2,3,4

Authors: Seiji Tomita
Comments: 5 Pages.

In this paper, we proved that there are infinitely many integer solutions of X^6 + Y^6 = W^n + Z^n, n=2,3,4.
Category: Number Theory

Approaching Goldbach's Conjecture Using the Asymmetric Relationship Between Primes and Composites Within a Limited Even Boundary

Authors: Junho Eom
Comments: 6 Pages. 1 figure

The wave of primes less than integer n is known to determine new primes by removing composites within a limited boundary between n and n2. As the boundary initiated from n also includes 2n, it is possible to apply the prime wave analysis to Goldbach’s conjecture within a limited 2n boundary. Prior to analysis, the 2n boundary is divided by two: a first boundary between 0 and n and a second boundary between n and 2n. From the point of n, the waves of primes in the first boundary directly but asymmetrically connect to composites in the second boundary. Due to asymmetrical connection between primes and composites, the composites are removed and the remaining numbers are all new primes. As a result, the passively remaining new primes in the second boundary, which is not affected by the continued waves of primes, are partially but symmetrically related to the primes in the first boundary. As long as primes and new primes are symmetrical through maintaining the same distance from n, the sum of a prime and a new prime is always equal to 2n.
Category: Number Theory

Verification of the Riemann Hypothesis Using a Novel Positive Coordinate System Approach

Authors: Bryce Petofi Towne
Comments: 12 Pages.

This paper presents a novel approach to verifying the Riemann Hypothesis us- ing a redefined positive coordinate system and polar representation of complex numbers. Inspired by discussions on the nature of negative numbers, zero, and imaginary numbers, we developed a coordinate system that exclusively uses pos- itive numbers. Through this innovative method, we recalculated and confirmed several known non-trivial zeros of the Riemann zeta function. Our results consis- tently support the hypothesis that all non-trivial zeros of the zeta function lie on the critical line where the real part is 1/2. This method provides a new perspective on the Riemann Hypothesis and opens potential avenues for further mathematical exploration.Furthermore, through rigorous mathematical proof and leveraging zero consis- tency theory in complex analysis, we demonstrate that in the polar coordinate system, the Riemann Hypothesis holds true. This proof provides a significant step towards a comprehensive understanding of this profound mathematical conjecture.
Category: Number Theory

There Are Infinitely Many Integers that Can be Expressed as the Sum of Four Cubes of Polynomials

Authors: Seiji Tomita
Comments: 3 Pages.

In this paper, we prove that there are infinitely many integers that can be expressed as the sum of four cubes of polynomials.
Category: Number Theory

How to Solve Diophantine Equations

Authors: Taha Sochi
Comments: 75 Pages.

We present in this article a general approach (in the form of recommendations and guidelines) for tackling Diophantine equation problems (whether single equations or systems of simultaneous equations). The article should be useful in particular to young "mathematicians" dealing mostly with Diophantine equations at elementary level of number theory (noting that familiarity with elementary number theory is generally required).
Category: Number Theory

Explanations of the Riemann Hypothesis

Authors: Dmitri Martila
Comments: 2 Pages.

Explanations why the real part of Zeta function zeroes is always being seen on the 1/2 line.
Category: Number Theory

Goldbach's Conjecture

Authors: Bassera Hamid
Comments: 1 Page. This article was submitted to the American Mathematical Society in 05 06 2024

In this article I try to make my modest contribution to the proof of Goldbach's conjecture and I propose to simply go through its negation.
Category: Number Theory

On the Diophantine Equation of A+b+c=1/a+1/b+1/c

Authors: Seiji Tomita
Comments: Pages.

In this paper, we proved that there are infinitely many integers n such that a+b+c=1/a+1/b+1/c=n has infinitely many rational solutions.
Category: Number Theory

Discover a Proof of Goldbach’s Conjecture

Authors: Budee U. Zaman
Comments: 9 Pages.

This paper presents a new proof of the Goldbach conjecture, which is a well-known problem originating from number theory that was proposedby Christian Goldbach back in 1742. Our way gives a simple but deep understanding of the even integers can be written as the sum of two prime numbers Through examining fully we show that every other even integer larger than two will essentially represent itself in form adding up two prime numbers. The revelation of a straightforward and elegant line to this enduring conjecture comes from the use of basic number theory concepts such as; by going a step further and coming up with creative strategies. There is more evidence and sound payments she makes for her assertion as we continue.The centuries-old mathematical puzzle has been solved paving way for the exploration of new possibilities in number theory and we are grateful for the perspective and the persistence accorded us by God, which enabled us to reach this milestone.
Category: Number Theory

Prime Number Theorem with Error Correction

Authors: Ricardo Gil
Comments: 3 Pages.

The distribution and density of these zeros affect the error term in the Prime Number Theorem. If the Riemann Hypothesis holds, it implies a tighter error bound in the Prime Number Theorem.
Category: Number Theory

Parametric Solutions to Sum & Difference of Two Cubes

Authors:
Comments: 4 Pages. (Author name added to the article by viXra Admin as required)

Since Fermat’s equation,[(a^3+b^3 )=(c)^3 ]does not have a solution,we are considering the below two Diophantine equations:∶(a^3+b^3 )=w(c)^3 -----(1)(a^3-b^3 )=w(c)^3 -----(2)Also, equation (2) above has been discussed in the book by Tito Piezas (Ref. # 3).
Category: Number Theory

A New Approach to the Goldbach Conjecture

Authors: Jim Rock
Comments: 2 Pages.

In 1742 Christian Goldbach suggested that any even number four or greater is the sum of two primes.The Goldbach conjecture remains unproven to the present day, although it has been verified for all even numbers up to 4 × 10^18. Previously this problem has been attacked using deep analytical methods and with complicated integer sieves. This paper takes an entirely new approach to the Goldbach conjecture using pairs of composite integers (composite pairs) that are used to find pairs of prime numbers (prime airs) that sum to the same even natural number.
Category: Number Theory

A Complete Proof of The abc Conjecture: It is Easy as abc!

Authors: Abdelmajid Ben Hadj Salem
Comments: 3 Pages.

In this paper, we consider the $abc$ conjecture. Assuming that the conjecture $c<rad^{1.63}(abc)$ is true, we give the proof that the $abc$ conjecture is true.
Category: Number Theory

The Explicit Formula of Bernoulli Numbers

Authors: Abdelhay Benmoussa
Comments: 11 Pages.

The aim of this paper is to give an elementary proof of a well-known explicit formula for Bernoulli numbers, with some remarks.
Category: Number Theory

Notes About Prime Numbers in Different Number Basis and the Goldbach Conjecture

Authors: Juan Elias Millas Vera
Comments: 2 Pages.

I share some thoughts about prime number and the use of basis in number theory.
Category: Number Theory

A Beautiful Geometric Property of the Complex Numbers: Statement and Proof in 6 Sentences

Authors: Lance Horner
Comments: 1 Page.

We relate the product of the vertices of a regular n-gon in the complex plane to the nth powers of the n-gon's center and complex radii.
Category: Number Theory

On an Identity Leading to the Fermat’s Last Theorem in a Short Computation

Authors: Pawel Piskorz
Comments: 3 Pages.

We propose a procedure which allows to compute the only acceptable natural exponents of the positive integers X, Y, Z in the equation of the Fermat’s Last Theorem. We use the approach similar to the one applied in computing of the expected value and the standard deviation of number of successes in Bernoulli trials presented by Kenneth S. Miller.
Category: Number Theory

Behavior of Non-trivial Zeros and Prime Numbers in Reimann Hypothesis

Authors: Ricardo Gil
Comments: 3 Pages.

The Riemann Hypothesis proposes a specific location (the critical line) for the non-trivial zeros of the Riemann zeta function. This paper argues that the aperiodicity observed in the distribution of these non-trivial zeros and the distribution of primes numbers is a fundamental property. A periodic zeta function would significantly alter its behavior, rendering it irrelevant to studying prime number distribution. Conversely, a periodic pattern in prime numbers or a deviation of non-trivial zeros from the critical line would disprove the Riemann Hypothesis. The observed aperiodicity in both prime number distribution and the zeta function's non-trivial zeros strengthens the case for the Hypothesis' validity. This aperiodicity suggests a deeper connection between prime numbers and the zeta function, one that wouldn't exist with a periodic structure.
Category: Number Theory

An Attempted Simple Proof Fermat’s Last Theorem

Authors: Geon Cho
Comments: 2 Pages. (Correction made by viXra Admin to conform with the requirements of viXra.org - Future non-compliant submission will not be accepted)

This paper attempts a simple proof for the Fermat’s Last Theorem.
Category: Number Theory

Function for Prime Numbers: Searching for Consecutive Prime Numbers with Billions of Digits

Authors: Massimo Russo
Comments: 10 Pages.

With the right equipment and a specially developed algorithm, I believe that my function could potentially find the largest prime number ever, even one with billions of digits. It might even be possible to find two or more consecutive prime numbers each with billions of digits.
Category: Number Theory

Applying Infinitesimals to Outputs from Sieve of Eratosthenes and Riemann Zeta Function When Treated as Infinite Series

Authors: John Yuk Ching Ting
Comments: 31 Pages.

We treat Sieve of Eratosthenes (algorithm that generates all prime numbers) and Dirichlet eta function (proxy function for Riemann zeta function that generates all nontrivial zeros) as infinite series. We apply infinitesimals to their outputs. We ignore even prime number 2. Based on every even Prime gaps 2, 4, 6, 8, 10...; the complete set and its derived subsets of Odd Primes all contain arbitrarily large number of elements while fully satisfying Prime number theorem for Arithmetic Progressions, Generic Squeeze theorem and Theorem of Divergent-to-Convergent series conversion for Prime numbers. With these theorems satisfied by all Odd Primes, Polignac's and Twin prime conjectures are proven to be true when usefully regarded as Incompletely Predictable Problems. Riemann hypothesis proposes all nontrivial zeros of Riemann zeta function are located on its critical line. It is separately proven to be true when usefully regarded as an Incompletely Predictable Problem.
Category: Number Theory

The Riemann Hypothesis Has Three Types of Non Trivial Zeros

Authors: Zhiyang Zhang
Comments: 8 Pages.

This paper classifies non trivial zeros based on the Riemann Zeta function. Through this operation, we can clearly understand the distribution pattern of non trivial zeros and predict the position of the next zero point. You can know that the Riemann hypothesis has three types of non trivial zeros, and the first type of non trivial zeros is located on the critical line, while the second and third types of zeros are not. Meanwhile, through a series of equation derivations, we can also understand why it is so difficult to find counterexamples of the Riemann hypothesis.
Category: Number Theory

N-Dimensional Representation of the Set of Prime Numbers

Authors: Viktor Strohm
Comments: 6 Pages. (Converted to pdf and author name added by viXra admin - Please only submist article in pdf format)

This paper reveals/discusses the connection between the difference of prime numbers and the remainder of division by 6, the periodicity of the remainder.
Category: Number Theory

Motzkin Numbers

Authors: Edgar Valdebenito
Comments: 5 Pages.

Motzkin numbers have many combinatorial interpretations. In particular, M(n) is the total number of ways in which it is possible to draw non-intersecting chords between n points on a circle.
Category: Number Theory

A Proof of Twin Prime Conjecture by Clement’s Theorem

Authors: Toshiro Takami
Comments: 6 Pages.

I proved the Twin Prime Conjecture by using Clement’s theorem. I was able to transform below. (n is positive integer) 4×(6n − 2)! + 6n + 3 ≡ 0 (mod (6n − 1)(6n + 1)). Even if the number(n) reaches the limit, use n=x+1,n=x+2...n=x+18 from n=x. By n=x+18, new twin prime numbers are found.In this way, even larger twin primes are born.Repeat this. That is, Twin Primes exist forever.
Category: Number Theory

The Mystery of Syracuse: Exploration of a Fascinating Mathematical Conjecture

Authors: Mostafa Senhaji
Comments: 24 Pages. In French

Delve into the fascinating depths of conjectureof Syracuse through this captivating work. Of its humble beginnings with complex ramifications, explore a mathematical problem that captured the imagination of researchers for decades. Discover the first empirical observations which led to the formulation ofthe conjecture, as well as the various attempts to solve, from classical approaches to modern methods. But the Syracuse conjecture is not limited to its mathematical aspects. Explore his connections with others fields of science, from cryptography to the theory of information, through musical analogy and geometric representations. At each step, discovernew ways this simple conjecture can illuminate complex concepts in other areas. This book offers an in-depth exploration of theSyracuse conjecture, combining mathematical rigor and creative imagination. It also offers aformal demonstration of convergence towards 1 for all initial value, thus consolidating our understanding of this intriguing phenomenon. Whether you are a researcher seasoned or passionate about mathematics, let yourselfinspire and challenge through this journey through a problem that continues to captivate the minds of the world's entire mathematicians.
Category: Number Theory

SCQ Two High Cycles of Links

Authors: Rolando Zucchini
Comments: 12 Pages.

With reference to the paper Syracuse Conjecture Quadrature (SCQ), this article reports two links of high horizons such that Ꝋ(l) < Ꝋ(m), calculated by Theorem of Independence. By application of the same theorem it’s confirmed the link of odd number 27. In this way it’s sure that the cycles of links can be managed to our liking. Moreover the procedure explains show the beauty and the magical harmony of odd numbers. At the same time it’s confirmed that SC (or CC) is not fully verifiable as further highlighted by the four illustrative patterns. There are no doubts: it’s a particular sort of the Circle Quadrature, but its initial statement is true. In other words: BIG CRUNCH (go back to 1) is always possible but BIG BANG (to move on) has no End.
Category: Number Theory

[An Attempted] Proof of Collatz Conjecture

Authors: Ryujin Choi
Comments: 4 Pages. In Russian (Correction made by viXra Admin - Future non-compliant submission will not be accepted))

[This paper gives an attempted proof of [the] Collatz conjecture[.]
Category: Number Theory

The Mathematical Expression and Approximate Numerical Value of the Counterexample of the Riemann Hypothesis

Authors: Zhiyang Zhang
Comments: 3 Pages.

In the process of searching for counterexamples of the Riemann hypothesis using a computer, I accidentally discovered the possibility of counterexamples in a region. After delving into the derivation of mathematical formulas, I found that a perfect mathematical expression can be used to describe them. The position of the counterexample is right next to the area where s=1.
Category: Number Theory

A Proof of Fermat’s Last Theorem by Relating to Two Polynomial Identity Conditions

Authors: Tae Beom Lee
Comments: 3 Pages.

Fermat's Last Theorem(FLT) states that there is no natural number set {a,b,c,n} which satisfies a^n+b^n=c^n or a^n=c^n-b^n, when n≥3. In this thesis, we related LHS and RHS of a^n=c^n-b^n to the constant terms of two monic polynomials f(x)=x^n-a^n and g(x)=x^n-(c^n-b^n). By doing so, the conditions to satisfy the number identity, a^n=c^n-b^n, are transferred to the conditions to satisfy the polynomial identity, f(x)=g(x), which leads to a trivial solution, a=c,b=0, when n≥3.
Category: Number Theory

Decimalising Integer Sequences

Authors: Julian Beauchamp
Comments: 9 Pages. (Author name added to the article by viXra Admin as required; also, please cite and list scientific references - Please conform in the future!)

In this paper, we observe how some well-known integer sequences when divided by powers of 10 and summed to infinity have a unique discrete value, similar to a person's `DNA'.
Category: Number Theory

The Distribution of Prime Numbers ≥ 5 Within Sequences of Natural Numbers

Authors: Adrian M. Stokes
Comments: 11 Pages.

The prime numbers ≥ 5 within a finite sequence of natural numbers can be found arithmetically by calculating all of the values of 6n-1 and 6n+1 that fall within the sequence and subtracting the composites given by (6n_1±1)(6n_2±1), where n is a natural number. For a given value of n_1, successive (6n_1-1)(6n_2+1), (6n_1-1)(6n_2-1) and (6n_1+1)(6n_2+1) composites occur at a regular interval, which increases by 36 from one value of n_1 to the next. When combined, these regular but different intervals create disorder in the sequences of 6n-1 and 6n+1 composites, which in turn creates the apparent randomness of the primes in the sequence of natural numbers. Furthermore, {(6n_1-1)(6n_2-1)} and {(6n_1+1)(6n_2+1)} numbers are subsets of 6n+1 composites whereas the only subset of the 6n-1 composites is {(6n_1-1)(6n_2+1)}. This creates a slight inequality in the proportions of composites and primes between the sets {6n-1} and {6n+1}, which otherwise have an equal number of members overall.
Category: Number Theory

Syracuse-Collatz Conjecture: Exploration, Analysis and Demonstration of a Mathematical Enigma (In English)

Authors: Mostafa Senhaji
Comments: 15 Pages.

