Number Theory

On the Study of a Fundamental Modular Equation for an Initial Theoretical Framework Concerning the Motivations of the Math Connections that Are Obtained Between Various Formulas of Ramanujan Maths and Different Parameters of Particle Physics/string Theory

Authors: Michele Nardelli, Antonio Nardelli
Comments: 37 Pages.

In this research thesis, we have analyzed a fundamental modular equation for an initial theoretical framework concerning the motivations of the mathematical connections that are obtained between various formulas of Ramanujan's mathematics and different parameters of Particle Physics and String Theory.
Category: Number Theory

Beal Conjecture Proved Finally

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

The author proves directly the original Beal conjecture (and not the equivalent conjecture) 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. The principles applied in the proof are based on the properties of the factored Beal equation. 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 greatest common power of the prime factors on the left side of the equation is the same as a power of the prime factor on the right side of the equation. High school students can learn and prove this conjecture as a bonus question on a final class exam.
Category: Number Theory

Naturally Numbers Are Three Plus One Dimensional Final Version 31.01.2020

Authors: Surajit Ghosh
Comments: 56 Pages.

Riemann hypothesis stands proved in three different ways.To prove Riemann hypothesis from the functional equation concept of Delta function is introduced similar to Gamma and Pi function. Other two proofs are derived using Eulers formula and elementary algebra. Analytically continuing gamma and zeta function to an extended domain, poles and zeros of zeta values are redefined. Hodge conjecture, BSD conjecture are also proved using zeta values. Other prime conjectures like Goldbach conjecture, Twin prime conjecture etc.. are also proved in the light of new understanding of primes. Numbers are proved to be multidimensional as worked out by Hamilton. Logarithm of negative and complex numbers are redefined using extended number system. Factorial of negative and complex numbers are redefined using values of Delta function.
Category: Number Theory

On the Number of Monic Admissible Polynomials in the Ring $\mathbb{z}[x]$

Authors: Theophilus Agama
Comments: 7 Pages.

In this paper we study admissible polynomials. We establish an estimate for the number of admissible polynomials of degree $n$ with coeffients $a_i$ satisfying $0\leq a_i\leq H$ for a fixed $H$, for $i=0,1,2, \ldots, n-1$. In particular, letting $\mathcal{N}(H)$ denotes the number of monic admissible polynomials of degree $n\geq 3$ with coefficients satisfying the inequality $0\leq a_i\leq H$, we show that \begin{align}\frac{H^{n-1}}{(n-1)!}+O(H^{n-2})\leq \mathcal{N}(H) \leq \frac{n^{n-1}H^{n-1}}{(n-1)!}+O(H^{n-2}).\nonumber \end{align} Also letting $\mathcal{A}(H)$ denotes the number of monic irreducible admissible polynomials, with coefficients satisfying the same condition , we show that \begin{align}\mathcal{A}(H)\geq \frac{H^{n-1}}{(n-1)!}+O\bigg( H^{n-4/3}(\log H)^{2/3}\bigg).\nonumber \end{align}
Category: Number Theory

The Prime Index Function

Authors: Theophilus Agama
Comments: 7 Pages.

In this paper we introduce the prime index function \begin{align}\iota(n)=(-1)^{\pi(n)},\nonumber \end{align} where $\pi(n)$ is the prime counting function. We study some elementary properties and theories associated with the partial sums of this function given by\begin{align}\xi(x):=\sum \limits_{n\leq x}\iota(n).\nonumber \end{align}
Category: Number Theory

Complete Sets

Authors: Theophilus Agama
Comments: 6 Pages.

In this paper we introduce the concept of completeness of sets. We study this property on the set of integers. We examine how this property is preserved as we carry out various operations compatible with sets. We also introduce the problem of counting the number of complete subsets of any given set. That is, given any interval of integers $\mathcal{H}:=[1,N]$ and letting $\mathcal{C}(N)$ denotes the complete set counting function, we establish the lower bound $\mathcal{C}(N)\gg N\log N$.
Category: Number Theory

On the Ramanujan’s Mathematics (Rogers-Ramanujan Continued Fractions, Hardy-Ramanujan Number and Sixth Order Mock Theta Functions) Applied to Various Parameters of Particle Physics: New Possible Mathematical Connections II.

Authors: Michele Nardelli, Antonio Nardelli
Comments: 57 Pages.

