Number Theory

The Riemann Hypothesis Proof

Authors: Isaac Mor
Comments: everything is the same as it was i just added another observation from a different point of view on the last two pages

I am using eta function because it extends the zeta function from Re(s) > 1 to the larger domain Re(s) > 0. I am going to use eta function spiral and its behavior of convergent points on the complex plane to get two functions f(x) and g(x). Then I am going to show when those two functions are eqaul to zero the spiral is converging to zero as well. I will then show that non trival zeros appears only when h(x) eqaul to zero. And also when function q(x) is eqauls to zeta(2a) then eta(a+ix) is a non trival zero. Then I am going to do convergence tests on the critical strip to show that there are no zeros on the strip other then the criticl line.
Category: Number Theory

A New Criterion for Riemann Hypothesis or a True Proof?

Authors: Dmitri Martila
Comments: 6 Pages. Rejected by many peer-review journals, submitted to Mathematika

There are tens of self-proclaimed proofs for Riemann Hypothesis and only 2 or 4 disproofs of it in arXiv. I am adding to the Status Quo my very short and clear evidence which uses the peer-reviewed achievement of Dr.Sole and Dr.Zhu, which they published just 4 years ago in a serious mathematical journal INTEGERS.
Category: Number Theory

Maximality Methods in Commutative Set Theory

Authors: Long Ngo
Comments: 10 Pages. Great for you!

Let f = w be arbitrary. Every student is aware that Kolmogorov’s criterion applies. We show that S ≤ |ρR,C |. J. Sasaki [34] improved upon the results of R. Thomas by deriving subsets. The goal of the present paper is to compute irreducible, generic random variables
Category: Number Theory

Improved Estimate for the Prime Counting Function

Authors: Theophilus Agama
Comments: 7 Pages.

Using some simple combinatorial arguments, we establish some new estimates for the prime counting function and its allied functions. In particular we show that \begin{align}\pi(x)=\Theta(x)+O\bigg(\frac{1}{\log x}\bigg), \nonumber \end{align}where \begin{align}\Theta(x)=\frac{\theta(x)}{\log x}+\frac{x}{2\log x}-\frac{1}{4}-\frac{\log 2}{\log x}\sum \limits_{\substack{n\leq x\\\Omega(n)=k\\k\geq 2\\2\not| n}} \frac{\log (\frac{x}{n})}{\log 2}.\nonumber \end{align}This is an improvement to the estimate \begin{align}\pi(x)=\frac{\theta(x)}{\log x}+O\bigg(\frac{x}{\log^2 x}\bigg)\nonumber \end{align}found in the literature.
Category: Number Theory

One page Proof of Riemann Hypothesis

Authors: Dmitri Martila
Comments: 3 Pages.

There are tenths of proofs for Riemann Hypothesis and 3 or 5 disproofs of it in arXiv. I am adding to the Status Quo my proof, which uses the achievement of Dr. Zhu.
Category: Number Theory

Riemann Hypothesis Using Just Integration

Authors: Shekhar Suman
Comments: 8 Pages.

Easy Proof. Must check
Category: Number Theory

Proof of the Beal Conjecture and Fermat Catalan Conjecture (Summary)

Authors: Quang Nguyen Van
Comments: 5 Pages.

This article inclucles the theorems anh the lemmas, using them to prove the Beal conjecture anh the Fermat- Catalan conjecture, through which we learn more about rational and irrational numbers. I think the method of proof will be useful for solving other Math- problems and they need more research.
Category: Number Theory

On the Pointwise Periodicity of Multiplicative and Additive Functions

Authors: Theophilus Agama
Comments: 8 Pages.

We study the problem of estimating the number of points of coincidences of an idealized gap on the set of integers under a given multiplicative function $g:\mathbb{N}\longrightarrow \mathbb{C}$ respectively additive function $f:\mathbb{N}\longrightarrow \mathbb{C}$. We obtain various lower bounds depending on the length of the period, by varying the worst growth rates of the ratios of their consecutive values
Category: Number Theory

Difference of Two Sums of Two Squares

Authors: Suaib Lateef
Comments: 1 page

This formula extends the difference of two squares theorem
Category: Number Theory

An Unintentional Repetition of the Ramanujan Formula for $\pi$, and Some Independent Mathematical Mnemonic Tools for Calculating with Exponentiation and with Dates

Authors: Janko Kokosar
Comments: 11 Pages.