In the infinite universe of numbers, the Syracuse conjecture emerges as a captivating enigma, defying mathematical conventions and arousing the curiosity of the most daring minds.
Category: Number Theory

The Distance Between Infinite Prime Numbers is ≥ 2

Authors: Giovanni Di Savino
Comments: 3 Pages. (Note by viXra Admin: Please cite and list scientific references)

Euclid and other mathematicians have emonstrated that prime numbers are infinite and, not being able to state how many prime numbers there are and how much time and space is needed to know their value, to satisfy the twin prime conjecture or Goldbach's conjecture , it will never be possible to elaborate all the possible combinations and values u200bu200bthat can be obtained by adding two or three of the infinite prime numbers but it is possible to know all the possible combinations and values u200bu200bthat can be obtained by adding two or three of the prime numbers, known, which are less than or equal to 2n+1.
Category: Number Theory

Minimal Dividing Odd Subsets Are Related to the Golden Ratio

Authors: Mathis Antonetti
Comments: 4 Pages.

In this note, we give a proof for a lower bound (though not a satisfying one) of the density of minimal dividing odd subsets where the goldenratio surprisingly appears. We also provide some other properties of such integer subsets and some new insight on the relationship between minimaldividing odd subsets and the Goldbach conjecture. We argue that the study of minimal dividing odd subsets is a new and interesting starting point to prove the Goldbach conjecture.
Category: Number Theory

Proof of Collatz Conjecture

Authors: Bob Ross
Comments: 4 Pages.

Proof of this conjecture has been elusive for over 60 years. The key to a proof was to find the right combination of logic and equations to complete the proof. We section the Natural numbers into 3 mutually exclusive sets. We first assume that for the first set that there is a number in the set that does not obey Collatz and show this cannot be true since it leads to a contradiction. Using this result we show that the other 2 sets must also obey Collatz.
Category: Number Theory

The Density of Minimal Dividing Odd Subsets for the Even Numbers is Asymptotically Normal

Authors: Mathis Antonetti
Comments: 5 Pages.

In this notice, we introduce the problem of minimal dividing odd subsets for the even numbers and we show that the density of such subsets of $n$ elements is asymptotically normal (that is at least decreasing as $frac{1}{n}$). We argue that understanding the problem of minimal dividing oddsubset might lead to new approaches to solving NP-hard problems.
Category: Number Theory

The Goldbach Conjecture: Eternal Mathematical Riddle

Authors: Mostafa Senhaji
Comments: 8 Pages. In French (Translation made by viXra Admin - Future non-compliant submission will not be accepted)

The Goldbach Conjecture is like a bewitching enigma, a melody whose first notes always elude our reach, challenging our most brilliant minds to unravel its mystery.
Category: Number Theory

Proofs for Collatz Conjecture and Kaakuma Sequence

Authors: Dawit Geinamo Bambore
Comments: 38 Pages.

The objective of this study is to present precise proofs of Collatz conjecture and to introduce some interesting conjectures on Kaakuma sequence. We Proposed a novel approach that tackles the Collatz conjecture in different techniques with different point of views. The study also shows some general truths and behaviors of Collatz conjecture. Finally, the proof discovers Qodaa ration test that works for all complex and complicated Kaakuma sequence and new characteristics of the sequence. This proof help to changes some views of researchers on unsolved problems and some point of views on probability in infinite range. In this study we discovered the dynamicity of Collatz sequence and reflection and interpretation of the probabilistic proof Collatz Conjecture.Our investigation culminates in the formulation of a set of conjecture encompassing lemmas and postulates, which we rigorously prove using a combination of analytical reasoning, numerical evidence, and exhaustive case analysis. These results provide compelling evidence for the veracity of the Collatz Conjecture and contribute to our understanding of the underlying mathematical structure. Finaly we derived some behaviors of Collatz sequence like translation, reflection, divisibility, constants, successive division, evenly distribution, iteration groups, huge untrivial cycle and we constructed point of views on Consistency of Constants, instantaneously falling values, contradictions on density and Qodaa ratio test.
Category: Number Theory

Contribution to Lonely Runner Conjecture

Authors: Radomir Majkic
Comments: 6 Pages.

The Lonely Runner conjecture finds its mathematical description in winding the runner's linear paths into the complete cycle, c-cycle, on the unit track circle. All runners, to finish the competition, must complete the c-cyclesimultaneously. Any collection BT mR_{cn}ET of the BT cnET integer speeds runners and maximum speed BT v_{cn} =nET is a subset of the enveloping collection BT mR_{n}={ 1,2,3,cdots,n}ET of the BT n,;n>cn,ETrunners with the maximum speed BT n.ET[0.50mm]The time period BT 1_{ct}=1/nET of the fastest runner, f-runner BT n,ET defines the set of BT nET right half open time f-segments of the measure BT 1_{ct},ET which cover the c-cycle time domain of the measure BT 1.ETThe winding mapping of the linear paths BT X(t)ET associates with the c-cycle Graph BT G,ET the union of the BT nET individual graphs BT g=(t,X(t)),ET reduced to the domain of the c-cycle. The time domain segmentation partitions the Graph BT GET into BT nET Subgraphs BT G_{i},ET each one on the one of the BT nET f-segments. The final Subgraph bundle sinks into the point BT (1,1).ETAt the end of the first f-segment, all the runners arranged into f-constellation at the BT nET fixed, stationary points on the unit circle in the sequence of the increasing speeds. However, at the final f-segment, the runners, on the way to the starting point, are arranged at the decreasing speed order at the same stationary points.The speed order inversion inverts the slope order of the graphs on the final Subgraph bundle.Finally, the infimum graph BT g_{n-1}ET of the Subgraph bundle of the BT n-1ET runner's mutual separation graphs, the graph of thelargest slope connects the points BT (0,n-1)1_{cl}ET and BT (1,1)1_{cl}.ETConsequently, the Lonely Runner conjecture is true on the set BT mR_{n},ET and must be true on any of its subset BT mR_{cn},ET
Category: Number Theory

A Formula About π

Authors: Rédoane Daoudi
Comments: 1 Page.

Here I present one formula about π.
Category: Number Theory

Study of the Riemann Zeta Function for Odd Integers

Authors: Jesús Sánchez
Comments: 34 Pages.

In this paper we will try to find a solution for the Riemann Zeta function for odd integers. We will start with ζ(3) (the Riemann Zeta function with s=3) emulating the "Basel problem". But instead of using a sine or cosine function, using functions similar to these:f(x)=1-x^3/3!+x^6/6!-x^9/9!+⋯f(x)=1/3 (e^(-x·e^(1/3 (2πi) ) ) 〖+e〗^(-x·e^(2/3 (2πi) ) )+e^(-x·e^(3/3 (2πi) ) ) )We will discover that the process itself seems ok, but with a problem. The solutions of the above functions are not periodic, so we cannot emulate the "Basel problem" perfectly, obtaining the following value:ζ(3)=(π^3/(3!(√3)^3 ))/(1-1/2^3 )=1.1366020≠1.202056903With a small correction, we arrive to:ζ(3)=(π^3/(3!(√3)^3 ))/(1-1/2^3 ) e^(2/√3)/3=1.202173775≈1.202056903 [7]But not getting the correct value anyhow. The only way of obtaining the correct value would be to find a function of the form:g(x)=1-r(3)·x^3/3!+r(6)·x^6/6!-r(9)·x^9/9!+⋯That has periodic zeros. Where r(n) is an unknown function to be calculate/discovered.We have also generalized this study to calculate a general ζ(k) where k can be higher odd numbers, or even numbers. Having ζ(k) for even numbers would lead to obtaining a closed equation for the Bernoulli numbers. If a generalization for k as a general complex number was possible, we could even consider k=½+it, obtaining a closed function for the zeros of the Riemann Zeta function.
Category: Number Theory

The Extension of the Riemann's Zeta Function

Authors: Mohamed Sghiar
Comments: 7 Pages.

Prime numbers [See 1-7] are used especially in information technology, such as public-key cryptography , and recall that the distribution of prime numbers is closely related to the non-trivial zeros of the zeta function therefore related to the Riemann hypothesis.Here I introduce the function $circledS$: $ (X,z) longmapsto prod_{pinmathcal{P}} frac1{1-X/p^{z}} $ which is a generalization of the function $zeta $ of Riemann that I will use to prove the Riemann hypothesis.
Category: Number Theory

A Short Proof of Fermat's Last Theorem Based on the Difference in Volume of Two Cubes

Authors: Sigrid M. L. Obenland
Comments: 5 Pages.

Over the centuries, numerous mathematicians have tried to proof Fermat’s Last Theorem. In the year 1994, Fermat’s Last Theorem in the form of a^m + b^m = c^m with a, b and c being natural numbers and m being a natural number > 2 was shown to be correct In this publication I demonstrate that the difference in volume of two cubes having different side lengths cannot be a cube in itself with a side length having the value of a natural number.This also holds for cubes having higher dimensions than three, since the surfaces of these cubes all consist of three-dimensional cubes.
Category: Number Theory

Syracuse-Collatz Conjecture: Exploration, Analysis and Demonstration of a Mathematical Enigma

Authors: Mostafa Senhaji
Comments: 12 Pages. In French

In the infinite universe of numbers, the Syracuse conjecture emerges as a captivating enigma, defying mathematical conventions and arousing the curiosity of the most daring minds.
Category: Number Theory

On the Convergence of the Generalized Collatz Function

Authors: Oussama Basta
Comments: 6 Pages.

The Collatz conjecture, which states that repeated application of the function ( f(n) = begin{cases} n/2, & text{if } n equiv 0 pmod{2} 3n+1, & text{if } n equiv 1 pmod{2} end{cases} ) to any positive integer ( n ) will eventually reach the number 1, has been a long-standing open problem in mathematics. In this paper, we investigate a generalized version of the Collatz function, denoted as ( f(E, T) ), where ( E ) is a positive integer and ( T ) is a fixed positive integer. We prove that for any positive integer ( E ), repeated application of ( f(E, T) ) will eventually lead to an even number. Furthermore, we show that any even number will eventually reach a power of 2 under repeated application of ( f(E, T) ), and once a power of 2 is reached, the sequence will enter the cycle ( 1 ightarrow 4 ightarrow 2 ightarrow 1 ). These results provide new insights into the behavior of the generalized Collatz function and its convergence properties.
Category: Number Theory

Attempt at a Proof of Collatz Conjecture

Authors: Ryujin Choi
Comments: 2 Pages. (Correction made by viXra Admin to conform with the requirements of viXra.org - Future non-compliant submission will not be accepted!)

This is an attempt of a proof of Collatz Conjecture.
Category: Number Theory

Contributions to the Langlands Program

Authors: Romain Viguier
Comments: 7 Pages. (Note by viXra Admin: Please cite and list scientific references as reminded previously - Future non-compliant submission/repalcement will not be accepted!)

This article is a contribution to the Langlands Program.
Category: Number Theory

Simple Arrangement in Which it is Possible to Arrange the Prime Numbers in Order to Obtain a Range of Occupied Consecutive Positions Greater Than Those Defined by the Legendre Conjecture

Authors: Andrea Berdondini
Comments: 4 Pages.

The Legendre conjecture tells us that there always exists a prime number between ��^2 and (�� + 1)^2. In this article, we will see that it is possible to arrange prime numbers less than or equal to N in order to obtain a consecutive number of occupied positions greater than the interval from ��^2to (�� + 1)^2. It is important to note that this result does not define a counterexample to the Legendre conjecture, but it represents an interesting theoretical result that can help us resolve many of the conjectures regarding the gap between two prime numbers
Category: Number Theory

Proving & Teaching Beal Conjecture

Authors: A. A. Frempong
Comments: 8 Pages. Copyright © by A. A. Frempong

By applying basic mathematical principles, the author surely, and instructionally, proves, directly, the original Beal conjecture which states that if A^x + B^y = C^z, where A, B, C, x, y, z are positive integers and x, y, z > 2, then A, B and C have a common prime factor. One will let r, s, and t be prime factors of A, B and C, respectively, such that A = Dr, B = Es, C = Ft, where D, E, and F are positive integers. Then, the equation A^x + B^y = C^z becomes D^xr^x + E^ys^y = F^zt^z. The proof would be complete after showing that the equalities, r^x = t^x, s^y = t^y and r = s = t, are true. More formally, the conjectured equality, r^x = t^x would be true if and only if (r^x /t^x) =1; and the conjectured equality s^y = t^y would be true if and only if (s^y/ t^y) = 1. These conjectures would be proved in the Beal conjecture proof. The main principle for obtaining relationships between the prime factors on the left side of the equation and the prime factor on the right side of the equation is that the power of each prime factor on the left side of the equation equals the same power of the prime factor on the right side of the equation. High school students can learn and prove this conjecture for a bonus question on a final class exam.
Category: Number Theory

Elementary Matrix Algorithm of Order 3

Authors: Ruslan Pozinkevych
Comments: 3 Pages. The author analyzes algorithm for 3 component vector bases

The aim of our research is to establish relationship between linearly independent vectors and the R^3 Space The means for that are going to be taken from Balanced Ternary analysis e.g triplets with entries (-1,0,1) This is not going to be any specific vector, for instance (-1,0,1) It’s going to be rather a Coordinate system with 8 quadrants and directional vectors The novelty of the proposed research lies in the fact that unlike traditional Cartesian system-vector basis our system uses 8 vectors which are at the same time a polyvector
Category: Number Theory

Collatz Conjecture Integer Series Has no Looping Except One

Authors: Tsuneaki Takahashi
Comments: 2 Pages. (Note by viXra Admin: Further repetition/regurgitation will not be accepted!)

If the series of Collatz Conjecture integer has looping in it, it is sure the members of the looping cannot reach to value 1. Here it is proven that the possibility of looping is zero except one case.
Category: Number Theory

Merging the Goldbach and the Bunyakovsky Conjecture into a Unified Second Order Prime Axiom and Investigating Much Beyond the Goldbach Conjecture and the Prime Number Theorem

Authors: Alexis Zaganidis
Comments: 30 Pages.

Merging the Goldbach and the Bunyakovsky conjecture into a Unified second order Prime Axiom and investigating much beyond the Goldbach conjecture and the prime number theorem.
Category: Number Theory

Kochanski's Approximation of pi

Authors: Edgar Valdebenito
Comments: 2 Pages.

The problem of the exact rectification of a circle cannot by solved by classical geometry. Many approximate methods have been developed. Such an elegant one is Kochanski's construction.
Category: Number Theory

Division by Zero is Incoherent and Contradictory

Authors: Paul Ernest
Comments: 3 Pages.

