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. 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

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: 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

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