The paper is consisted of contentually unrelated sections, where each section could be one short paper. Section 1 deals with the process of an unintentional repetition of one of the Ramanujan formulas, and the author did not know it before. The process of the unintentional repetition of the Ramanujan formula is interesting for estimating, for instance, the physical background for guessing of dimensionless physical constants. Section 2 shows an approximation which helps at memorizing the square roots of integers up to 10. Section 3 shows the specialities of the squares of integers at the last digits. Section 4 shows how the last two digits are repeated for 2 to the sequential integer powers, and how to find out this. Section 5 contains some mathematical peculiarities at calculating dates. All sections contain mnemonic methods to help us memorize and calculate. These sections belong to pedagogical and recreational mathematics, maybe even something more is here.
Category: Number Theory

The abc Conjecture is False: The End of The Mystery

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

In this note, I give the proof that the abc conjecture is false because, in the case c>rad(abc), for 0<\epsilon<1, presenting a counterexample that implies a contradiction for c very large.
Category: Number Theory

A Probabilistic Approach to some Problems of Number Theory

Authors: Gregory Sobko
Comments: 109 Pages.

Some classical questions and problems of Number Theory, like the Goldbach conjecture, distributions of twin- and d–primes and primes among arithmetic sequences, are addressed here from an entirely probabilistic point of view. We discuss the concept of ‘independence’ relevant to number-theoretic problems and interpret the basic concepts of divisibility of natural number in terms of probability spaces and appropriate probability distributions on classes of congruence. We analyze and demonstrate the importance of Zeta probability distribution and prove, in particular, theorems stating the equivalence of probabilistic independence of divisibility by co-prime factors, and the fact that random variables with the property of independence of co-prime factors must have Zeta probability distribution. Multiplicative and additive models with recurrent equations for generating sequences of prime numbers allow to interpret such sequences as realizations of random walks on set of natural numbers and on multiplicative semigroups generated by set of prime numbers , representing paths of stochastic dynamical systems. We discuss some limit theorems related to distribution of primes and their residuals. More specifically, we provide a continuous-time description of the distribution of counting function of primes in terms of diffusion approximation of non-Markov random walks.
Category: Number Theory

Les Nombres Ultimes et le Ratio 3/2

Authors: Boulay Jean-Yves
Comments: Pages.

Selon une nouvelle définition mathématique, les nombres entiers naturels se divisent en deux ensembles dont l’un est la fusion de la suite des nombres premiers et des nombres zéro et un. Trois autres définitions, déduites de cette première, subdivisent l’ensemble des nombres entiers naturels en quatre classes de nombres aux propriétés arithmétiques propres et uniques. La distribution géométrique de ces différents types d’entiers naturels, dans de diverses matrices fermées, s’organise en ratios exacts de valeur 3/2 ou 1/1.
Category: Number Theory

Riemann Hypothesis Disproof Using Simple Inequalty

Authors: Shekhar Suman
Comments: 4 Pages.

The Riemann Zeta function is defined as \\\\ \large \zeta(s)= $$\sum_{n=1}^{\infty} 1/n^{s}$$ ,\ \Re(s)$$>$$1 \\\\ The \ Zeta \ function \ is \ holomorphic \ in \ the \ complex \ plane \ except \ for \ a \ pole \ at \\\\ s=1. \ The \ trivial \ zeros \ of \ \zeta(s)\ are \ -2,-4,-6,... . \ Its \ non \ trivial \ zeros \ lie \ in \\\\ the \ critical \ strip \ 0< Re(s)< 1 .\\\\ The \ Riemann \ Hypothesis \ states \ that \ all \ the \ non \ trivial \ zeros \ lie \ on \ the \ critical \ line \\\\ Re(s)=1/2.\\\\
Category: Number Theory

Riemann Hypothesis - A Disproof

Authors: Shekhar Suman
Comments: 4 Pages.