A number of authors have claimed that Division by Zero and in particular the Division of Zero by Zero (0/0) can be computed and has a definite value (Mwang 2018, Saitoh & Saitoh 2024). I refute these claims. This is trivial, but despite its elementary standing, some peripheral or recreational mathematicians make claims about 0/0 or k/0 having some value, or in some cases, several values in different contexts, according to the author’s whim. Division by zero is undefined and attempts to define it lead to contradiction.
Category: Number Theory

Collatz Conjecture, Pythagorean Triples, and the Riemann Hypothesis: Unveiling a Novel Connection Through Dropping Times

Authors: Darcy Thomas
Comments: 14 Pages. (Note by viXra Admin: Please cite and list scientific references in the future))

In the landscape of mathematical inquiry, where the ancient and the modern intertwine, few problemscaptivate the imagination as profoundly as the Collatz conjecture and the quest for Pythagorean triples. The former, a puzzle that has defied solution since its inception in the 1930s by Lothar Collatz, asks us to consider a simple iterative process: for any positive integer, if it is even, divide it by two; if it is odd, triple it and add one. Despite its apparent simplicity, the conjecture leads us into a labyrinth ofdiverse complexity, where patterns emerge and dissolve in an unpredictable dance. On the other hand, Pythagorean triples, sets of three integers that satisfy the ancient Pythagorean theorem, have been a cornerstone of geometry since the time of the ancient Greeks, embodying the harmony of numbers and the elegance of spatial relationships. This exploratory paper embarks on an unprecedented journey to bridge these seemingly disparatedomains of mathematics. At the heart of this exploration is the discovery of a novel connectionbetween Collatz dropping times and Pythagorean triples. I will demonstrate how the dropping timeof each odd number can be uniquely associated with a Pythagorean triple. As you will see, the triplesseem to be encoding spatial information about Collatz trajectories. As we begin to work with triples, we’ll be motivated to move from the number line to the complex plane where we find structure and behavior resembling that of the Riemann Zeta function and it’s zeros.
Category: Number Theory

The Symmetry of D2n+2n 、D2n×2n 、D1/2×1/2、D∞+i and Numbers Conjectures

Authors: Yajun Liu
Comments: 12 Pages. (Auther name re-ordered by viXra Admin - Future non-compliant submission/repalcement will not be accepted)

In this paper, we discuss the symmetry of D2n+2n 、D2n×2n 、D1/2×1/2、D∞+i and we find that using the symmetry characters of Natural Numbers we can give proofs of the Prime Conjectures: Goldbach Conjecture、Polignac’s conjecture (Twins Prime Conjecture) and Riemann Hypothesis. . We also gave a concise proofs of Collatz Conjecture in this paper. And we found that if the Goldbach Conjecture、Polignac’s conjecture (Twins Prime Conjecture) were proofed, we also can get a concise proof of Fermat Last Theorem and get an Unified Field Theory for physic.
Category: Number Theory

Syracuse Conjecture

Authors: Mostafa Senhaji
Comments: 7 Pages. In French

IL s'agit d'une séquence très simple d'opérations sur les nombres qui ramène toujours au même endroit, le nombre 1. D'abord un amusement, cette étonnante suite est devenue troublante pour les mathématiciens qui ne se lassent pas de l'explorer sans avoir encore réussi à la domestique.

This is a very simple sequence of number operations that always returns to the same place, the number 1. At first an amusement, this astonishing sequence has become disturbing for mathematicians who never tire of exploring it without having yet succeeded in domesticating it.
Category: Number Theory

A Machine Learning Guided Proof of Beal's Conjecture

Authors: Jonathan Wilson
Comments: 15 Pages.

This paper presents a proof of Beal's conjecture, a long-standing open problem in number theory, guided by insights from machine learning. The proof leverages a novel combination of techniques from modular arithmetic, prime factorization, and the theory of Diophantine equations. Key lemmas, including an expanded version of a modular constraint and a pairwise coprimality condition, are derived with the help of patterns discovered through computational experiments. These lemmas, together with a refined conjecture based on the distribution of prime factors in the dataset, are used to derive a contradiction, proving that any solution to Beal's equation must have a common prime factor among its bases. The proof demonstrates the potential of machine learning in guiding the discovery of mathematical proofs and opens up new avenues for research at the intersection of artificial intelligence and number theory.
Category: Number Theory

Geometric Interpretations of Riemann Hypothesis and the Proof

Authors: Tae Beom Lee
Comments: 5 Pages.

The Riemann zeta function(RZF), ζ(s), is a function of a complex variable s=x+iy. Riemann hypothesis(RH) states that all the non-trivial zeros of RZF lie on the critical line, x=1/2. The symmetricity of RZF zeros implies that if ζ(α+ iβ)=0, then ζ(1-α+ iβ)=0, too. In geometric view, if RH is false, two trajectories ζ(α+ iy) and ζ(1-α+ iy) must intersect at the origin when y=β. But, according to the functional equations of RZF, two trajectories ζ(α+ iy) and ζ(1-α+ iy) can’t intersect except when α=1/2. So, they can’t intersect at the origin, too, proving RH is true.
Category: Number Theory

Set Theory Can’t be Directly Representative of Algebraic Constructions in Goldbach Conjecture and Other NT Problems

Authors: Juan Elias Millas Vera
Comments: 3 Pages.

In this paper I want to express my thoughts on the non possible link between set theory arguments in Number Theory. It is maybe good a first approximation to a problem to think in relation to sets, but my actual thinking is that you can’t solve an algebraic construction with a direct implication between the set and the algebraic variable. As example I will analyze my past trying to prove strong Goldbach conjecture. Finally I explain other version of this topic also with Goldbach conjectures as examples.
Category: Number Theory

Isolating the Prime Numbers

Authors: Adrian M. Stokes
Comments: 4 Pages.

Prime numbers greater than 3 belong to the number sequences 6n ± 1 wheren ≥ 1. These sequences also include the composites that are not divisible by 2 and/or3 and therefore their factors must also be of the form 6n ± 1. This allows all of the6n±1 composites to be equivalently written in the form of factors (6n1±1)(6n2±1),where n1 and n2 ≥ 1, creating three sub-sequences that exclude prime numbers.Finding and isolating the prime numbers can be achieved by selecting a numberrange and creating a set of 6n ± 1 numbers for that range before subtracting thesubsets (6n1 ± 1)(6n2 ± 1) to isolate and identify all the primes in the set.
Category: Number Theory

The Zeta Function and the Euler-Maclaurin Formula

Authors: Marco Burgos
Comments: 15 Pages. (Note by viXra Admin: Author name is required on the article in pdf)

The Riemann Zeta function is very famous because hidden within it lies the much-desired prime counting function. In this paper, we will unlock the door using the Euler-Maclaurin formula and present the proof of the Riemann Hypothesis.
Category: Number Theory

The First Counterexample of Riemann Hypothesis Found Through Computer Calculation

Authors: Zhiyang Zhang
Comments: 7 Pages.

The counterexample of the Riemann hypothesis causes a significant change in the image of the Riemann Zeta function, which can be distinguished using mathematical judgment equations. The first counterexample can be found through this equation.
Category: Number Theory

Incomplete Gamma Function and pi

Authors: Edgar Valdebenito
Comments: 3 Pages.

In this note we give three double series for Pi.
Category: Number Theory

Collatz Conjecture [Being] Truly a Definite Conclusion is Drawn by Using the Principle of Net Induction Rate and Net Reduction Rate

Authors: Chandan Chattopadhyay
Comments: 10 Pages.

This research work establishes a theory for concluding an affirmative answer to the famous, long-standing unresolved problem "TheCollatz Conjecture".
Category: Number Theory

Further Investigations on Euler's Odd Perfect Numbers

Authors: Chandan Chattopadhyay
Comments: 11 Pages.

It is a long-standing question whether there exists an odd perfect number. This article establishes a complete theory in order to prove that if an oddperfect number n exists then n = pm^2 with p prime and p is congruent to 1 (mod 4), andgcd (p, m) = 1.
Category: Number Theory

Diophantine Nth-tuples

Authors: Claude Michael Cassano
Comments: 2 Pages. (Note by viXra Admin: Please use complete sentences in the abstract)

Diophantine equations of sums of terms of various degrees [are explored.]
Category: Number Theory

Riemann Hypothesis is [Claimed to Be] Proven

Authors: Dmitri Martila
Comments: 2 Pages. (Note by viXra Admin: Please use complete sentences and do not use grandiose title - Future non-compliant submission will be rejected))

A short research about the Riemann Hypothesis [is given].
Category: Number Theory

Euler-Mascheroni Constant is Irrrational

Authors: Dmitri Martila
Comments: 2 Pages.

[The attempted proof that the Euler-Mascheroni constant is irrrational is given.]
Category: Number Theory

Proof of Riemann Hypothesis via Robin's Theorem

Authors: Dmitri Martila
Comments: 3 Pages.

I show that the minimum of the function F=e^gamma*ln(ln n)-sigma(n)/n isfound to be positive. Therefore, F>0 holds for any n>5040.
Category: Number Theory

The Signature of Abc Conjecture is Proven

Authors: Dmitri Martila
Comments: 4 Pages.

Equivalent view of abc conjecture is proven. Some crucial properties of the abc conjecture are presented and proven. For example, there exist three numbers (a, b, c) that satisfy the abc conjecture for an arbitrary value c.
Category: Number Theory

An Attempt to Prove the Riemann Hypothesis Simply

Authors: Dmitri Martila
Comments: 1 Page.

This work says that Riemann Hypothesis is true.
Category: Number Theory

Proof of Some Conjectures and Riemann Hypothesis

Authors: Dmitri Martila
Comments: 5 Pages.

A simple proof confirms Riemann, Generalized Riemann, Collatz, Swinnerton-Dyer conjectures and Fermat's Last Theorem.
Category: Number Theory

Proof of Strong Golbach Conjecture

Authors: Dmitri Martila
Comments: 2 Pages. (Note by viXra Admin: Please use complete sentences in the abstract)

Proof of Strong Golbach Conjecture [is explored in this article].
Category: Number Theory

A New Attempt to Check Whether Ramanujan's Formula Pi^4 Approx 97.5-1/11 is a Part of Some Completely Accurate Formula

Authors: Janko Kokošar
Comments: 7 Pages.

Intuitively, it seems that Ramanujan's formula $pi^4approx 97.5-1/11$ is an approximation for some perfectly accurate formula for $pi$. Here is one attempt to prove this. The principle of proof, however, is based on closeness of the every rest term to the inverse of integers. Although it is indeed somewhat closer to integers than it is on average, this proof is not complete. So we cannot say for sure whether this proves or disproves that this Ramanujan's formula has higher approximations; however, it gives hints and opens up space for further research.Moreover, this attempted proof is quite original. Also, such a method could also help in physics.
Category: Number Theory

A Curious Family of Integrals that Give pi

Authors: Edgar Valdebenito
Comments: 2 Pages.

In this note we give a set of integrals for Pi.
Category: Number Theory

Validating Collatz Conjecture Through Binary Representation and Probabilistic Path Analysis

Authors: Budee U. Zaman
Comments: 15 Pages.

The Collatz conjecture, a longstanding mathematical puzzle, posits that, regardless of the starting integer, iteratively applying a specific formula will eventually lead to the value 1. This paper introduces a novelapproach to validate the Collatz conjecture by leveraging the binary representation of generated numbers. Each transition in the sequence is predetermined using the Collatz conjecture formula, yet the path of transitionsis revealed to be intricate, involving alternating increases and decreases for each initial value. The study delves into the global flow of the sequence, investigating thebehavior of the generated numbers as they progress toward the termination value of 1. The analysis utilizes the concept of probability to shed light on the complex dynamics of the Collatz conjecture. By incorporatingprobabilistic methods, this research aims to unravel the underlying patterns and tendencies that govern the convergence of the sequence.The findings contribute to a deeper understanding of the Collatz conjecture,offering insights into the inherent complexities of its trajectories. This work not only validates the conjecture through binary representation but also provides a probabilistic framework to elucidate the global flow ofthe sequence, enriching our comprehension of this enduring mathematical mystery.
Category: Number Theory

Derivation/Correction of Hardy-Littlewood Twin Prime Constant using Prime Generator Theory (PGT)

Authors: Jabari Zakiya
Comments: 7 Pages.

The Hardy-Littlewood twin prime constant is a metric to compute the distribution of twin primes. Using Prime Generator Theory (PGT), it is shown it is more easily mathematically and conceptually derived, and the correct value is a factor of 2 larger.
Category: Number Theory

Adaptive Polynomial Factorization (APF) Method: Enhanced Factorization using Modified Pollard Rho Algorithm

Authors: Anil Sharma
Comments: 3 Pages. (Note by viXra Admin: Please list scientific references in future submissions)

This research paper introduces the Adaptive Polynomial Factorization (APF) Method, an enhanced factorization technique based on the Modified Pollard Rho Algorithm. The method incorporates adaptive polynomial evaluation, providing efficiency in factorization tasks. The paper presents a mathematical representation, performance analysis, and examples showcasing the APF Method’s versatility and superiority over the original Pollard Rho algorithm.
Category: Number Theory

The Riemann Hypothesis Assumes that the First Counterexample is Located Near S=0.383+(1.578 * 10 ^ 16) I

Authors: Zhiyang Zhang
Comments: 10 Pages.

We already know the distribution of non trivial zeros in the Riemann hypothesis, and there is a formula for calculating counterexamples. The first counterexample can be obtained using a computer, and its value is s=0.383+15786867949799975i
Category: Number Theory

On the Generation of Odd Prime Numbers Through a Modified Composite Expression

Authors: Anil Sharma
Comments: 2 Pages. (Note by viXra Admin: Please list scientific references in future submissions)

This research paper investigates a distinctive mathematical expression involving natural numbers, unveiling its remarkable property of generating odd prime numbers. The expression, given by N+1 / N × (N! mod PN k=1 k) for natural positive integers N ranging from 2 to infinity, serves as the focal point of our exploration. The paper formulates a formal conjecture, provides a comprehensive proof, and elucidates the claim through stepwise examples.
Category: Number Theory

Beyond Fibonacci: the Sequence of Integer Powers of Numbers

Authors: Jean-Philippe Vassan
Comments: 15 Pages. In French

Neighboring triangles of Pascal's triangle, cousin numbers of the golden ratio, a simplified formula giving the numbers of generalized Fibonacci sequences, associated generating function, chaos theory and tent function equation.

Triangles voisins du triangle de Pascal, nombres cousins du nombre d'or, une formule simplifiée donnant les nombres des suites de Fibonacci généralisées, fonction génératrice associée, théorie du chaos et équation de la fonction tente.
Category: Number Theory

The Symmetry of S∞+i and Number Conjectures

Authors: Yajun Liu
Comments: 4 Pages. (Author name reversed by viXra Admin - Future non-compliant submission will not be accepted!)

In this paper, we discuss the symmetry of S∞+i and we find that using the symmetry characters of S∞+i , we can give proofs of the Hodge Conjecture and the Prime Conjectures: Goldbach Conjecture、Polignac’s conjecture and Twins Prime Conjecture. And we also give a proof of Collatz conjecture.
Category: Number Theory

Function for Prime Numbers

Authors: Massimo Russo
Comments: 66 Pages. In Italian (Correction made by viXra Admin to conform with the requirements of viXra.org)

The Function 5*(1+1/x) + 1 for every value of x determined by Sequence A x = (5^2)+5*2*(n(n+1)/2) where n ≥ 0 determines an infinite series of fractional numbers N/d: 5*(1+1/x) + 1 = N/d such that N and d are prime. numbers.
Category: Number Theory

Analyzing the Connection from (H(z)) to the Riemann Zeta Function

Authors: Oussama Basta
Comments: 6 Pages.

This paper explores the intriguing connection between the function (H(z) = ln(|sec(pi z/log(z))|)) and the Riemann Zeta Function (zeta(s)). The journey begins by investigating the zeros of (H(z)) and employing advanced mathematical tools such as the Taylor series expansion, the argument principle, and the inverse Mellin transform. Through this exploration, we establish a relationship that leads to a complex integral representation connecting (H(z)) to the Riemann Zeta Function (zeta(s)).
Category: Number Theory

A Mathematical Criterion for the Validity of the Riemann Hypothesis

Authors: Zhiyang Zhang
Comments: 7 Pages.

We already know in what situations there will be counterexamples for the Riemann hypothesis, but simply increasing Im (s) to find counterexamples for the Riemann hypothesis is still very slow. If there is only a counterexample when Im (s)=10 ^ 1000, or even 10 ^ 10000, then the performance requirements for the computer are very demanding. So, we must create a numerical order determinant to determine whether the Riemann hypothesis holds.
Category: Number Theory

Pythagorean Triples and Fermat's Theorem N = 4

Authors: Rolando Zucchini
Comments: 7 Pages.

This article contains a theorem to build the Primitive Pythagorean triples and the proof of the last Fermat’s Theorem for n = 4.
Category: Number Theory

Periodic Function Approach to Prime Number Analysis with Graphical Illustrations

Authors: Budee U. Zaman
Comments: 15 Pages.