In this research thesis, we have analyzed and deepened further Ramanujan expressions (Rogers-Ramanujan continued fractions, Hardy-Ramanujan number and sixth order mock theta functions) applied to various parameters of Particle Physics. We have therefore described new possible mathematical connections.
Category: Number Theory

On the Ramanujan’s Mathematics (Rogers-Ramanujan Continued Fractions, Taxicab Numbers and Sixth Order Mock Theta Functions) Applied to Various Parameters of Particle Physics: New Possible Mathematical Connections.

Authors: Michele Nardelli, Antonio Nardelli
Comments: 84 Pages.

In this research thesis, we have analyzed and deepened further Ramanujan expressions (Rogers-Ramanujan continued fractions, Taxicab numbers and sixth order mock theta functions) applied to various parameters of Particle Physics. We have therefore described new possible mathematical connections.
Category: Number Theory

Break time

Authors: Yuji Masuda
Comments: 1 Page.

This is a interesting relationship.
Category: Number Theory

On the Ramanujan’s Mathematics (Rogers-Ramanujan Continued Fractions, Hardy-Ramanujan Number and Manuscript Book 1 Formulae) Applied to Various Sectors of String Theory: Further New Possible Mathematical Connections Xiii.

Authors: Michele Nardelli, Antonio Nardelli
Comments: 78 Pages.

In this research thesis, we have analyzed and deepened further Ramanujan expressions (Rogers-Ramanujan continued fractions, Hardy-Ramanujan number and Manuscript Book 1 formulae) applied to some sectors of String Theory. We have therefore described other new possible mathematical connections.
Category: Number Theory

Correspondence Between the Solutions of an Equation and the Divisors of Odd Numbers

Authors: Edoardo Gueglio
Comments: 6 Pages.

There is a correspondence between the positive solutions of a diophantine equation and the divisors of odd numbers
Category: Number Theory

Correspondence Between the Solutions of an Equation and the Divisors of Natural Numbers

Authors: Edoardo Gueglio
Comments: 5 Pages.

There is a correspondence between the positive solutions of a diophantine equation and the divisors of natural numbers.
Category: Number Theory

On The Infinitude of Twin Prime Pairs And The Generalized Goldbach's Conjecture

Authors: Siddhartha Shree kaushik
Comments: 4 Pages.

We prove that there are infinitely many Twin Prime pairs and further propose the generalized version of Goldbach's Conjecture
Category: Number Theory

The Prime Pairs Are Equidistributed Among the Coset Lattice Congruence Classes

Authors: T. Agama, M. Bortolamasim, A. Tapia
Comments: 15 Pages. submitted to Journal

In this paper we show that for some constant $c>0$ and for any $A>0$ there exist some $x(A)>0$ such that, If $q\leq (\log x)^{A}$ then we have \begin{align}\Psi_z(x;\mathcal{N}_q(a,b),q)&=\frac{\Theta (z)}{2\phi(q)}x+O\bigg(\frac{x}{e^{c\sqrt{\log x}}}\bigg)\nonumber \end{align}for $x\geq x(A)$ for some $\Theta(z)>0$. In particular for $q\leq (\log x)^{A}$ for any $A>0$\begin{align}\Psi_z(x;\mathcal{N}_q(a,b),q)\sim \frac{x\mathcal{D}(z)}{2\phi(q)}\nonumber \end{align}for some constant $\mathcal{D}(z)>0$ and where $\phi(q)=\# \{(a,b):(p_i,p_{i+z})\in \mathcal{N}_q(a,b)\}$.
Category: Number Theory

The Compression Method and Applications

Authors: Theophilus Agama
Comments: 13 Pages. submitted to Journal

In this paper we introduce and develop the method of compression of points in space. We introduce the notion of the mass, the rank, the entropy, the cover and the energy of compression. We leverage this method to prove some class of inequalities related to Diophantine equations. In particular, we show that for each $Ln-1$, there exist some $(x_1,x_2,\ldots,x_n)\in \mathbb{N}^n$ with $x_i\neq x_j$ for all $1\leq in-1$ there exist some $(x_1,x_2,\ldots,x_n)$ with $x_i\neq x_j$ for all $1\leq i<j\leq n$ and some $s\geq 2$ such that \begin{align}\sum \limits_{j=1}^{n}\frac{1}{x_j^s}\gg s\frac{n}{L^{s-1}}.\nonumber \end{align}
Category: Number Theory

The Riemann Hypothesis is false

Authors: Viola Maria Grazia
Comments: 1 Page.

In this page I talk about convergence of zeta function.
Category: Number Theory

On the Various Ramanujan Equations (Mock Theta Functions and Taxicab Numbers) Linked to Some Sectors of String Theory (Black Branes) and Black Hole Physics: Further New Possible Mathematical Connections Vii.