The Riemann Zeta function is defined as \\\\ \large \zeta(s)= $$\sum_{n=1}^{\infty} 1/n^{s}$$ ,\ Re(s)$$>$$1 \\\\ The \ Zeta \ function \ is \ holomorphic \ in \ the \ complex \ plane \ except \ for \ a \ pole \ at \\\\ s=1. \ The \ trivial \ zeros \ of \ \zeta(s)\ are \ -2,-4,-6,... . \ Its \ non \ trivial \ zeros \ lie \ in \\\\ the \ critical \ strip \ 0< Re(s)< 1 .\\\\ The \ Riemann \ Hypothesis \ states \ that \ all \ the \ non \ trivial \ zeros \ lie \ on \ the \ critical \ line \\\\ Re(s)=1/2.\\\\
Category: Number Theory

Ramification Theory

Authors: Theophilus Agama
Comments: 8 Pages.

In this paper we introduce and develop the concept of ramification in a given modulus. We study some properties in relation to this concept and it's connection to some important problems in mathematics, particularly Goldbach's conjecture.
Category: Number Theory

Prove Collatz Conjecture via Operations for Integer Expressions plus Few Integers

Authors: Zhang Tianshu
Comments: 15 Pages.

First, let us set forth certain of basic concepts related to proving Collatz conjecture. After that, list the mathematical induction that proves the conjecture, and prepare two theorems plus one lemma, which are used to judge relevant operational results. Next, classify integers successively and prove a class by the theorem 1, after each classification. Until the last two classes are proved bidirectionally, which are to start with several proven kinds to expand successively the scope of proven kinds up to all kinds are proven and star with each unproven kind to a proven kind via operations.
Category: Number Theory

Links Between String Theory and the Riemann’s Zeta Function

Authors: Rosario Turco, Maria Colonnese, Michele Nardelli
Comments: 23 Pages.

In this paper are descibed several Mathematical connections between some sectors of String Theory and the Riemann zeta function
Category: Number Theory

On the Riemann Hypothesis. Formulas Explained ψ(x) as Equivalent RH. Mathematical Connections with “Aurea” Section and Some Sectors of String Theory

Authors: Rosario Turco, Maria Colonnese, Michele Nardelli
Comments: 25 Pages.

In this work the authors will examine the themes of RH, equivalent RH and GRH already presented in [25]. The authors will explain some formulas and will show other special functions that are usually introduced with the PNT and useful to investigate other ways.
Category: Number Theory

Relations Between the Gauss-Eisenstein Prime Numbers and Their Correlation with Sophie Germain Primes

Authors: Pier Francesco Roggero, Michele Nardelli, Francesco Di Noto
Comments: 35 Pages.

In this paper we examine the relations between the Gauss prime numbers and the Eisenstein prime numbers and their correlation with Sophie Germain primes. Furthermore, we have described also various mathematical connections with some equations concerning the string theory.
Category: Number Theory

Proof of Fermat Last Theorem Based on Successive Presentations of Pairs of Odd Numbers

Authors: Yuri K. Shestopaloff
Comments: 26 Pages.

A simpler proof of Fermat Last Theorem (FLT), based mostly on new concepts, is suggested. FLT was formulated by Fermat in 1637, and proved by A. Wiles in 1995. The initial equation x^n + y^n = z^n is considered not in natural, but in integer numbers. It is subdivided into four equations based on parity of terms and their powers. Cases 1, 3 and 4 can be converted to case 2, which is studied using presentations of pairs of odd numbers with a successively increasing presentation factor of 2^r. The proposed methods and ideas can be used for studying other problems in number theory.
Category: Number Theory

On the Mathematical Connections Between Some Formulas Concerning the Shapiro-Virasoro Model in String Theory, Ramanujan Equations, ϕ, ζ(2) and Various Parameters of Particle Physics. III

Authors: Michele Nardelli, Antonio Nardelli
Comments: 70 Pages.