This paper introduces a novel approach employing periodic functions for the comprehensive analysis of prime numbers. The method ncompassesprimality testing, factor counting and listing, prime distribution calculation, and the determination of the Nth prime. The exposition of the technique is presented in a clear and sequential manner, guiding the reader through each step with explicit equations. Graphs are strategically incorporated between crucial stages to facilitate a rapid and intuitive visualization of the rationale and outcomes of each maneuver. The paper concludes with concise reflections and ongoing inquiries into the potential applications and refinements of the proposed method.
Category: Number Theory

Rsa 2048 Factoring NPQ

Authors: Ricardo Gil
Comments: 5 Pages.

The purpose of this paper is to provide an algorithm that has 5 lines of code and that finds P & Q when N is given. It will work for RSA 2048 & RSA-617.
Category: Number Theory

The Riemann Hypothesis Has no Counterexamples When Imaginary Part Below One Million Billion

Authors: Zhiyang Zhang
Comments: 2 Pages.

This article aims to identify counterexamples of the Riemann hypothesis. Although no upper bound was found for the counterexample, at least it was proven that there were no counterexamples when imaginary part below one million billion, significantly increasing the lower bound of the counterexample.
Category: Number Theory

Continued Fraction Generalization [Part 3]

Authors: Isaac Mor
Comments: 18 Pages.

This is a list of ten types of continued fraction generalization. (This is [Part 3] , every volume contains 10 formulas) I am using Euler's continued fraction formula in order to find some nice continued fraction generalization.
Category: Number Theory

A Simple Proof of The ABC Conjecture

Authors: Oussama Basta
Comments: 3 Pages.

This work analyzes the ABC conjecture, which states that for any positive real number ε, there exists a constant Kε such that for all coprime positive integer triples (a, b, c) with a + b = c, c < Kε * rad(abc)^(1 + ε). We focus on the case where a > F, F > ε, and b = (a + F - ε), c = (a + F + ε), where F and ε are positive real numbers with F > ε.
Category: Number Theory

Riemann Hypothesis: Direct Demonstration Proposal

Authors: Vincent Koch
Comments: 3 Pages.

In his 1859 article "On the number of prime numbers less than a given quantity", Bernhard Riemann formulated the hypothesis that all non-trivial zeros of the Zeta function have the real part 1/2.This assertion, known as the "Riemann Hypothesis", remains unproven to this day. The present paper is an attempt at a direct demonstration.
Category: Number Theory

Hypothesis: Distribution of Primes and the Logarithmic Expression

Authors: Anil Sharma
Comments: 4 Pages.

This research explores the distribution of prime numbers using a novel ljogarithmic expression. The hypothesis suggests that an expression, partitions natural numbers into groups, revealing a systematic distribution of primes. Experimental results demonstrate an intriguing pattern as the range of N increases, with the average number of primes in each group stabilizing around 15. The paper discusses thebackground, mathematical expression, experimental results, and potentialavenues for future research.
Category: Number Theory

Revolutionizing Prime Factorization: A Time Complexity-Optimized Approach for Efficient Composite Number Analysis

Authors: Anil Sharma
Comments: 2 Pages.

This research investigates patterns in prime number distributions and proposes an optimized factorization method. A novel approach is introduced to explore the position of the first prime factor in composite numbers, focusing on a specific range for potential computational time savings.
Category: Number Theory

An Efficient Method to Prove that the Riemann Hypothesis Is Not Valid

Authors: Zhiyang Zhang
Comments: 12 Pages. (Name added to article by viXra Admin - Please conform!)

Analytical number theory is a combination of trigonometric functions and polynomial symbols, which can be solved no matter how difficult it is. Therefore, I believe that the Riemann hypothesis is not unsolvable. In the field of number theory, the mathematical community tends to seek a maximum number to overturn the conclusion. Whether it is the Riemann hypothesis or the Goldbach conjecture, this should be the solution.
Category: Number Theory

On the Sum of Reciprocals of Primes

Authors: Young Deuk Kim
Comments: 5 Pages.

Suppose that $y>0$, $0leqalpha<2pi$ and $0K$ and $P^-$ the set of primes $p$ such that $cos(yln p+alpha)<-K$ . In this paper we prove $sum_{pin P^+}frac{1}{p}=infty$ and $sum_{pin P^-}frac{1}{p}=infty$.
Category: Number Theory

Redefining Mathematical Structure: From the Real Number Non-Field to the Energy Number Field

Authors: Parker Emmerson
Comments: 8 Pages.

The traditional classification of real numbers (R) as a complete ordered field is contested throughcritical examination of the field axioms, with a focus on the absence of a multiplicative inverse for zero. We propose an alternative mathematical structure based on Energy Numbers (E), deriving from quantum mechanics, which addresses the classical anomalies and fulfills field properties universally, including an element structurally analogous but functionally distinctive from the zero in R.
Category: Number Theory

Charles Hutton's Formula

Authors: Edgar Valdebenito
Comments: 3 Pages.

Some remarks about a formula of Charles Hutton.
Category: Number Theory

Contribution to Goldbach's Conjectures

Authors: Radomir Majkic
Comments: 6 Pages.

The internal structure of the natural numbers reveals the relation between the weak and strong Goldbach's conjectures. Explicitly, if the weak Goldbach's conjecture is true, the strong Goldbach's conjecture is, and Goldbach's conjectures are true.
Category: Number Theory

Goldbach's Number Construction

Authors: Radomir Majkic
Comments: 6 Pages.

Goldbach’s numbers, all-natural integers which satisfy Goldbach’s conjectures are all odd integers and a subset of the even integers. Naturally, they appear in the proof of Goldbach’s conjectures. In this paper, the construction of Goldbach’s numbers approach is used to prove Goldbach’s conjectures, hopefully, it will bring a happy end.
Category: Number Theory

Multivariate Circle of Partitions and the Squeeze Principle

Authors: Theophilus Agama
Comments: 5 Pages.

The goal of this paper is to extend the squeeze principle to circle of partitions with at least two resident points on their axes.
Category: Number Theory

Investigation on Brocard-Ramanujan Problem

Authors: Akash Shivaji Pawar
Comments: 7 Pages. 6 Lemma,2 Theorem

Exploring n! + 1 = m^2 for natural number solutions beyond n = 4, 5, 7 confirms no further solutions exist,validated by using GCD Linear Combination Theorem
Category: Number Theory

On the Nonexistence of Solutions to a Diophantine Equation Involving Prime Powers

Authors: Budee U. Zaman
Comments: 7 Pages.

This paper investigates the Diophantine equation pr + (p + 1)s = z2 Where p > 3, s ≥ 3 , z is an even integer. The focus of the study is to establish rigorous results concerning the existence of solutions within this specific parameter space. The main result presented in this paper demonstrates the absence of solutions under the stated conditions. The proof employs mathematical techniques to systematically address the case when the prime p exceeds 3, and the exponent s is equal to or greater than2, while requiring the solution to conform to the constraint of an even z. This work contributes to the understanding of the solvability of the given Diophantine equation and provides valuable insights into the interplay between prime powers and the resulting solutions.
Category: Number Theory

Euler’s Totient Function, Sum of Divisors and Primes

Authors: Rédoane Daoudi
Comments: (Note by viXra Admin: Future stub page/paper will not be accepted!)

Here I present a conjecture about Euler’s totient function, sum of divisors andprimes.
Category: Number Theory

On the Notion of Carries of Numbers 2^n-1 and Scholz Conjecture

Authors: Theophilus Agama
Comments: 16 Pages.

Applying the pothole method on the factors of numbers of the form $2^n-1$, we prove that if $2^n-1$ has carries of degree at most $$kappa(2^n-1)=frac{1}{2(1+c)}lfloor frac{log n}{log 2}floor-1$$ for $c>0$ fixed, then the inequality $$iota(2^n-1)leq n-1+(1+frac{1}{1+c})lfloorfrac{log n}{log 2}floor$$ holds for all $nin mathbb{N}$ with $ngeq 4$, where $iota(cdot)$ denotes the length of the shortest addition chain producing $cdot$. In general, we show that all numbers of the form $2^n-1$ with carries of degree $$kappa(2^n-1):=(frac{1}{1+f(n)})lfloor frac{log n}{log 2}floor-1$$ with $f(n)=o(log n)$ and $f(n)longrightarrow infty$ as $nlongrightarrow infty$ for $ngeq 4$ then the inequality $$iota(2^n-1)leq n-1+(1+frac{2}{1+f(n)})lfloorfrac{log n}{log 2}floor$$ holds.
Category: Number Theory

A Proof of the Wen-Yao Conjecture

Authors: David Adam
Comments: 24 Pages. In French

In this article, we characterize monomials in de facto values.Carlitz-Goss rielle defined on the complement of Fq (T) in a finite place which arealgebraic on Fq (T ). In particular, this confirms Wen-Yao's conjecturestated in 2003. This gives a necessary and sufficient condition on an en-p-adic tier so that the value of the Carlitz-Goss factorial in it is algebraic on Fq (T ). When restricted to rational arguments, we determinenot all algebraic relations between the values u200bu200btaken by this function, this which gives the counterpart for finite places of a result of Chang, Papanikolas, Thakur and Yu obtained in the case of infinite place.

Dans cette article, nous caractérisons les monômes en les valeurs de la facto-rielle de Carlitz-Goss définie sur le complété de Fq (T ) en une place finie qui sont algébriques sur Fq (T ). En particulier, cela confirme la conjecture de Wen-Yaoénoncée en 2003 . Celle-ci donne une condition necessaire et suffisante sur un en-tier p-adique pour que la valeur de la factorielle de Carlitz-Goss en celui-ci soit algébrique sur Fq (T ). Lorsque restreint aux arguments rationnels, nous détermi-nons toutes les relations algébriques entre les valeurs prises par cette fonction, cequi donne le pendant pour les places finies d’un résultat de Chang, Papanikolas, Thakur et Yu obtenu dans le cas de la place infinie.
Category: Number Theory

Complete Operations

Authors: Pith Peishu Xie
Comments: 26 Pages.

The Operator axioms have produced complete operations with real operators. Numerical computations have been constructed for complete operations. The classic calculator could only execute 7 operator operations: + operator operation(addition), - operator operation(subtraction), $times$ operator operation(multiplication), $div$ operator operation(division), ^{} operator operation(exponentiation), $surd$ operator operation(root extraction), log operator operation(logarithm). In this paper, we invent a complete calculator as a software calculator to execute complete operations. The experiments on the complete calculator could directly prove such a corollary: Operator axioms are consistent.
Category: Number Theory

A Proof of a Result of James Stirling

Authors: Hervé Gandran-Tomeng
Comments: 2 Pages.

A recent paper contains a proof of a result of James Stirling,$sum_{n=1}^inftyfrac{1}{n^2binom{2n}{n}}=frac{1}{3}sum_{n=1}^infty frac{1}{n^2}$What is following is another proof of this equality.
Category: Number Theory

Rediscovering Ramanujan

Authors: Edgar Valdebenito
Comments: 14 Pages.

In this note, we revisit Ramanujan-type series for 1/pi .
Category: Number Theory

Collatz Conjecture: a Countably Infinite Sequence

Authors: Vishesh Mangla
Comments: 2 Pages. CC BY-SA (Note by viXra admin: The article needs an abstract and scientific references are required

In this paper, I have explored the Collatz conjecture and presented a new result regarding the behavior of the sequence. The proof demonstrated that for natural numbers n subjected to the Collatz Algorithm, the sequence can potentially havea countably infinite number of terms.
Category: Number Theory

The Twin Prime Conjecture - An Analytical Approach

Authors: Patrick DiCarlo
Comments: 37 Pages.

This paper offers a proof of the twin prime conjecture. The basic strategy is to first establish that there is no highest prime number by calculating the rates at which the multiples of each successive prime preclude higher numbers from being prime, and then proving that this rate (in the aggregate) can never reach 100%. The same basic methodology is then used to show that there can also be no highest twin prime.
Category: Number Theory

One Theorem Complementary to the Fundamental Theorem of Arithmetic

Authors: Juan Elias Millas Vera
Comments: 2 Pages.

In this paper I want to show a complementary theorem of the Fundamental Theorem of Arithmetic. Using the delta notation (Δ) I was able to deduce a generic formula involving prime numbers and natural numbers.
Category: Number Theory

New Equivalent of the Riemann Hypothesis

Authors: Leonardo de Lima
Comments: 10 Pages.

In this article, it is demonstrated that if the zeta function does not have a sequence of zeros whose real part converges to 1, then it cannot have any zeros in the critical strip, showing that the Riemann Hypothesis is false.
Category: Number Theory

Cracking the Collatz Code: a Journey from Conjecture to Certainty Through Infinity

Authors: Eric Lough
Comments: 7 Pages.

The Collatz conjecture, a puzzle that has intrigued mathematicians for decades, has met its resolution through a simple combination of trial and error combined with a spreadsheet. While the primary goal was to definitively prove the conjecture, the exploration revealed unexpected insights into the underlying patterns of the Collatz sequence. This study introduces a novel approach to organizing and analyzing Collatz data, laying bare the formulas governing the sequence's behavior. The results present a conclusive proof of the Collatz conjecture and unveil a spectrum of sets and formulas reaching to infinity, each contributing to a deeper understanding of its workings. Comprehensive instructions are provided, encouraging collaboration and furthering the collective understanding of mathematical sequences.
Category: Number Theory

The Magic of Mirror Composite Numbers

Authors: Emilio Sánchez, Óscar E. Chamizo
Comments: 6 Pages.

In this paper, continuation and completion of some previous researches, we fully develop the new concept of mirror composite numbers. Mirror composite numbers are composite numbers of the form 2n-p for some n natural number and p prime. We shall show that the factorization of these numbers have interesting properties in order to face the Goldbach conjecture [2][3] by the divide et impera method.
Category: Number Theory

Proof for Specific Type of Continued Fraction

Authors: Isaac Mor
Comments: 7 Pages.

I am going to use telescoping series and then a proof by induction. I am using Lambert's continued fraction for the base case.
Category: Number Theory

Elliptic Curve

Authors: Ian Connell
Comments: 327 Pages.

The first version of this handbook was a set of notes of about 100 pages handed out to the class of an introductory course on elliptic curves given in the 1990 fall semester at McGill University in Montreal. Since then I haveadded to the notes, holding to the principle: If I look up a certain topic a year from now I want all the details right at hand, not in an "exercise", so if I’ve forgotten something I won’t waste time. Thus there is much that anordinary text would either condense, or relegate to an exercise. But at the same time I have maintained a solid mathematical style with the thought of sharing the handbook.
Category: Number Theory

Fermat's Last Theorem for Odd Primes

Authors: Minho Baek
Comments: 26 Pages.

It was already proved right that xn+yn=zn, (n>2) has no solutions in positive integers which we called Fermat’s Last Theorem (FLT) by Andrew Wiles. But his proof would be impossible in the 17th century. Since Fermat showed he proved n=even by leaving proof for n=4, many people have tried to prove odd primes. I took the idea from Euler proof and proved in case of n=odd primes by simple method.
Category: Number Theory

A Proof of Riemann Hypothesis by Symmetry and Circular Properties of Riemann Zeta Function

Authors: Tae Beom Lee
Comments: 9 Pages.

Riemann zeta function(RZF) z(s) is a function of a complex variable s=x+iy. Riemann hypothesis(RH) states that all the non-trivial zeros of RZF lie on the critical line, 0.5+iy. The symmetricity of RZF zeros implies that if z(a+ib)=0,0<a<0.5, then z(1-a+ib)=0, too. The graphs of RZF are similar to the graphs of circles with non-uniform radius and argument. These two, symmetry and circular properties of RZF, are the basis of our proof.
Category: Number Theory

Solution of a Five Degree Equation

Authors: Ait saadi Ahcene
Comments: 4 Pages. (Correction made by viXra Admin to conform with scholarly norm)

In this article, I solve the general equation of degree 5 of the a particular form. For this I used Mathematics that I invented. The method I invented allows me to solve all equations of degrees greater than 4, as well as equations of another form: The principle, is to find for all these equations, an equation of degree three (3) which has at least one solution in common with those of degree greater than 4. With my method of course this is possible.
Category: Number Theory

New Bounds on Mertens Function

Authors: Juan Moreno Borrallo
Comments: 7 Pages.