Authors: Michele Nardelli, Antonio Nardelli
Comments: 82 Pages.

In this research thesis, we have analyzed and deepened further Ramanujan expressions (mock theta functions and taxicab numbers) applied to some sectors of String Theory (black branes) and Black Hole Physics. We have therefore described other new possible mathematical connections.
Category: Number Theory

Remarks on Birch and Swinnerton-Dyer Conjecture

Authors: Algirdas Antano Maknickas
Comments: 1 Page.

These short remarks show derivation of Birch and Swinnerton-Dyer conjecture. As a consequence new one resulting constant free equality of Birch and Swinnerton-Dyer conjecture proposed
Category: Number Theory

On the Ramanujan Mathematics Applied to Some Sectors of String Theory and Particle Physics: Further New Possible Mathematical Connections V.

Authors: Michele Nardelli, Antonio Nardelli
Comments: 94 Pages.

In this research thesis, we have analyzed and deepened further Ramanujan expressions applied to some sectors of String Theory and Particle Physics. We have therefore described new possible mathematical connections.
Category: Number Theory

Twin Primes Conjecture

Authors: Nikos Mantzakouras
Comments: 7 Pages.

Twin prime conjecture, also known as Polignac’s conjecture, in number theory, assertion that there are infinitely many twin primes, or pairs of primes that differ by 2. The first statement of the twin prime conjecture was given in 1846 by French mathematician Alphonse de Polignac, who wrote that any even number can be expressed in infinite ways as the difference between two consecutive primes
Category: Number Theory

Proof of Golbach's Conjecture

Authors: Nikos Mantzakouras
Comments: 8 Pages.

Every even integer > 2 is the sum of two prime numbers & equivalent Each odd integer > 5 is the sum of three prime numbers USING THE SIEVE OF ERATOSTHENES.
Category: Number Theory

On Some Ramanujan Equations Concerning the Continued Fractions. Further Possible Mathematical Connections with Some Parameters of Particle Physics and Cosmology Vi.

Authors: Michele Nardelli, Antonio Nardelli
Comments: 115 Pages.

In this research thesis, we have analyzed and deepened some equations concerning the Ramanujan continued fractions. We have described further possible mathematical connections with some parameters of Particle Physics and Cosmology.
Category: Number Theory

Proof of the Riemann Hypothesis

Authors: Nikos Mantzakouras
Comments: 71 Pages.