In this paper we describe and analyze the mathematical connections between some formulas concerning the Shapiro-Virasoro model in String Theory, Ramanujan equations, ϕ, ζ(2) and various parameters of Particle Physics.
Category: Number Theory

On the Existence of Prime Numbers in Constant Gaps

Authors: Juan Moreno Borrallo
Comments: 11 Pages.

This paper studies the existence of prime numbers on constant gaps, establishing a lower bound for the number of consecutive constant gaps for which the existence of some prime number contained in them is necessary.
Category: Number Theory

On Some Equations Concerning the Casimir Effect Between World-Branes in Heterotic Mtheory and the Casimir Effect in Spaces with Nontrivial Topology. Mathematical Connections with Some Sectors of Number Theory.

Authors: Michele Nardelli, Francesco Di Noto
Comments: 59 Pages.

The present paper is a review, a thesis of some very important contributes of P. Horava, M. Fabinger, M. Bordag, U. Mohideen, V.M. Mostepanenko, Trang T. Nguyen et al. regarding various applications concerning the Casimir Effect. In conclusion, we have described some mathematical connections between the equation of the energy negative of the Casimir effect, the Casimir operators and some sectors of the Number Theory, i.e. the triangular numbers, the Fibonacci’s numbers, Phi, Pigreco and the partition of numbers.
Category: Number Theory

On the Mathematical Connections Between Some Formulas Concerning Modular Forms, Elliptic Curves, Ramanujan Equations, ϕ, ζ(2) and Various Topics and Parameters of String Theory and Particle Physics. II

Authors: Michele Nardelli, Antonio Nardelli
Comments: 65 Pages.

In this paper we describe and analyze the mathematical connections between some formulas concerning Modular forms, Ramanujan equations, ϕ, ζ(2) and various topics and parameters of String Theory and Particle Physics.
Category: Number Theory

The Operative Set Theory with Application to Goldbach Conjecture

Authors: Anze Zhou
Comments: 12 Pages.

We defined the arithmetic operations on the integer sets. These new set operators bring the new properties and theorems to the integer sets. Therefore, we called the integer sets with these new arithmetic operators as the "operative set" and proved its properties and theorems, which are the building blocks of the "operative set theory". Then we used the operative set theory to prove the Goldbach Conjecture.
Category: Number Theory

On Some Equations Concerning the Cusp Anomalous Dimension from a Tba Equation and Generalized Quark-Antiquark Potential at Weak and Strong Coupling and the Complete Four-Loop 4-Point Amplitude of N=4 Sym Theory. Mathematical Connections with Number Theory

Authors: Michele Nardelli, Francesco Di Noto, Roberto Servi
Comments: 77 Pages.

In the present paper in the Section 1, we have described some equations concerning the cusp anomalous dimension in the planar limit of N = 4 super Yang-Mills from a Thermodynamic Bethe Antsaz (TBA) system, the Luscher correction at strong coupling and the strong coupling expansion of the function F. In the Section 2, we have described some equations concerning a two-parameter family of Wilson loop operators in N = 4 supersymmetric Yang-Mills theory which interpolates smoothly between the 1/2 BPS line or circle, principally some equations concerning the one-loop determinants. In the Section 3, we have described some results and equations of the mathematician Ramanujan concerning some definite integrals and an infinite product and some equations concerning the development of derivatives of order n (n positive integer) of various trigonometric functions and divergent series. Thence, we have described some mathematical connections between some equations concerning this Section and the Sections 1 and 2. In the Section 4, we have described some equations concerning the relationship between Yang-Mills theory and gravity and, consequently, the complete four-loop four-point amplitude of N = 4 super-Yang-Mills theory including the nonplanar contributions regarding the gauge theory and the gravity amplitudes. In conclusion, in the Appendix A and B, we have described a new possible method of factorization of a number and various mathematical connections with some sectors of Number Theory (Fibonacci's numbers, Lie's numbers, triangular numbers, Phi, Pigreco, etc...).
Category: Number Theory

On the Mathematical Connections Between Some Formulas Concerning Ramanujan Modular Forms, ϕ, ζ(2) and Various Topics and Parameters of String Theory and Particle Physics.