In this brief paper we study and bound Mertens function. The main breakthrough is theobtention of a Möbius-invertible formulation of Mertens function, which with some transformationsand the application of the generalization of Möbius inversion formula, allows us to reach anasymptotic equivalence of the absolute value of Mertens function that proves the Riemann Hypothesis.
Category: Number Theory

Generalization for Specific Type of Continued Fraction

Authors: Isaac Mor
Comments: 2 Pages. (Correction made by viXra Admin to conform with scholarly norm - Future non-comforming submission will not be accepted.))

I came across "The Ramanujan Machine" on the Internet and, using my intuition on those kind of stuff, I found some interesting results.
Category: Number Theory

No Collatz Conjecture Integer Series Have Looping

Authors: Tsuneaki Takahashi
Comments: 3 Pages. Note by viXra Admin: Future repetition/regurgitation will not be accepted.

If the series of Collatz Conjecture integer has looping in it, it is sure the members of the loop cannot reach to value 1. Here it is proven that the possibility of looping is zero except one.
Category: Number Theory

Zeta Function

Authors: Leonardo de Lima
Comments: 8 Pages.

This article delves into the properties of the Riemann zeta function, providing a demonstration of the existence of a sequence of zeros ${z_k}$, where $lim operatorname{Re}(z_k) = 1$. The exploration of these mathematical phenomena contributes to our understanding of complex analysis and the behavior of the zeta function on the critical line.
Category: Number Theory

Riemann Hypothesis

Authors: Bertrand Wong
Comments: 14 Pages.

This paper discusses the distribution of the non-trivial zeros of the Riemann zeta function ζ. It looks into the question of whether any non-trivial zeros would ever possibly be found off the critical line Re(s) = 1/2 on the critical strip between Re(s) = 0 and Re(s) = 1, e.g., at Re(s) = 1/4, 1/3, 3/4, 4/5, etc., and why all the non-trivial zeros are always found at the critical line Re(s) = 1/2 on the critical strip between Re(s) = 0 and Re(s) = 1 and not anywhere else on this critical strip, with the first 1013 non-trivial zeros having been found only at the critical line Re(s) = 1/2. It should be noted that a conjecture, or, hypothesis could possibly be proved by comparing it with a theorem that has been proven, which is one of the several deductions utilized in this paper. Through these several deductions presented, the paper shows how the Riemann hypothesis may be approached to arrive at a solution. In the paper, instead of merely using estimates of integrals and sums (which are imprecise and may therefore be of little or no reliability) in the support of arguments, where feasible actual computations and precise numerical facts are used to support arguments, for precision, for more sharpness in the arguments, and for "checkability" or ascertaining of the conclusions. This paper is the revised and expanded version of a paper [5] published in 2022.
Category: Number Theory

Proof of the Collatz Conjecture

Authors: Wiroj Homsup, Nathawut Homsup
Comments: 8 Pages. (Note by vXra Admin: Future repetition/regurgitation will not be accepted!)

The Collatz conjecture considers recursively sequences of positive integers where n is succeeded by n/2 , if n is even, or (3n+1)/2 , if n is odd. The conjecture states that for all starting values n the sequence eventually reaches the trivial cycle 1, 2, 1, 2u2026u2026The inverted Collatz sequences can be represented as a tree with 1 as its root node. In order to prove the Collatz conjecture, one must demonstrate that the tree covers all natural numbers. In this paper, we construct a Collatz tree with 1 as its root node by connecting infinite number of basic trees. Each basic tree relates to each natural number. We prove that a Collatz tree is a connected binary tree and covers all natural numbers.
Category: Number Theory

Zero Times Zero Equals Nonzero

Authors: Michael Graham
Comments: 3 Pages.

The current Multiplication and Division Properties of Zero are flawed and illogical. This paper illustrates why and presents logical solutions that resolve the issue of dividing by zero.
Category: Number Theory

On the Largest Prime Factor of the K-Generalized Lucas Numbers

Authors: Herbert Batte, Florian Luca
Comments: 14 Pages.

Let $(L_n^{(k)})_{ngeq 2-k}$ be the sequence of $k$--generalized Lucas numbers for some fixed integer $kge 2$ whose first $k$ terms are $0,ldots,0,2,1$ and each term afterwards is the sum of the preceding $k$ terms. For an integer $m$, let $P(m)$ denote the largest prime factor of $m$, with $P(0)=P(pm 1)=1$. We show that if $n ge k + 1$, then $P (L_n^{(k)} ) > (1/86) log log n$. Furthermore, we determine all the $k$--generalized Lucas numbers $L_n^{(k)}$ whose largest prime factor is at most $ 7$.
Category: Number Theory

3n+1 Conjecture: A Proof or Almost

Authors: A. Makarenko
Comments: 4 Pages.

The Collatz algorithm is rewritten to remove divisions by two and to transform it from a hailstone to a steadily growing value. In contrast with the original problem this new sequence becomes reversible and it is reverted in combinatorial way to find all integers leading to the sequence end. Computer programs are available for demonstrations and experimenting.
Category: Number Theory

A Proof of Fermat’s Last Theorem by Relating to Monic Polynomial Properties

Authors: Tae Beom Lee
Comments: 5 Pages.

Fermat's Last Theorem(FLT) states that there is no natural number set {a,b,c,n} which satisfies a^n+b^n=c^n or a^n=c^n-b^n when n≥3. In this thesis, we related LHS and RHS of a^n=c^n-b^n to the constant terms of two monic polymials x^n-a^n and x^n-(c^n-b^n). By doing so, we could inspect whether these two polynomials can be identical when n≥3, i.e., x^n-a^n=x^n-(c^n-b^n), which satisfies a^n=c^n-b^n. By inspecting the properties of two polynomials such as factoring, root structures and graphs, we found that x^n-a^n and x^n-(c^n-b^n) can’t be identical when n≥3, except when trivial cases.
Category: Number Theory

Divisible Cyclic Numbers

Authors: Julian Beauchamp
Comments: 5 Pages. (Author name added to the article by viXra Admin - Please conform!)

There are known to exist a number of (multiplicative) cyclic numbers (CNs), but in this paper I introduce what appears to be a new kind of number, which we call divisible cyclic numbers (DCNs) and wonder what properties they may possess. Strangely, I can find no reference to them anywhere. Given that they are simple to understand and quite commonplace, it would be remarkable if they were hitherto unknown to the mathematical world.
Category: Number Theory

For a Number k, Can (2[k]m)+1 Always be Prime for All Number m?

Authors: Juan Elias Millas Vera
Comments: 2 Pages.

This paper is about hyperoperators. In this paper I ask myself and the mathematical community if there is possible that a k-ation of the number 2 will be always a number prime for any number m if we add the number one to the result.
Category: Number Theory

On the Incompletely Predictable Problems of Riemann Hypothesis, Modified Polignac's and Twin Prime Conjectures

Authors: John Yuk Ching Ting
Comments: 72 Pages.

We validly ignore even prime number 2. Based on all arbitrarily large number of even prime gaps 2, 4, 6, 8, 10...; the complete set and its derived subsets of Odd Primes fully comply with Prime number theorem for Arithmetic Progressions. With this condition being satisfied by all Odd Primes, we argue that Polignac's and Twin prime conjectures are proven to be true with these conjectures treated as Incompletely Predictable Problems. In so doing [and with the famous Riemann hypothesis being a special case], the generalized Riemann hypothesis formulated for Dirichlet L-function is also supported. Riemann hypothesis is separately proven to be true with this hypothesis treated as an Incompletely Predictable Problem.
Category: Number Theory

Mathematics for Incompletely Predictable Problems Required to Prove Riemann Hypothesis, Modified Polignac's and Twin Prime Conjectures

Authors: John Yuk Ching Ting
Comments: 69 Pages.

As two different but related infinite-length equations through analytic continuation, Hasse principle is satisfied by Riemann zeta function as a certain type of equation that generates all infinitely-many trivial zeros but this principle is not satisfied by its proxy Dirichlet eta function as a dissimilar type of equation that generates all infinitely-many nontrivial zeros. Based on two seemingly different location that are in fact identical, all nontrivial zeros are mathematically located on critical line or geometrically located on Origin point. Thus we prove location for complete Set nontrivial zeros to be critical line confirming Riemann hypothesis to be true. Sieve of Eratosthenes as a certain type of infinite-length algorithm is exactly constituted by an Arbitrarily Large Number of (self-)similar infinite-length sub-algorithms that are specified by every even Prime gaps. Modified Hasse principle is satisfied by this algorithm and its sub-algorithms that perpetually generate the Arbitrarily Large Number of all Odd Primes. Thus we prove Set even Prime gaps with corresponding Subsets Odd Primes all have cardinality Arbitrarily Large in Number confirming Modified Polignac's and Twin prime conjectures to be true.
Category: Number Theory

Collatz Conjecture Proved Ingeniously & Very Simply

Authors: A. A. Frempong
Comments: 12 Pages. Copyright © by A. A. Frempong

Collatz conjecture states that beginning with a positive integer, if one repeatedly performs the following operations to form a sequence of integers, the sequence will eventually reach the integer one; the operations being that if the integer is even, divide it by 2, but if the integer is odd, multiply it by 3 and add one; and also, use the result of each step as the input for the next step. One would note the patterns of the sequence terms as the Collatz process reaches the equivalent powers, 2^(2k) (k = 2, 3, . . . ) and continues as 2^(2k-1), 2^(2k-2), 2^(2k-3), . . ., 2^(2k-2k). Two main cases are covered. In Case 1, the integer can be readily written as a power of 2 as 2^(k) (k=1,2,3, . . . ), and the sequence would reach the integer one by following 2^(k-1), 2^(k-2), 2^(k-3,), . . . , 2^(k-k). In Case 2, the integer cannot be readily written as a power of 2, but the sequence terms reach the equivalent power, 2^(2k) (k = 2, 3, . . . ) which will continue as 2^(2k-1), 2^(2k-2), 2^(2k-3), . . . , 2^(2k-2k). In Case 2, when the sequence terms reach some particular integers such as 5, 21 and 85, the application of 3n + 1 to these integers will result in the powers, 2^(2k). One would call these integers, the 2k-power converters. There are infinitely many 2k-power converters as there are 2^(2k) powers. A term of the sequence must be converted to 2^(2k). There are infinitely many paths for converting integers to 2^(2k) powers. Of these paths, the integer 5-path, is the nearest 2^(2k) converter path to the integer 1 on the 2^(2k)-route. For the 5-path, when a sequence terms reach the integer, 5, the next term would be 16. Other integers can follow the integer 5-path to 16 as follows: Let n be an integer whose sequence terms would reach 16, and let n ± r = 5, where r is the net change in the sequence terms before the integer 5; and one uses the positive sign if n < 5, but the negative sign if n > 5. One will call the following, the 5-path 2k-converter template: 3(n ± r) + 1 = 16. By the substitution axiom, using this template, the sequence of every positive integer would reach 16; and applying repeated division by 2, the sequence will reach the integer 1.
Category: Number Theory

Mirror Composite Numbers: Their Factorization and Their Relationship with Goldbag Conjecture.

Authors: Ángeles Jimeno Yubero, Óscar E. Chamizo Sánchez
Comments: 5 Pages.

Mirror composite numbers are composite numbers of the form 2n-p for some n positive natural number and p prime. We shall show that the factorization of these numbers have interesting properties in order to face the Goldbach conjecture by the divide et impera method.
Category: Number Theory

A Simple Proof that E^(p/q) is Irrational

Authors: Timothy W. Jones
Comments: 3 Pages.

Using a simple application of the mean value theorem, we show that rational powers of e are irrational.
Category: Number Theory

Euler's Identity, Leibniz Tables, and the Irrationality of Pi

Authors: Timothy W. Jones
Comments: 6 Pages.

Using techniques that show show that e and pi are transcendental, we give a short, elementary proof that pi is irrational based on Euler's formula. The proof involves evaluation of a polynomial using repeated applications of Leibniz formula as organized in a Leibniz table.
Category: Number Theory

Convergence and Computation of Sum of a Series on the Riemann Zeta Function

Authors: HaeRyong Kim, HyonChol Kim, YongHun Jo
Comments: 13 Pages.

In this paper, we present a new method of evaluating the convergence and sum of a series with the Riemann zeta function in its general term.We consider the convergence and sum of a series by means of difference other than previous methods.
Category: Number Theory

Optimal FractionalPIβ(t)Dα(t) Controllers and Numerical Simulation for DC Motor Speed Control

Authors: Ji-Song Ro, Myong-Hyok Sin, Yong-Ho Kim, Sung-Il Gang
Comments: 12 Pages.

We model the rotation process of the motor for variable-order fractional control, which has been active in recent research, and perform numerical simulation of its optimal control and automatic control process. In this paper, we verify numerical method and error estimation of variable order fractional linear dynamic system with time-varying coefficients, a variable-order fractional PID controller design method where the integral of the absolute error with time weight is minimized is proposed using particle swarm optimization algorithm and demonstrate its effectiveness through numerical simulation for DC motor speed control. Numerical experiments show that the performance of the VFPID controller is superior to PID and FPID, especially VFPIDB (B-type variable order FPID) controller has the best performance. Finally, when the differential order varies, the subtypes of variable-order fractional derivatives are analyzed for the effects on the control objective, its effectiveness is newly clarified, and their research and practice is highlighted.　
Category: Number Theory

Reverse Chebyshev Bias in the Distribution of Superprimes

Authors: Waldemar Puszkarz
Comments: 12 pages. Originally posted on ResearchGate in September 2023.

We study the distribution of superprimes, a subsequence of prime numbers with prime indices, mod 4. Rather unexpectedly, this subsequenceexhibits a reverse Chebyshev bias: terms of the form 4k + 1 are more common than those of the form 4k + 3, whereas the opposite is the case in the sequence of all primes. The effect, while initially weak and easy to overlook, tends tobe several times larger than the Chebyshev bias for all primes for samples of comparable size, at least, by one simple measure. By two other measures, it can be seen as fairly strong; by the same measures the ordinary Chebyshev effectis very strong. Both of these measures also imply that the reverse Chebyshev bias for superprimes is more volatile than the ordinary Chebyshev bias.
Category: Number Theory

On the Binomial and Fermat's Last Theorem

Authors: Carlos Villacres
Comments: 9 Pages.

An approach to the classic problem of Fermat's last theorem. Using the binomial theorem and the cases where n is even or odd, we find a solution as well as a Pythagorean triple generator.
Category: Number Theory

The Symmetry of N-domain and Numbers Conjuctures

Authors: Yajun Liu
Comments: 11 Pages.

In this paper, we discuss the symmetry of N-domain and we find that using the symmetry characters of Natural Numbers we can give proofs of the Prime Conjectures: Goldbach Conjecture、Polignac’s conjecture (Twins Prime Conjecture) and Riemann Hypothesis. . We also gave a concise proofs of Collatz Conjecture in this paper.
Category: Number Theory

Proving the Goldbach Conjecture

Authors: Jim Rock
Comments: 2 Pages.

In 1742 Christian Goldbach suggested that any even number four or greater is the sum of two primes. The Goldbach Conjecture remains unproven to the present day though it has been verified for all even numbers up to 4 x 1018. This paper suggests an algorithm for checking the Goldbach conjecture for individual even numbers and a generalization that could be used to prove the Goldbach conjecture.
Category: Number Theory

An Truly Easy Proof: Pi is Irrational

Authors: Timothy W. Jones
Comments: 1 Page.

Using the derivative of an integer polynomial composed with Euler's formula we prove that pi is irrational.
Category: Number Theory

Guessing that the Riemann Hypothesis Is Unprovable

Authors: T. Nakashima
Comments: 3 Pages.

Riemann Hypothesis has been the unsolved conjecture for 164 years. This conjecture is the last one of conjectures without proof in "Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse"(B.Riemann). The statement is the real part of the non-trivial zero points of the Riemann Zeta function is 1/2. Very famous and difficult this conjecture has not been solved by many mathematicians for many years. In this paper, I guess the independence (unprovability) of the Riemann Hypothesis.
Category: Number Theory

Proof of Collatz Conjecture Using Division Sequence Ⅳ

Authors: Masashi Furuta
Comments: 8 Pages.

This paper is positioned as a sequel edition of [1]. First, as in [1], define "division sequence", "complete division sequence", "star conversion", and "extended star conversion". Next, we use Well-Founded Induction and Peirce's law to prove the Collatz conjecture. This proof uses the theorem proving system Idris.
Category: Number Theory

Proof of ABC Conjecture

Authors: Xiaohui Li
Comments: 4 Pages.