The Riemann zeta function is one of the most Euler's important and fascinating functions in mathematics. By analyzing the material of Riemann's conjecture, we divide our analysis in the ζ(z) function and in the proof of the conjecture, which has very important consequences on the distribution of prime numbers. The proof of the Hypothesis of Riemann result from the simple logic, that when two properties are associated, (the resulting equations that based in two Functional equations of Riemann), if we zero these equations, ie ζ(z) = ζ (1-z)= 0 and simultaneously they to have the proved property 1-1 of the Riemann function ζ(z), the hypothesis will be true.Thus, there is not margin for to non apply that the Re (z)=1/2 {because simple apply ζ (z) = ζ (1-z)=0 and also ζ(z) as and ζ(1-z) are 1-1}.This, after it stands, will gives the direction of all the non-trivial roots to be all in on the critical line, with a value in the real axis equal 1/2. Part I. Proof Riemann Hypothesis from Theorems 1,2,3. The R-Hypothesis focuses on the point where we must prove that if s = Re(s) + Im(s)*I .. i) The functions ζ(s) and ζ(1-s) are 1-1 on the critical strip. ii) The common roots of the equations ζ(s)-ζ(1-s) = 0 they have Re(s)=Re(1-s)=1/2 within the interval (0,1) and determine unique position, which is called critical line. In Part II, we examine the distribution of prime numbers. The complexity of the Riemann Hypothesis is not limited to any method of proving that Re(z)=1/2 for the function ζ( ). It is a multidimensional issue that has many dimensions in physics as well as in other sciences. Anyone who says he has completely solved the hypothesis will be incomplete as to the possibilities it contains and the methods he can use to get some insight that comes close to some truths concerning it. With this in mind if we ask to prove that the hypothesis is correct, we must ask for the values of the critical lines that confirm the Riemann hypothesis. However, the Riemann hypothesis does not hold only in the interval (0,1) in the generalized form of the Zeta-functions ,but in the generalized form several critical lines arise within the intervals (-1,0) or (1,+∞) or (-∞,-1) specifically example, in the cases of the general equation ζ(q*z)=0,q>1/2 or q<-1/2 or (01/2 in R, G-z(z,q)=0 for 3 general cases out of 5, and for case Davenport Heilbronn case, which has zeros of a Dirichlet series periodic.Therefore, whether or not the Riemann Hypothesis holds depends on whether or not we accept the fact if we consider only ζ(z)=0. If in the definition of the Hypothesis the other forms or part of them also hold , i.e. ζ(q*z)=0 or ζ(q,z)=0 or ζ(z,q)=0 with q∊R in general or the Epstein Function or the Dirichlet series and others, then the Hypothesis does not hold in general, but only in special cases which are stated in detail.But we have to accept an indisputable fact that in order to have a visual perception of which ζ-functions have zeros with Re(z)=1/2 or have Re(z) different from 1/2 or other critical lines outside the interval (0,1), we have to solve transcendental equations with accuracy. This can only be achieved with the generalized theorem , and with the Periodic Radicals or Lagrange Method (https://arxiv.org/search/ query=mantzakouras +&searchtype =all&source=header) , work that has already been published. This makes it a unique reality and in this respect it has to be clarified , which ζ-functions we accept when we talk about proving the Riemann Hypothesis. "The question, dear colleagues, is not simply whether or not Riemann's Hypothesis is valid, but what ζ- functions do we accept that the Hypothesis as defined by Riemann includes? Or must we admit the general view of existing ζ-functions that Hypothesis includes. So if we take as a general assumption of the Hypothesis all the parallel ζ-functions then it is not 100% true. Of course it is also true that the Riemann hypothesis has multiple interconnections with other sciences and gives directions appropriate and grouped, depending on the function that we will accept to be valid in each specific case of application.The R-H applies only when we have specific cases mentioned in this document and perhaps and in others, but not in general, If we accept the generalized ζ- functions .This is considered as the only real answer to the conjecture of the Riemann Hypothesis that has stunned the mathematical community for more 200 years", Cases I,II of the characteristic generalized equation G-ζ(z,q)=0, {q=(a+b)/a, q>0} ,|q|<1 or |q|>1,q∊R as well as the cases of ζ(q*z)=0 with q>1/2 are subclass of the generalized one mentioned above, and some cases of the Dirihlet series reject the Riemann Hypothesis. So if the hypothesis is formulated as follows: "For any z-function, the zeros of the resulting equation coincide on the critical line with Re(z)=1/2, the formulation is complicated as a Latent and the Hypothesis does NOT hold". But if it is formulated as "The zeros only of ζ(z)=0 coincide on the critical line for Re(z)=1/2 is correct and the Hypothesis is Valid", as has been shown. Obviously this happened because the equations of other related functions i.e. other related equations arising from related functions with the function z were not completely or adequately solved or not at all and there was a large knowledge gap related to where their zeros are located with respect mainly to the interval (0,1).These equations are forms of Transcendental equations quite difficult and to solve them a strong method and a generalized Theorem for solving such equations is needed.
Category: Number Theory

On Some Ramanujan Equations Concerning the Continued Fractions. Further Possible Mathematical Connections with Some Parameters of Particle Physics and Cosmology V.

Authors: Michele Nardelli, Antonio Nardelli
Comments: 82 Pages.

In this research thesis, we have analyzed and deepened some equations concerning the Ramanujan continued fractions. Further possible mathematical connections with some parameters of Particle Physics and Cosmology.
Category: Number Theory

The Theory of the Collatz Process

Authors: Theophilus Agama
Comments: 7 Pages.

In this paper we introduce and develop the theory of the Collatz process. We leverage this theory to study the Collatz conjecture. This theory also has a subtle connection with the infamous problem of the distribution of Sophie germain primes. We also provide several formulation of the Collatz conjecture in this language.
Category: Number Theory

On Some Ramanujan’s Equations of Manuscript Book 2. Further New Possible Mathematical Connections with Some Parameters of Particle Physics and Cosmology. V

Authors: Michele Nardelli, Antonio Nardelli
Comments: 59 Pages.

In this research thesis, we continue to analyze and deepen further Ramanujan’s equations of Manuscript Book 2 and describe new possible mathematical connections with some parameters of Particle Physics and Cosmology.
Category: Number Theory

Assuming C

Authors: Abdelmajid Ben Hadj Salem
Comments: 7 Pages. Submitted to the journal "Publications of the Research Institute for Mathematical Sciences (PRIMS),Japan.

In this paper, assuming that c0, for a,b,c positive integers relatively prime with c=a+b, we have c Category: Number Theory

Naturally Numbers Are Three Plus One Dimensional Final

Authors: Surajit Ghosh
Comments: 40 Pages.