Authors: Michele Nardelli, Antonio Nardelli
Comments: 46 Pages.

In this paper we describe and analyze the mathematical connections between some formulas concerning Ramanujan Modular Forms, ϕ, ζ(2) and various topics and parameters of String Theory and Particle Physics.
Category: Number Theory

Further Mathematical Connections Between Some Number Theory Formulas, ϕ, ζ(2) and Various Topics and Parameters of D-Branes and Particle Physics. Vii

Authors: Michele Nardelli, Antonio Nardelli
Comments: 52 Pages.

In this paper we describe and analyze some Number Theory expressions. Furthermore, we have obtained several mathematical connections with ϕ, ζ(2) and various topics and parameters of D-branes and Particle Physics.
Category: Number Theory

The Fermat Classes and the Proof of Beal Conjecture

Authors: Mohamed sghiar
Comments: 9 pages, accepted version

If after 374 years the famous theorem of Fermat-Wiles was demonstrated in 150 pages by A. Wiles , The purpose of this article is to give a proofs both for the Fermat last theorem and the Beal conjecture by using the Fermat class concept.
Category: Number Theory

Une Preuve Relativiste de la Conjecture de Goldbach et Des Nombres Premiers Jumeaux

Authors: Mohamed sghiar
Comments: 3 Pages. Version published on May 11, 2020.

Relativistic techniques [3,4] have made it possible to give the conjectures of Goldbach and De Polignac new and hitherto unknown versions. Relativistic techniques have also made it possible to demonstrate them in their new versions. This shows the importance of the theory of mathematical relativity in the theory of numbers and that the mathematical community must finally admit.
Category: Number Theory

Some Unknown Methods of Converting Numbers to Gray Code

Authors: George Plousos
Comments: 2 Pages.

I discovered the methods I present here as I was working on a different mathematical object. I checked the relevant sequence in OEIS and found that it is similar to the A003188 sequence. That's when I learned about the existence of the Gray code. Searching the Internet, I found no reference to these methods. I do not have strict proof that guarantees the accuracy of these methods. The burden of control and proof falls on you. Once all of this is proven, I think they will make the lives of the Gray code developers easier.
Category: Number Theory

On Some Possible Mathematical Connections Between Various Equations Concerning the Mock Modularity Closely Related to N = 4 Super Yang-Mills, ϕ, ζ(2) and Some Parameters of Particle Physics.

Authors: Michele Nardelli, Antonio Nardelli
Comments: 83 Pages.

In this paper we have described some possible mathematical connections between various equations concerning the Mock Modularity closely related to N = 4 super Yang-Mills, ϕ, ζ(2) and some parameters of Particle Physics.
Category: Number Theory

The Special Functions and the Proof of the Riemann’s Hypothesis

Authors: Mohamed sghiar
Comments: 2 Pages. improved version

By studying the $ \circledS $ function whose integer zeros are the prime numbers, and being inspired by the article [2], I give a new proof of the Riemann hypothesis.
Category: Number Theory

Periodic Tables of Prime Numbers

Authors: George Plousos
Comments: 4 Pages.

When we divide any integer n by a prime number p on a base B of the arithmetic system, we get a decimal extension that has a constant period length, which is equal to λ. For many values ​​of n we get the same period of p but from another digit. p has exactly r=(p-1)/λ different periods. So we need to test many values ​​of n to find all the periods of p. However, we can easily locate and manage all periods of p if we know the values ​​of two numbers, b and λ, because then we can construct a table that allows easy access to any digit of any period of p. To do this we use the relationship (x, y) = b^(rx + y) mod p. This relationship gives the position (x, y) of the table an integer value n such that the n/p fraction forms the y-th period of p from the x-th digit.
Category: Number Theory

Riemann Hypothesis Simple Proof

Authors: Shekhar Suman
Comments: 8 Pages.

Simple proof
Category: Number Theory

Proof of Goldbach Conjecture

Authors: Anze Zhou
Comments: 6 Pages.