This paper utilizes the fact that the prime factor among all factors in the root number rad (c) can only be a power of 1. Then, analyze all combinations of c that satisfy rad (c)=c, calculate the value of the combination, and find the maximum and minimum values of the root number rad, as well as the maximum exponent between them. Using this maximum exponent then an equivalent inequality is constructed to prove the ABC conjecture.
Category: Number Theory

New Maximum Interval Between Any Number and the Nearest Prime Number and Related Conjectures

Authors: Juan Moreno Borrallo
Comments: 5 Pages.

In this short paper we prove that for n ≥ 2953652287 it exists some prime number between nand n + log(n), improving the best known proved bounds for the maximum interval between anynumber and the nearest prime number, as well as the maximum difference between two consecutiveprime numbers (prime gap). We note that this result proves some open conjectures on prime gapsand maximum intervals between any number and the nearest prime number.
Category: Number Theory

Sufficiently Large Number N Makes ∑_(k=n)^(2n-1) C/(ak+b) = C/a Lnu20612

Authors: Tai-Choon Yoon, Yina Yoon
Comments: 5 Pages.

According to the Riemann rearrangement theorem, when a sequence converges, the sum can be changed by rearranging the order of the sequence. However, the result cannot be changed simply by rearranging the order of any sequences. In the case of the alternating harmonic series exemplified by Riemann, even if the result was the same by chance, the sum of the series was obtained by ignoring the sum oflim┬( n→∞)u2061∑_(k=n)^(2n-1) 1/(k+1)=lnu20612.
Category: Number Theory

Les Preuves de Syracuse (Evidence from Syracuse)

Authors: Pierre Lamothe
Comments: 15 Pages. In French

The monoid algebra of transition functions between numbers of generalized Collatzsequences has revealed the universal cause of cycles and were used to demonstrate, both :— The absolute truth of the Syracuse conjecture cannot be verified mathematically, as a random cycle is always possible when the ratio (d/m) of d divisions by 2 to m multiplications by 3 is a rational approximation of log 3/ log 2.— The Syracuse conjecture remains true with a zero probability status in practice dueto the exponential decay of its probability as a function of cycle length.
Category: Number Theory

An Algorithm for Finding the Factors of Fermat Numbers

Authors: Emmanuil Manousos
Comments: 3 Pages.

In this article we present an algorithm for finding the factors Q of composite Fermat numbers. The algorithm finds the Q factors with less tests than required through the equation Q=2n K+1.
Category: Number Theory

The Philosophical and Mathematical Implications of Division by 0/0 = 1 in Light of Einstein’s Theory of Special Relativity

Authors: Budee U. Zaman
Comments: 9 Pages.

The enigma of dividing zero by zero 0 0 has perplexed scholars across philosophy, mathematics, and physics, remaining devoid of a clear-cut solution. This lingering conundrum leaves us in an unsatisfactory position,as there emerges a genuine necessity for such divisions, particularly in scenarios involving tensor components that are both set at zero. This article endeavors to grapple with this profound issue by leveraging the insights of Einstein’s theory of special relativity. Surprisingly, when we wholeheartedly embrace the ramifications of this theory, it becomes evidentthat zero divided by zero must equate to one 00 = 1. Essentially, we are confronted with a pivotal decision: either embrace the feasibility and definition of dividing zero by zero, in accordance with Einstein’s theory of special relativity, or reevaluate the integrity of this fundamental theory itself. This exploration delves into the profound consequences arisingfrom this critical choice.
Category: Number Theory

Exist[ence Of] a Prime in Interval N^2 and "N^2+epsilon n"

Authors: Hashem Sazegar
Comments: 5 Pages.

Oppermance’ conjecture states that there is a prime number between n^2 and n^2 + n for every positive integer n,first we show that , All integer numbers between x^2 and x^2 + ϵx can be written as x^2 + i > 4p that 1 ≤ i ≤ ϵx andp = (x − m − 2)2 + j in which j is a number in intervals 1 ≤ j ≤ ϵ(x − m − 2),and then we prove generalization of Oppermance’ conjecture i.e there is a prime number in interval n^2 and n^2 + ϵn such that 0 < ϵ ≤ 1.
Category: Number Theory

A Theorem on the Golden Section and Fibonacci Numbers

Authors: Rolando Zucchini
Comments: 13 Pages.

In the chapter 12° of his most significant book LIBER ABACI, Leonardo Pisano known as Fibonacci (Pisa 1170-1240 (?)) proposed a problem on the reproduction of rabbits [*]. So many scholars deduced that he arrives to his famous numerical sequence starting from this problem. In this article is explained a new hypothesis. The Fibonacci sequence was generated by the iteration of a theorem on the Golden Section, and it is presumably that was the great Italian mathematician to state and demonstrate it. The theorem allow us to proof a lot of properties of Fibonacci numbers. [*] A man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced by the initial couple in a year supposing every couple each month produces a new pair that can reproduce itself from the second month?
Category: Number Theory

A New Formulation of Mertens Function

Authors: Juan Moreno Borrallo
Comments: 2 Pages.

In this brief note there are showed original formulations for the reciprocal of the Riemann zeta function evaluated at 1, and Mertens function.
Category: Number Theory

Infinity Tensors, the Strange Attractor and the Riemann Hypothesis

Authors: Parker Emmerson
Comments: 6 Pages.

The Riemann Hypothesis can be reworded to indicate that the real part of one half always balanced at the infinity tensor by stating that the Riemann zeta function has no more than an infinity tensor's worth of zeros on the critical line.
Category: Number Theory

A New Closed Formula for the Riemann Zeta Function at Prime Numbers

Authors: Oussama Basta
Comments: 2 Pages. (Note by viXra Admin: Please only submit complete/finalized paper)

The Riemann zeta function is one of the most important functions in mathematics, but it is also one of the most difficult to compute. In this paper, we present a new closed formula for the Riemannzeta function at prime numbers. Our formula is based on a new function.
Category: Number Theory

Flows of the Riemann Hypothesis

Authors: Tai-Choon Yoon, Yina Yoon
Comments: 4 Pages.

The Riemann hypothesis is a mathematical conjecture that relates to the calculation of prime numbers through the Riemann product formula, which represents the product of Riemann zeta function and factorial. There were flows in deriving ∫x^(s-1)/(e^x-1) dx from Riemann product formula and, in attempting to represent the negative region by substituting x with —x. Furthermore, asserting that the Riemann zeta function, in the absence of a definition for negative factorial, obtains trivial zeros for negative even numbers through the Bernoulli exponential generating formula in the negative domain is also incorrect.
Category: Number Theory

About an Integral in Valean's Book

Authors: Edgar Valdebenito
Comments: 3 Pages.

The evaluation of integrals is an important subject in mathematics, physics and applied sciences. In this note we give some integrals fot pi^3 .
Category: Number Theory

Sum of Three Cubes Explored - Proof

Authors: James DeCoste
Comments: 15 Pages. Contact: jbdecoste@eastlink.ca

Using already known techniques along with some not so obvious innovations on my part, I was able to show (prove) that there are solutions for all K (except those of the form 9m+/-4 and 9m+/-5 which are impossible) for +/-K = +/- (x^3) +/- (y^3) +/- (z^3). A further stipulation is that x, y and z must be whole numbers that can be a combination of positives and negatives. This is achieved through simple subtraction. Setting up a table showing that all K can be represented using a multiple of 27 plus a mask lends validity to a portion of the proof. These representations may and often do contain many more than the required number of cubes summed up. I side step that problem by showing that no matter the K picked and how ever many cubes are required to create it in my representations, they can all be reduced to a maximum of cubes summed. Exactly what we require for the proof. Having done that we are complete. The three new cubes we have just reduced to are already included in table. They are items I have already represented in the above format.
Category: Number Theory

The Proof of the Riemann Conjecture

Authors: Liao Teng
Comments: 34 Pages.

In order to strictly prove the hypothesis and conjectures in Riemann's 1859 paper on the Number of Prime Numbers Not greater than x from a pure mathematical point of view, and in order to strictly prove the Generalized hypothesis and the Generalized conjectures, this paper uses Euler's formula to study the relationship between symmetric and conjugated zeros of Riemann's ζ(s) function and Riemann's ξ(s) function, and proves that Riemann's hypothesis and Riemann's conjecture are completely correct.
Category: Number Theory

Proof of the Collatz Conjecture Based on a Directed Graph

Authors: Wiroj Homsup, Nathawut Homsup
Comments: 6 Pages. (Note by viXra Admin: Both author's names should be filled on the Replacement Form))

The Collatz conjecture considers recursively sequences of positive integers where n is succeeded by n/2 , if n is even or (3n+1)/2 , if n is odd. The conjecture states that for all starting values n the Collatz sequence eventually reaches a trivial cycle 1, 2, 1, 2u2026u2026. If the Collatz conjecture is false, then either there is a nontrivial cycle, or one sequence goes to infinity. In this paper, we construct a directed graph based on the union of infinite number of basic Collatz directed graphs. Each basic Collatz directed graph relates to each positive integer. We show that the directed graph is connected and covers all positive integers. There is only a trivial cycle and no sequence goes to infinity.
Category: Number Theory

"Infinitely Often"="infinity" [:] a Statistics Approach to Small Gaps Between Primes

Authors: Yung Zhao
Comments: 2 Pages.

We construct a sequence of consecutive primes. From the perspective of statistics, we analyze and handle them by the combination of the fundamental property of primes with James Maynard's result. It reveals that there are infinitely many pairs of primes which differ by two.
Category: Number Theory

Proof of the Collatz Conjecture Using the Reverse Algorithm

Authors: Ugur Pervane
Comments: 7 Pages.

The Collatz conjecture has remained unsolved for a long time. In this paper, a proof of this conjecture will be presented. We know that almost all numbers will eventually reach one through the steps of the Collatz algorithm. Terence Tao has previously proven this. This paper demonstrates that all numbers will reach one by using Tao's proof. If almost all numbers reach one, then the probability that a randomly chosen number from an infinite set of numbers will reach one is one. The probability that a number will not reach one is zero. The probability of selecting the elements of the number sequences associated with a number n that violates the conjecture from an infinite set of numbers is a non-zero value such as c. However, this contradicts the proof that almost all numbers reach one. Therefore, there is no such number n that violates the conjecture, and the conjecture is true for all numbers. In order to prove that the probability of selecting the elements of the number sequences associated with a number n that violates the conjecture from an infinite set of numbers is a non-zero value like c, we examine the sequences associated with one. If the probability of selecting the elements of each branch of these sequences from an infinite set of numbers is a non-zero value, we reach the desired proof.
Category: Number Theory

A Simple Probabilistic Heuristic Supporting the Collatz Conjecture

Authors: Shreyansh Jaiswal
Comments: 11 Pages.

The Collatz Conjecture, also known as the 3x+1 problem, posits that for any positive integer n, the sequence defined by the Collatz functionwill eventually reach the number 1. This conjecture has been extensively tested for a vast range of values, consistently supporting its validity. Inthis paper, we explore a probabilistic perspective to provide additional support for the conjecture. We focus on the probability that the Collatzsequence T(n), for any starting value n, reaches a power of 2—an essential step in the sequence’s progression toward 1. Our approach suggests that as n tends to infinity, the likelihood of the Collatz conjecture being satisfied becomes very high. This probabilistic argument aligns with the extensive empirical evidence supporting the conjecture and offers a novel perspective on its validity. While not a formal proof, our findings contribute to the broader understanding of the Collatz Conjecture and reinforce the conjecture’s plausibility through probabilistic reasoning.
Category: Number Theory

A Simple Framework for Improving the Prime Number Theorem Regarding Estimating the Nth Prime Number

Authors: Shreyansh Jaiswal
Comments: 20 Pages. Distributed under CC BY-SA 4.0

The Prime Number Theorem (PNT) offers a foundational approximation for the distribution of prime numbers and aids in estimating the nth prime number p(n) through p(n) ∼ n log n. This paper proposes enhancements to this approximation by introducing a correction factor C(k), refining the estimate to p(n) ≈ C(k)·n·log n. The derivation of C(k) is explored, alongside its asymptotic behavior and empiricalanalysis. A generalized formula for p(n) is also derived, eliminating variables other than n and e (Euler’s number). Empirical comparisons with traditional methods demonstrate the accuracy and computational efficiency of these new approaches. Ideal conditions for optimal performance of C(k) are examined. Graphical representations and statistical analyses support the validity of the proposed refinements. The paper concludes with a discussion on the implications of these findings and potential areas for future research.
Category: Number Theory

On the L-Functions from Generalized Riemann Hypothesis, Birch and Swinnerton-Dyer Conjecture, and the Prime Numbers from Polignac's and Twin Prime Conjectures

Authors: John Yuk Ching Ting
Comments: 64 Pages. LMFDB input on BSD conjecture, Riemann hypothesis, Polignac's and Twin prime conjectures

We analyze L-functions of elliptic curves (and apply Sign normalization) which support simplest version of Birch and Swinnerton-Dyer conjecture to be true. Dirichlet eta function (proxy function for Riemann zeta function as generating function for all nontrivial zeros) and Sieve of Eratosthenes (generating algorithm for all prime numbers) are essentially infinite series. We apply infinitesimals to their outputs. Riemann hypothesis asserts the complete set of all nontrivial zeros from Riemann zeta function is located on its critical line. It is proven to be true when usefully regarded as an Incompletely Predictable Problem. The complete set with derived subsets of Odd Primes contain arbitrarily large number of elements and satisfy Prime number theorem for Arithmetic Progressions, Generic Squeeze theorem and Theorem of Divergent-to-Convergent series conversion for Prime numbers. Having these theorems being satisfied, Polignac's and Twin prime conjectures are separately proven to be true when usefully regarded as Incompletely Predictable Problems.
Category: Number Theory

The Strict Proof That the Riemann Zeta Function Equation Has No Non-Trivial Zeros

Authors: Xiaochun Mei
Comments: 13 Pages.

A standard method is proposed to prove strictly that the Riemann Zeta function equation has no non-trivial zeros. The real part and imaginary part of the Riemann Zeta function equation are separated completely. By comparing the real part and the imaginary part of Zeta function equation individually, a set of equation is obtained. It is proved that this equation set only has the solutions of trivial zeros. So the Riemann Zeta function equation has no non-trivial zeros. The Riemann hypothesis does not hold.
Category: Number Theory

Pi's Irrationality Using Maclaurin Polynomials

Authors: Timothy Jones
Comments: 7 Pages. Got some suggestions for improving.

After reviewing Maclaurin series and the Alternating Series Estimation Theorem (ASET), we show how these can be combined with some algebraic observations to prove that pi is irrational.
Category: Number Theory

On Prime Cycles in Directed Graphs Built with Primality Test

Authors: Marcin Barylski
Comments: 4 Pages. Small updates in paragraph 2 - adding example and reference to the framework

One of the most famous unsolved problems in mathematics is Collatz conjecture which is claiming that all positve numbers subjected to simple 3x + 1/2 formula will eventually result in 1, with only one known cycle (1, 4, 2, 1) present in the calculations. This work is devoted to finding cycles in other interesting sequences of integer numbers, constructed with the use of some aspect of primality test.
Category: Number Theory

Clarifying an Early Step in Hardy's Transcendence of Pi Proof

Authors: Timothy Jones
Comments: 3 Pages. Got some suggestions for improving.

We clarify and strengthen Hardy's footnote proof of an essential step in his proof of the transcendence of pi. We show that ri is algebraic if and only if r is algebraic.
Category: Number Theory

Approaching Goldbach’s Conjecture Using the Asymmetric Relationship Between Primes and Composites Within a Limited Even Boundary

Authors: Junho Eom
Comments: 10 Pages.