Riemann hypothesis stands proved in three different ways.To prove Riemann hypothesis from the functional equation concept of Delta function is introduced similar to Gamma and Pi function. Other two proofs are derived using Eulers formula and elementary algebra. Analytically continuing gamma and zeta function to an extended domain, poles and zeros of zeta values are redefined. Hodge conjecture, BSD conjecture are also proved using zeta values. Other prime conjectures like Goldbach conjecture, Twin prime conjecture etc.. are also proved in the light of new understanding of primes. Numbers are proved to be multidimensional as worked out by Hamilton. Logarithm of negative and complex numbers are redefined using extended number system. Factorial of negative and complex numbers are redefined using values of Delta function.
Category: Number Theory

On Various Ramanujan’s Equations of Manuscript Book 2. New Possible Mathematical Connections with Some Parameters of Particle Physics and Black Holes Physics. IV

Authors: Michele Nardelli, Antonio Nardelli
Comments: 59 Pages.

In this research thesis, we continue to analyze and deepen further Ramanujan’s equations of Manuscript Book 2 and described new possible mathematical connections with some parameters of Particle Physics and Black Holes Physics.
Category: Number Theory

Fermat_port_8_de_jan_2020

Authors: OttoAltorfer
Comments: 7 Pages.

A introdução de uma função de números inédita facilitou a solução do último Teorema de Fermat.
Category: Number Theory

On Some Formulas of Manuscript Book 1 of Srinivasa Ramanujan: New Possible Mathematical Connections with Various Parameters of Particle Physics and Cosmology Part II.

Authors: Michele Nardelli, Antonio Nardelli
Comments: 87 Pages. UPDATED VERSION

In this research thesis, we have analyzed further formulas of Manuscript Book 1 of Srinivasa Ramanujan and described new possible mathematical connections with various parameters of Particle Physics and Cosmology (Cosmological Constant, some parameters of Dark Energy)
Category: Number Theory

Definitive Tentative of a Proof of the \textit{abc} Conjecture

Authors: Abdelmajid Ben Hadj Salem
Comments: 11 Pages. Submitted to the journal Inventiones Matemathicae

In this paper, we consider the $abc$ conjecture. Firstly, we give anelementaryproof that $c<3rad^2(abc)$. Secondly, the proof of the $abc$ conjecture is given for $\epsilon \geq 1$, then for $\epsilon \in ]0,1[$. We choose the constant $K(\epsilon)$ as $K(\epsilon)=\frac{3}{e}.e^{ \left(\frac{1}{\epsilon^2} \right)}$ for $0<\epsilon <1$ and $K(\epsilon)=3$ for $\epsilon \geq 1$. Some numerical examples are presented.
Category: Number Theory

Goldbach Conjecture

Authors: Xuan Zhong Ni
Comments: 2 Pages.

In this article, we use method of a modified sieve of Eratosthenes to prove the Goldbach conjecture.
Category: Number Theory

On Some Formulas of Manuscript Book 1 of Srinivasa Ramanujan: New Possible Mathematical Connections with Various Parameters of Particle Physics and Cosmology.

Authors: Michele Nardelli, Antonio Nardelli
Comments: 119 Pages.

In this research thesis, we have analyzed further formulas of Manuscript Book 1 of Srinivasa Ramanujan and described new possible mathematical connections with various parameters of Particle Physics and Cosmology (Cosmological Constant, some parameters of Dark Energy)
Category: Number Theory

Twin Prime Conjecture

Authors: Xuan Zhong Ni
Comments: 2 Pages.

In this article, we use method of a modified sieve of Eratosthenes to prove the twin prime conjecture.
Category: Number Theory

On Some Ramanujan Formulas: New Possible Mathematical Connections with Various Parameters of Particle Physics and Cosmology Iv.

Authors: Michele Nardelli, Antonio Nardelli
Comments: 105 Pages.

In this research thesis, we have analyzed further Ramanujan formulas and described new possible mathematical connections with various parameters of Particle Physics and Cosmology
Category: Number Theory

On Some Ramanujan Formulas Concerning Highly Composite Numbers: New Possible Mathematical Connections with Various Parameters of Particle Physics, Dark Matter, Dark Energy and Cosmology III.

Authors: Michele Nardelli, Antonio Nardelli
Comments: 130 Pages.

In this research thesis, we have analyzed further Ramanujan formulas inherent Highly composite numbers and described new possible mathematical connections with various parameters of Particle Physics, Dark Matter, Dark Energy and Cosmology
Category: Number Theory