We developed the addition operation on the integer sets and used the theorems of the set addition operations to prove the Goldbach Conjecture.
Category: Number Theory

On Some Possible Mathematical Connections Between Various Equations Concerning the Riemann Zeta Function, the Riemann’s Hypothesis, the Einstein’s Type Universes, ϕ, ζ(2) and Some Parameters of Particle Physics.

Authors: Michele Nardelli, Antonio Nardelli
Comments: 79 Pages.

In this paper we have described some possible mathematical connections between various equations concerning the Riemann zeta function, the Riemann’s Hypothesis, the Einstein’s type Universes, ϕ, ζ(2) and some parameters of Particle Physics.
Category: Number Theory

Beal’s Conjecture is Tenable ( Revised Version)

Authors: Zhang Tianshu
Comments: 17 Pages.

In this article, first classify A, B and C according to their odevity, and thereby get rid of two kinds of AX+BY≠CZ. Then, exemplify AX+BY=CZ under the given requirements. After that, divide AX+BY≠CZ into 4 inequalities under the known requirements, and that apply the mathematical induction, the odd-even relations on the symmetry or the method that takes apart integers to prove each inequality. Finally, reach the conclusion that Beal’s conjecture is tenable via the comparison between AX+BY=CZ and AX+BY≠CZ under the given requirements.
Category: Number Theory

A Note on Primality of Ap^k + 1 Numbers

Authors: Ariko Stephen Philemon
Comments: 10 Pages.

In 1876, Edouard Lucas showed that if an integer b exists such that b^(n-1)≡1 (mod n) and b^((n-1)/q)≢1 (mod n) for all prime divisors q of n-1, then n is prime, a result known as Lucas’s converse of Fermat’s little theorem. This result was considerably improved by Henry Pocklington in 1914 when he showed that it’s not necessary to know all the prime factors of n-1 to determine the primality of n. In this paper we optimize Pocklington’s primality test for integers of the form ap^k+1 where p is prime, a<4(p+1), k≥1. Precisely, this paper shows that if an integer b exists such that b^(n-1)≡1 (mod n) and n∤b^((n-1)/p)-1, then n is prime as opposed to Pocklington’s primality test that imposes the more stringent hypothesis that n and b^((n-1)/p)-1 be relatively prime. We also present a conjecture whose proof will significantly reduce the computations required to determine the primality of these integers.
Category: Number Theory

New Mathematical Connections Between Riemann-Ricci-Einstein Models and String Theory, Cosmological Constant, Dark Matter and Dark Energy

Authors: Michele Nardelli
Comments: Summary and models in Italian (number of pages in Italian 3)

In this paper we describe the possible new mathematical connections between Riemann-Ricci-Einstein models and String Theory, Cosmological Constant, Dark Matter and Dark Energy.
Category: Number Theory

On Some Possible Mathematical Connections Between Various Equations Concerning the Dirichlet Boundary Conditions of the D-Branes and Several Equations Inherent the Zeros of Certain Dirichlet Series

Authors: Michele Nardelli, Antonio Nardelli
Comments: 66 Pages.

In this paper we have described some possible mathematical connections between various equations concerning the Dirichlet boundary conditions of the D-branes and several equations inherent the zeros of certain Dirichlet series
Category: Number Theory

Poincaré and Geometrization Conjectures: Mathematical Connections Between String Theory, Ricci Flow and Number Theory

Authors: Michele Nardelli
Comments: 32 Pages. Summary in Italian

In this thesis are highlighted the mathematical connections obtained between the Poincarè conjecture, the Thurston geometrization conjecture, some topics of String Theory and various sectors of the Number Theory.
Category: Number Theory

Assuming C

Authors: Abdelmajid Ben Hadj Salem
Comments: 6 Pages. Submitted to the journal Bulletin of The London Mathematical Society. Comments welcome.

In this paper, assuming the conjecture $c<rad^2(abc)$ true, I give, using elementary calculus, the proof of the $abc$ conjecture proposing the constant $K(\ep)$. Some numerical examples are given.
Category: Number Theory