When an arbitrary integer n is chosen, the set of consecutive numbers from 0 to n is considered the 1st boundary and it expands using an arithmetic sequence with n elements but limited to n2, or the nth boundary. Except for 1, each number in the 1st boundary can be expressed in the wave form of ‘y_n=sinu2061(180/n∙x)’, it connects to composites located between the 2nd and nth boundaries. Through wave multiplication, the wave of a composite, such as y4, overlaps with the wave of a prime, y2, can be eliminated, leaving only the waves of primes. Except for the prime factors of n, the waves of primes in the 1st boundary now have an asymmetrical relationship with the composites located between the 2nd and nth boundaries. If ‘y1’ is divided by the product of prime waves, the asymmetrical relationship between primes and composites is eliminated, leaving only new primes passively remaining between the 2nd and nth boundaries. Under the same conditions, the boundary can be limited to 2n (or 2nd boundary) instead of n2 (or nth boundary). When n ≥ 4, there is at least one prime in the 1st boundary, excluding the even prime 2. This ensures that a new prime, which maintains the same distance from n and forms a symmetric relationship, can be paired. The sum of this pair - a caused prime in the 1st boundary and an effected new prime in the 2nd boundary - is always 2n, thereby satisfying Goldbach’s conjecture.
Category: Number Theory

Analyzing Non-Trivial Zeros of the Riemann Zeta Function Using Polar Coordinates

Authors: Bryce Petofi Towne
Comments: 22 Pages.

This paper presents an approach to analyzing the non-trivial zeros of the Riemann zeta function using polar coordinates. We investigate whether the real part of all non-trivial zeros can be determined to be a constant value. By transforming the traditional complex plane into a polar coordinate system, we recalculated and examined several known non-trivial zeros of the zeta function. Our findings provide an alternative framework for understanding this profound mathematical conjecture.Through mathematical proof and leveraging analytic continuation and holomorphic function theory, we explore the nature of (sigma) in the polar coordinate system. This analysis transforms the problem into a geometric one, allowing for simpler and more intuitive calculations. This approach provides a step towards an alternative understanding of the properties of the Riemann zeta function's non-trivial zeros. The findings of this work indicates that wit this geometric perspective, the Riemann Hypothesis holds true.
Category: Number Theory

Verification of the Riemann Hypothesis Using a Novel Positive Coordinate System Approach

Authors: Bryce Petofi Towne
Comments: 19 Pages.

This paper presents a novel approach to verifying the Riemann Hypothesis using a redefined positive coordinate system and polar representation of complex numbers. Inspired by discussions on the nature of negative numbers, zero, and imaginary numbers, we developed a co- ordinate system that exclusively uses positive numbers. Through this innovative method, we recalculated and confirmed several known non- trivial zeros of the Riemann zeta function. Our results consistently support the hypothesis that all non-trivial zeros of the zeta function lie on the critical line where the real part is 1/2. This method provides a new perspective on the Riemann Hypothesis and opens potential avenues for further mathematical exploration.Furthermore, through rigorous mathematical proof and leveraging zero consistency theory in complex analysis, we demonstrate that in the polar coordinate system, the Riemann Hypothesis holds true for all non-trivial zeros. This proof provides a significant step towards a comprehensive understanding of this profound mathematical conjecture.
Category: Number Theory

Goldbach's Conjecture

Authors: Bassera Hamid
Comments: 1 Page. Sent to American Mathematical Society in June 05 2024

In this article I try to make my modest contribution to the proof of Goldbach’s conjecture and I propose to simply go through its negation.
Category: Number Theory

The abc Conjecture Is True

Authors: Abdelmajid Ben Hadj Salem
Comments: 5 Pages. Submitted to the Czechoslovak Mathematical Journal. Comments welcome.

In this paper, we consider the abc conjecture. Assuming the conjecture c1. For the case epsilon in ]0,1[, we consider that the abc conjecture is false, from the proof, we arrive in a contradiction.
Category: Number Theory

The Explicit Formula of Bernoulli Numbers

Authors: Abdelhay Benmoussa
Comments: 10 Pages.

The aim of this paper is to provide an elementary proof to a well-known explicit formula of Bernoulli numbers.
Category: Number Theory

The Explicit Formula of Bernoulli Numbers

Authors: Abdelhay Benmoussa
Comments: 10 Pages.

The aim of this paper is to give an elementary proof to a well-known explicit formula for Bernoulli numbers.
Category: Number Theory

Both the 1-Dimensional Line from Sieve of Eratosthenes and 2-Dimensional Trajectory from Riemann Zeta Function Have Centroids as Perfect Point Symmetry

Authors: John Yuk Ching Ting
Comments: 31 Pages. Finalized Preprint Version for submission to JNT dated July 7, 2024.

From perspective of Number theory, Dirichlet eta function (proxy function for Riemann zeta function as generating function for all nontrivial zeros) and Sieve of Eratosthenes (as generating algorithm for all prime numbers) are essentially infinite series. We apply infinitesimals to their outputs. Riemann hypothesis asserts the complete set of all nontrivial zeros from Riemann zeta function is located on its critical line. It is proven to be true when usefully regarded as an Incompletely Predictable Problem. We ignore even prime number 2. The complete set with derived subsets of Odd Primes all contain arbitrarily large number of elements while satisfying Prime number theorem for Arithmetic Progressions, Generic Squeeze theorem and Theorem of Divergent-to-Convergent series conversion for Prime numbers. Having these theorems satisfied by all Odd Primes, Polignac's and Twin prime conjectures are separately proven to be true when usefully regarded as Incompletely Predictable Problems.
Category: Number Theory

Applying Infinitesimals to Outputs from Sieve of Eratosthenes and Riemann Zeta Function When Treated as Infinite Series

Authors: John Yuk Ching Ting
Comments: 33 Pages. Revised version of ANT Submission

In essence, Sieve of Eratosthenes (as generating algorithm for all prime numbers) and Dirichlet eta function (the proxy function for Riemann zeta function as generating function for all nontrivial zeros) are infinite series. We apply infinitesimals to their outputs. We ignore even prime number 2. All the complete set and its derived subsets of Odd Primes contain arbitrarily large number of elements that satisfy Prime number theorem for Arithmetic Progressions, Generic Squeeze theorem and Theorem of Divergent-to-Convergent series conversion for Prime numbers. With these theorems satisfied by all Odd Primes, Polignac's and Twin prime conjectures are proven to be true when usefully regarded as Incompletely Predictable Problems. Riemann hypothesis proposes all nontrivial zeros of Riemann zeta function are located on its critical line. It is separately proven to be true when usefully regarded as an Incompletely Predictable Problem.
Category: Number Theory

An Ephemeral Approach to Solving Fermat’s Last Theory

Authors: D. Ross Randolph
Comments: 22 Pages.

This paper approaches FLT for the Sophie Germain Case 2 set of conditions, and shows through infinite iteration that no finite non-trivial integer value solutions may exist.
Category: Number Theory

SCQ Two Cycles of Links High Horizons

Authors: Rolando Zucchini
Comments: 11 Pages.

With reference to the Syracuse Conjecture Quadrature (SCQ), this article contains two links of high main horizons and their corresponding lower horizons such that Ꝋ(l) < Ꝋ(m), calculated by Theorem of Independence. A further confirmation that cycles of links can be managed to our liking. Moreover the procedure explains show the beauty and the magical harmony of odd numbers. At the same time it’s confirmed that SC (or CC) is not fully verifiable as additional highlighted by the four illustrative patterns. There are no doubts: it’s a particular sort of the Circle Quadrature, but its initial statement is true. In other words: BIG CRUNCH (go back to 1) is always possible but BIG BANG (to move on) has no End.
Category: Number Theory

Proofs for Collatz Conjecture and Kaakuma Sequence

Authors: Bambore Dawit Geinamo
Comments: 47 Pages. 47

The objective of this study is to present precise proofs of the Collatz conjecture and introduce some interesting behavior on Kaakuma sequence. We propose a novel approach that tackles the Collatz conjecture using different techniques and angles. The Collatz Conjecture, proposed by Lothar Collatz in 1937, remains one of the most intriguing unsolved problems in mathematics. The conjecture posits that, for any positive integer, applying a series of operations will eventually lead to the number 1. Despite decades of rigorous investigation and countless computational verification, a complete proof has eluded mathematicians. In this research endeavor, we embark on a comprehensive exploration of the Collatz Conjecture, aiming to shed light on its underlying principles and ultimately establish its validity. Our investigation begins by defining the Collatz function and investigating some behavior like. transformation, selective mapping, successive division, constant growth rate of inverse tree, and more. Using and analyzing the discovered properties of Collatz sequence we can show there are contradiction. In addition to this we investigate Qodaa ratio test that validates the reality of discovered behavior of Collatz sequence and works for infinite and distinct Kaakuma sequences. Our investigation culminates in the formulation of a set of conjectures encompassing lemmas and postulates, which we rigorously prove using a combination of analytical reasoning, numerical evidence, and exhaustive case analysis. These results provide compelling evidence for the veracity of the Collatz Conjecture and contribute to our understanding of the underlying mathematical structure. This proof helps to change some researchers' views on unsolved problems and offers new perspectives on probability in infinite range. In this study, we uncover the dynamic nature of the Collatz sequence and provide a reflection and interpretation of the probabilistic proof of the Collatz Conjecture.
Category: Number Theory

Proving & Teaching Beal Conjecture

Authors: A. A. Frempong
Comments: 8 Pages. Copyright © by A. A. Frempong

By applying basic mathematical principles, the author surely, and instructionally, proves, directly, the original Beal conjecture which states that if A^x + B^y = C^z, where A, B, C, x, y, z are positive integers and x, y, z > 2, then A, B and C have a common prime factor. One will let r, s, and t be prime factors of A, B and C, respectively, such that A = Dr, B = Es, C = Ft, where D, E, and F are positive integers. Then, the equation A^x + B^y = C^z becomes D^xr^x + E^ys^y = F^zt^z. The proof would be complete after showing that the equalities, r^x = t^x, s^y = t^y and r = s = t, are true. More formally, the conjectured equality, r^x = t^x would be true if and only if (r^x /t^x) =1; and the conjectured equality s^y = t^y would be true if and only if (s^y/ t^y) = 1. These conjectures would be proved in the Beal conjecture proof. The main principle for obtaining relationships between the prime factors on the left side of the equation and the prime factor on the right side of the equation is that the power of each prime factor on the left side of the equation equals the same power of the prime factor on the right side of the equation. High school students can learn and prove this conjecture for a bonus question on a final class exam
Category: Number Theory

Merging the Goldbach and the Bunyakovsky Conjecture into a Unified Prime Axiom of Second-Order Logic and Investigating Much Beyond the Goldbach's Conjecture and the Prime Number Theorem

Authors: Alexis Zaganidis
Comments: 31 Pages.

Merging the Goldbach and the Bunyakovsky conjecture into a Unified Prime Axiom of second-order logic and investigating much beyond the Goldbach's conjecture and the prime number theorem.
Category: Number Theory

The Collatz Conjecture, Pythagorean Triples, and the Riemann Hypothesis: Unveiling a Novel Connection Through Dropping Times

Authors: Darcy Thomas
Comments: 15 Pages.

In the landscape of mathematical inquiry, where the ancient and the modern intertwine, few problems captivate the imagination as profoundly as the Collatz conjecture and the quest for Pythagorean triples. The former, a puzzle that has defied solution since its inception in the 1930s by Lothar Collatz, asks us to consider a simple iterative process: for any positive integer, if it is even, divide it by two; if it is odd, triple it and add one. Despite its apparent simplicity, the conjecture leads us into a labyrinth of diverse complexity, where patterns emerge and dissolve in an unpredictable dance. On the other hand, Pythagorean triples, sets of three integers that satisfy the ancient Pythagorean theorem, have been a cornerstone of geometry since the time of the ancient Greeks, embodying the harmony of numbers and the elegance of spatial relationships. This exploratory paper embarks on an unprecedented journey to bridge these seemingly disparatedomains of mathematics. At the heart of this exploration is the discovery of a novel connection between Collatz dropping times and Pythagorean triples. I will demonstrate how the dropping time of each odd number can be uniquely associated with a Pythagorean triple. As you will see, the triples seem to be encoding spatial information about Collatz trajectories. As we begin to work with triples, we’ll be motivated to move from the number line to the complex plane where we find structure andbehavior resembling that of the Riemann Zeta function and it’s zeros.
Category: Number Theory

Derivation|Correction of Hardy-Littlewood Twin Prime Constant using Prime Generator Theory (PGT)

Authors: Jabari Zakiya
Comments: 7 Pages. Corrected data value in Figure 2 for m = 11.

The Hardy—Littlewood twin prime constant is a metric to compute the distribution of twin primes. Using Prime Generator Theory (PGT) it is shown it is more easily mathematically and conceptually derived, and the correct value is a factor of 2 larger.
Category: Number Theory

Function for Prime Numbers

Authors: Massimo Russo
Comments: 58 Pages.

The Function [5*(1+1/x) + 1] for every value of x determined by Sequence A: x = (5^2)+5*2*(n(n+1)/2)where n ≥ 0 determines an infinite series of fractional numbers N/d: 5*(1+1/x) + 1 = N/dsuch that N and d are prime numbers.
Category: Number Theory

Pythagorean Triples and Fermat's Theorem N = 4

Authors: Rolando Zucchini
Comments: 7 Pages.

This article contains a theorem to build the Primitive Pythagorean triples and the proof of the last Fermat’s Theorem for n = 4.
Category: Number Theory

Riemann Hypothesis: Direct Demonstration Proposal

Authors: Vincent KOCH
Comments: 3 Pages.

In his 1859 article "On the number of prime numbers less than a given quantity", Bernhard Riemann formulated the hypothesis that all non-trivial zeros of the Zeta function have the real part 1/2.This assertion, known as the "Riemann Hypothesis", remains unproven to this day. The present paper is an attempt at a direct demonstration.
Category: Number Theory

A Convergent Subsequence of $theta_n(x+iy)$ in a Half Strip

Authors: Young Deuk Kim
Comments: 8 Pages. Typos are fixed.

For $frac{1}{2}0$ and $ninmathbb{N}$, let $displaystyletheta_n(x+iy)=sum_{i=1}^nfrac{{mbox{sgn}}, q_i}{q_i^{x+iy}}$,where $Q={q_1,q_2,q_3,cdots}$ is the set of finite product of distinct odd primes and${mbox{sgn}}, q=(-1)^k$ if $q$ is the product of $k$ distinct primes.In this paper we prove that there exists an ordering on $Q$ such that $theta_n(x+iy)$ has a convergent subsequence.
Category: Number Theory

Goldbach's Number Construction

Authors: Radomir Majkic
Comments: 7 Pages.

The internal structure of the natural numbers reveals the relation between the weak and the strong Goldbach's conjectures. The three prime integers structure of the odd integers alreadycontains the two prime integers base of the even integers. An explicit one-to-one correspondence between these two structures, defined asGoldbach's numbers exist. Thus, if the weak Goldbach's conjecture is true, the strong Goldbach'sconjecture should be. Hopefully, this will bring a happy end to Goldbach'sconjecture problem.
Category: Number Theory

Complete Operations

Authors: Pith Peishu Xie
Comments: 27 Pages.

The Operator axioms have produced complete operations with real operators. Numerical computations have been constructed for complete operations. The classic calculator could only execute 7 operator operations: + operator operation(addition), - operator operation(subtraction), $times$ operator operation(multiplication), $div$ operator operation(division), ^{} operator operation(exponentiation), $surd$ operator operation(root extraction), log operator operation(logarithm). In this paper, we invent a complete calculator as a software calculator to execute complete operations. The experiments on the complete calculator could directly prove such a corollary: Operator axioms are consistent.
Category: Number Theory

Proof for Specific Type of Continued Fraction

Authors: Isaac Mor
Comments: 9 Pages.

I am going to use telescoping series and then a proof by induction. I am using Lambert's continued fraction for the base case.
Category: Number Theory

Proof for Specific Type of Continued Fraction

Authors: Isaac Mor
Comments: 9 Pages.

I am going to use telescoping series and then a proof by induction. I am using Lambert's continued fraction for the base case.
Category: Number Theory

Proof of Fermat’s Last Theorem for Odd Primes

Authors: Minho Baek
Comments: 27 Pages.

It was already proved right that xn+yn=zn, (n>2) has no solutions in positive integers which we called Fermat’s Last Theorem (FLT) by Andrew Wiles. But his proof would be impossible in the 17th century. Since Fermat showed he proved n=even by leaving proof for n=4, many people have tried to prove the odd primes. I took the idea from Euler proof and proved in case of n=odd primes by simple method.
Category: Number Theory

New Bounds on Mertens Function

Authors: Juan Moreno Borrallo
Comments: 6 Pages.

In this brief paper we study and bound Mertens function. The main breakthrough is the obtention of a Möbius-invertible formulation of Mertens function, which with some transformations and the application of a generalization of Möbius inversion formula, allows us to reach an asymptotic rate of growth of Mertens function that proves the Riemann Hypothesis.
Category: Number Theory

Generalization for Specific Type of Continued Fraction

Authors: Isaac Mor
Comments: 13 Pages.

I came across "The Ramanujan Machine" on the Internet and, using my intuition on those kind of stuff, I found some interesting results.
Category: Number Theory

Generalization for Specific Type of Continued Fraction

Authors: Isaac Mor
Comments: 7 Pages.

I came across "The Ramanujan Machine" on the Internet and, using my intuition on those kind of stuff, I found some interesting results.
Category: Number Theory

No Collatz Conjecture Integer Series Have Looping

Authors: Tsuneaki Takahashi
Comments: 2 Pages.

If the series of Collatz Conjecture integer has looping in it, it is sure the members of the loop cannot reach to value 1. Here it is proven that the possibility of looping is zero except one.
Category: Number Theory

No Collatz Conjecture Integer Series Have Looping

Authors: Tsuneaki Takahashi
Comments: 3 Pages.

If the series of Collatz Conjecture integer has looping in it, it is sure the members of the loop cannot reach to value 1. Here it is proven that the possibility of looping is zero except one.
Category: Number Theory

Riemann Hypothesis

Authors: Bertrand Wong
Comments: 14 Pages.

This paper discusses the distribution of the non-trivial zeros of the Riemann zeta function ζ. It looks into the question of whether any non-trivial zeros would ever possibly be found off the critical line Re(s) = 1/2 on the critical strip between Re(s) = 0 and Re(s) = 1, e.g., at Re(s) = 1/4, 1/3, 3/4, 4/5, etc., and why all the non-trivial zeros are always found at the critical line Re(s) = 1/2 on the critical strip between Re(s) = 0 and Re(s) = 1 and not anywhere else on this critical strip, with the first 1013 non-trivial zeros having been found only at the critical line Re(s) = 1/2. It should be noted that a conjecture, or, hypothesis could possibly be proved by comparing it with a theorem that has been proven, which is one of the several deductions utilized in this paper. Through these several deductions presented, the paper shows how the Riemann hypothesis may be approached to arrive at a solution. In the paper, instead of merely using estimates of integrals and sums (which are imprecise and may therefore be of little or no reliability) in the support of arguments, where feasible actual computations and precise numerical facts are used to support arguments, for precision, for more sharpness in the arguments, and for "checkability" or ascertaining of the conclusions. [Published in international mathematics journal.]
Category: Number Theory

Proof of the Collatz Conjecture

Authors: Wiroj Homsup, Nathawut Homsup
Comments: 9 Pages.

The Collatz conjecture considers recursively sequences of positive integers where n is succeeded by n/2 , if n is even, or (3n+1)/2 , if n is odd. The conjecture states that for all starting values n the sequence eventually reaches the trivial cycle 1, 2, 1, 2u2026u2026The inverted Collatz sequences can be represented as a tree with 1 as its root node. In order to prove the Collatz conjecture, one must demonstrate that the tree covers all positive integers. In this paper, we construct a Collatz tree with 1 as its root node by connecting infinite number of basic trees. Each basic tree relates to each positive integers. We prove that a Collatz tree is a connected tree and covers all positive integers.
Category: Number Theory

Divisible Cyclic Numbers

Authors: Julian Beauchamp
Comments: 4 Pages.

There are known to exist a number of (multiplicative) cyclic numbers, but in this paper I introduce what appears to be a new kind of number, which we call divisible cyclic numbers (DCNs), examine some of their properties and give a proof of their cyclic property. It seems remarkable that I can find no reference to them anywhere. Given their simplicity, it would be extraordinary if they were hitherto unknown.
Category: Number Theory

On the Incompletely Predictable Problems of Riemann Hypothesis, Modified Polignac's and Twin Prime Conjectures

Authors: John Yuk Ching Ting
Comments: 82 Pages. Now incorporating Hodge conjecture, Grothendieck period conjecture and Pi-Circle conjecture

We validly ignore even prime number 2. Based on all arbitrarily large number of even prime gaps 2, 4, 6, 8, 10...; the complete set and its derived subsets of Odd Primes fully comply with Prime number theorem for Arithmetic Progressions. With this condition being satisfied by all Odd Primes, we argue that Modified Polignac's and Twin prime conjectures are proven to be true when these conjectures are treated as Incompletely Predictable Problems. In so doing [and with Riemann hypothesis being a special case], this action also support the generalized Riemann hypothesis formulated for Dirichlet L-function. By broadly applying Hodge conjecture, Grothendieck period conjecture and Pi-Circle conjecture to Dirichlet eta function (which acts as proxy function for Riemann zeta function), Riemann hypothesis is separately proven to be true when this hypothesis is treated as Incompletely Predictable Problem.
Category: Number Theory

On the Incompletely Predictable Problems of Riemann Hypothesis, Modified Polignac's and Twin Prime Conjectures

Authors: John Yuk Ching Ting
Comments: 74 Pages. Now incorporating Hodge conjecture and Grothendieck period conjecture

We validly ignore even prime number 2. Based on all arbitrarily large number of even prime gaps 2, 4, 6, 8, 10...; the complete set and its derived subsets of Odd Primes fully comply with Prime number theorem for Arithmetic Progressions. With this condition being satisfied by all Odd Primes, we argue that Modified Polignac's and Twin prime conjectures are proven to be true with these conjectures treated as Incompletely Predictable Problems. In so doing [and with the famous Riemann hypothesis being a special case], the generalized Riemann hypothesis formulated for Dirichlet L-function is also supported. By broadly applying Hodge conjecture and Grothendieck period conjecture to Dirichlet eta function (as proxy function for Riemann zeta function), Riemann hypothesis is separately proven to be true with this hypothesis treated as Incompletely Predictable Problem.
Category: Number Theory

Collatz Conjecture Proved Ingeniously & Very Simply

Authors: A. A. Frempong
Comments: 12 Pages. Copyright © by A. A. Frempong

Collatz conjecture states that beginning with a positive integer, if one repeatedly performs the following operations to form a sequence of integers, the sequence will eventually reach the integer one; the operations being that if the integer is even, divide it by 2, but if the integer is odd, multiply it by 3 and add one; and also, use the result of each step as the input for the next step One would note the patterns of the sequence terms as the Collatz process reaches the equivalent powers, 2^(2k) (k = 2, 3,...) and the sequence reaches the integer 1 by repeated division by 2. Two main cases are covered. In Case 1, the integer can be written as a power of 2 as 2^(k) (k=1,2,3,u2026), and in this case, the sequence would reach the integer one by repeated division by 2, i.e., 2^(k-1), 2^(k-2), 2^(k-3,),u2026,2^(k-k). In Case 2, the integer cannot be written as a power of 2, but the sequence terms reach the equivalent power, 2^(2k) (k = 2, 3,...) and by repeated division by 2, the sequence will reach the integer 1. In Case 2, when the sequence terms reach some particular integers such as 5, 21 and 85, the application of 3n + 1 to these integers will result in the powers, 2^(2k). One would call these integers, the 2k-power converters. There are infinitely many 2k-power converters as there are 2^(2k) powers. There are infinitely many paths for converting integers to 2^(2k) powers. Of these paths, the integer 5-path, is the nearest 2^(2k) converter path to the integer 1 on the 2^(2k)-route. Other integers can follow the integer 5-path to 16 as follows: Let n be an integer whose sequence terms would reach 16, and let n ± r = 5, where r is the net change in the sequence terms before the integer 5; and one uses the positive sign if n<5, but the negative sign if n > 5. One will call the following, the 5-path 2k-converter formula: 3(n ± r) + 1 = 16. By the substitution axiom, using this formula, the sequence of every positive integer that cannot be written as a power of 2, would reach the integer, 16, and continue to reach the integer 1. Therefore, the sequence of every positive integer would reach the integer 1.
Category: Number Theory

Collatz Conjecture Proved Ingeniously & Very Simply

Authors: A. A. Frempong
Comments: 12 Pages. Copyright © by A. A. Frempong

To prove Collatz conjecture, one would apply a systematic observation of the sequences produced by the (3n + 1)/2 process, Two main cases are covered. In Case 1, the integer can be written as a power of 2 as 2^(k) (k = 1, 2, 3,u2026), and the sequence would reach the integer 1 by repeated division by 2, In Case 2, the integer cannot be written as a power of 2, but the sequence terms of the integers reach integers equivalent to 2^(2k) (k = 2, 3,...),and by repeated division by 2, the sequences would reach the integer 1. In Case 2, when the sequence terms reach some particular odd integers such as 5, 21 and 85, the application of 3n +1 operation to these integers will result in the integers equivalent to the powers, 2^(2k) (k = 2, 3, 4,u2026). One would call these integers, the 2k-power converters. A term of the sequence must be converted to an integer equivalent to 2^(2k) (k = 2, 3, 4,u2026). There are infinitely many paths for converting integers to 2^(2k) powers on the 2^(2k)-route, a route on which a 2^(2k)-power can be divided repeatedly by 2 until the sequence reaches the integer 1. Of these conversion paths, the integer 5-path is the nearest 2^(2k) (k=2) converter path to the integer 1 on the 2^(2k)-route. Other paths to the 2^(2k)-route include the 21-path, (k = 3), the 85-path (k = 4), and infinitely, many paths. For the 5-path, when a sequence terms reach the integer, 5, the next term would be 16. Similarly, for the integers 21, and 85, the next terms, respectively, would be 64 and 256. Some non-2k power converters can follow the integer 5-path to the to 2^(4) power. The integers 2^(p)(5) (p = 1, 2, 3,u2026) will take the integer 5-path to convert to a 2k-power. The integers, 2^(p)(21), 2^(p)(85) with (p = 1, 2, 3,u2026), will take, respectively, the integer 21-path, and the 85-path, to reach 2k-powers. There are infinitely many 2^(2k)-power converters, 2k-powers, 2k-power converter paths, and descendants, 2^(p)(C) (p = 1, 2, 3,u2026) of C, By repeated division by 2, the sequence of every positive integer that reaches the equivalent integer, 2^(2k), would reach the integer 1, Therefore, using the approaches in Cases 1 and 2, the sequence of every positive integer would eventually reach the integer 1.
Category: Number Theory

Collatz Conjecture Proved Ingeniously & Very Simply

Authors: A. A. Frempong
Comments: 12 Pages. Copyright © by A. A. Frempong

Collatz conjecture states that beginning with a positive integer, if one repeatedly performs the following operations to form a sequence of integers, the sequence will eventually reach the integer one; the operations being that if the integer is even, divide it by 2, but if the integer is odd, multiply it by 3 and add one; and also, use the result of each step as the input for the next step.One would note the patterns of the sequence terms as the Collatz process reaches the equivalent powers, 2^(2k) (k = 2, 3, ...), and the sequence reaches the integer 1 by repeated division by 2. Two main cases are covered. In Case 1, the integer can be written as a power of 2 as 2^(k) (k=1,2,3,...), and in this case, the sequence would reach the integer one by repeated division by 2, i.e., 2^(k-1), 2^(k-2), 2^(k-3,),...,2^(k-k). In Case 2, the integer cannot be written as a power of 2, but the sequence terms reach the equivalent power, 2^(2k) (k = 2, 3,...) and by repeated division by 2, the sequence will reach the integer 1. In Case 2, when the sequence terms reach some particular integers such as 5, 21 and 85, the application of 3n + 1 to these integers will result in the powers, 2^(2k). One would call these integers, the 2k-power converters. There are infinitely many 2k-power converters as there are 2^(2k) powers. There are infinitely many paths for converting integers to 2^(2k) powers. Of these paths, the integer 5-path, is the nearest 2^(2k) converter path to the integer 1 on the 2^(2k)-route. Other integers can follow the integer 5-path to 16 as follows: Let n be an integer whose sequence terms would reach 16, and let n ± r = 5, where r is the net change in the sequence terms before the integer 5; and one uses the positive sign if n < 5, but the negative sign if n > 5. One will call the following, the 5-path 2k-converter formula: 3(n ± r) + 1 = 16. By the substitution axiom, using this formula, the sequence of every positive integer that cannot be written as a power of 2, would reach the integer, 16, and continue to reach the integer 1. Therefore, the sequence of every positive integer would reach the integer 1.
Category: Number Theory

A Truly Easy Proof: Pi is Irrational

Authors: Timothy Jones
Comments: 5 Pages. Expanded content per comments received.

Using the sum of the derivatives of an integer polynomial with Euler's formula we prove that pi is irrational. We show how the technique can be used to show e and pi's transcendence.
Category: Number Theory

An Truly Easy Proof: Pi is Irrational

Authors: Timothy W. Jones
Comments: 2 Pages. There was an error in the previous: conflated composition with evaluation.

Using the derivative of an integer polynomial composed with Euler's formula we prove that pi is irrational.
Category: Number Theory

Nuanced Truth of Collatz Conjecture

Authors: Pierre Lamothe
Comments: 15 Pages. In French

The algebra of transition functions between elements of generalized Collatz sequences has revealed the true nature of cycles. The structure of the cyclic invariant enables us to demonstrate both:
a) The Syracuse conjecture cannot be substantiated as absolute veracity because no matter the length, a random cycle is theoretically possible.
b) The Syracuse conjecture can only hold true in practice with exceedingly probable status due to the exponential decay of a cycle’s probability as a function of length.
Category: Number Theory

An Algorithm for Finding the Factors of Fermat Numbers

Authors: Emmanuil Manousos
Comments: 3 Pages. Dear Editor, I am replacing the article due to errors in Example 2. Thank you for hosting my work on vixra.org. Best regards Emmanuil Manousos

In this article we present an algorithm for finding the factors Q of composite Fermat numbers. The algorithm finds the Q factors with less tests than required through the equation 2n×K+1.
Category: Number Theory

A New Closed Formula for the Riemann Zeta Function at Prime Numbers

Authors: Oussama Basta
Comments: 2 Pages. A better version

The Riemann zeta function is one of the most important functions in mathematics, but it is also one of the most difficult to compute. In this paper, we present a new closed formula for the Riemann zeta function at prime numbers. Our formula is based on a new function.
Category: Number Theory

The Geometric Collatz Correspondence

Authors: Darcy Thomas
Comments: 28 Pages. This is a final, and more complete, version of the paper. It is better categorized number theory.

The Collatz Conjecture, one of the most renowned unsolved problems in mathematics, presents adeceptive simplicity that has perplexed both experts and novices. Distinctive in nature, it leaves manyunsure of how to approach its analysis. My exploration into this enigma has unveiled two compellingconnections: firstly, a link between Collatz orbits and Pythagorean Triples; secondly, a tie to theproblem of tiling a 2D plane. This latter association suggests a potential relationship with PenroseTilings, which are notable for their non-repetitive plane tiling. This quality, reminiscent of theunpredictable yet non-repeating trajectories of Collatz sequences, provides a novel avenue to probethe conjecture’s complexities. To clarify these connections, I introduce a framework that interpretsthe Collatz Function as a process that maps each integer to a unique point on the complex plane.In a curious twist, my exploration into the 3D geometric interpretation of the Collatz Function has nudged open a small, yet intriguing door to a potential parallel in the world of physics. A subtle link appears to manifest between the properties of certain objects in this space and the atomic energy spectral series of hydrogen, a fundamental aspect in quantum mechanics. While this connection is in its early stages and the depth of its significance is yet to be fully unveiled, it subtly implies a simple merging where pure mathematics and applied physics might come together.The findings in this paper have led me to pursue development of a new type of number I call a Cam number, which stands for "complex and massive", indicating that it is a number with properties that on one hand act like a scalar, but on the other hand act as a complex number. Cam numbers can be thought of as having somewhat dual identities which reveal their properties and behavior under iterations of the Collatz Function. This paper serves as a motivator for a pursuit of a theory of Cam numbers.
Category: Number Theory