Number Theory

Riemann Hypothesis Disproof Using a Contradiction

Authors: Shekhar Suman
Comments: 5 Pages.

Very easy proof
Category: Number Theory

A^x + B^y = C^z Part 2: Another Version of my Theorem and Infinite Ascent

Authors: Quang Nguyen Van
Comments: 3 Pages.

We give another version of my theorem that was submitted in the previous article, which is the theorem leading to the proof of the Beal's conjecture and the Fermat - Catalan conjecture. We also give a view to prove whether the equation has infinitely many solutions in integer or not related to parametric solution and infinite ascent.
Category: Number Theory

Further Mathematical Connections Between Some Ramanujan Formulas, ϕ, ζ(2) and Various Topics and Parameters of LQG, Open Strings and Particle Physics. vi

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

In this paper we continue to describe and analyze some Ramanujan expressions. Furthermore, we have obtained several mathematical connections with ϕ, ζ(2) and various topics and parameters of LQG, Open Strings and Particle Physics.
Category: Number Theory

Mathematical Connections Between Some Ramanujan Formulas, ϕ, ζ(2) and Various Topics and Parameters of Open Strings and Particle Physics. V

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

In this paper we have described and analyzed some Ramanujan expressions. We have obtained several mathematical connections with ϕ, ζ(2) and various topics and parameters of Open Strings and Particle Physics.
Category: Number Theory

Riemann Hypothesis Disproof

Authors: Shekhar Suman
Comments: 5 Pages.

Two equivalent conditions are used to get a contradiction
Category: Number Theory

Collatz Conjecture is Proved in this Article

Authors: Zhang Tianshu
Comments: 18 Pages.

First, it is necessary to expound certain of basic concepts relating to prove this conjecture, after that, the author applies the mathematical induction to complete this proof. Also, before the beginning of this proof, prepares several judging criteria, so as to use them in following classified proofs after operations according to the operational rule. After accomplish proofs of certain of stepwise classification for integers, emphatically proves 15+12c and 19+12c, so the entire proof for the conjecture was finished.
Category: Number Theory

A Proof of the Erdös-Straus Conjecture

Authors: Zhang Tianshu
Comments: 14 Pages.

In this article, we classify positive integers except for 1 into certain classes gradually, and formulate each of majority’s classes after each classification to a sum of three unit fractions, until all classes.
Category: Number Theory

Prove Grimm’s Conjecture by Stepwise Forming Consecutive Composite Number’s Points

Authors: Zhang Tianshu
Comments: 14 Pages.

Regard positive integers which have a common prime factor as a kind, then the positive half line of the number axis consists of infinitely many recurring line segments which have same permutations of c kinds of integer’s points, where c≥1. Thus, the author first makes a positive half line of the number axis, yet the positive half line is marked only with integer’s points. Then, change gradually symbols of each kind of composite number’s points thereon according to the order of common prime factors from small to large, so as to form consecutive composite number’s points within limits that proven Bertrand's postulate restricts to prove Grimm’s conjecture.
Category: Number Theory

Pi and Golden Number: not Random Occurrences of the Ten Digits.

Authors: BOULAY Jean-Yves
Comments: 30 Pages.

This paper demonstrates that the order of first appearance of the ten digits of the decimal system in the two most fundamental mathematical constants such as the number Pi and the Golden Number is not random but part of a arithmetical logic. This arithmetical logic is identical to Pi to its inverse and to the Golden Number. The same arithmetical phenomenon also operates in many other constants whose square roots of numbers 2, 3 and 5, the first three prime numbers.
Category: Number Theory

The Ultimate Numbers and the 3/2 Ratio

Authors: BOULAY Jean-Yves
Comments: 38 Pages. You will understand, from reading the article, that it does not require any reference.

According to a new mathematical definition, whole numbers are divided into two sets, one of which is the merger of the sequence of prime numbers and numbers zero and one. Three other definitions, deduced from this first, subdivide the set of whole numbers into four classes of numbers with own and unique arithmetic properties. The geometric distribution of these different types of whole numbers, in various closed matrices, is organized into exact value ratios to 3/2 or 1/1.
Category: Number Theory

Algoritmo de Números Primos Y Números Primos Gemelos

Authors: Zeolla Gabriel Martin
Comments: 22 Pages. Español

Este documento desarrolla una fórmula para obtener números primos y números primos gemelos por debajo de un número cualquiera dado. La formula se desprende del uso de 4 cribas triangulares.
Category: Number Theory

Mysteries of the Prime Numbers

Authors: Andrzej Lukaszewicz
Comments: 8 Pages.

The elaboration contains a method of determining prime numbers and confirms the Riemann hypothesis
Category: Number Theory

On Some Ramanujan Formulas: Mathematical Connections with ϕ, ζ(2) and Several Parameters of String Theory and Particle Physics V.

Authors: Michele Nardelli, Antonio Nardelli
Comments: 63 Pages.

In this paper we have described and analyzed some Ramanujan expressions. We have obtained several mathematical connections with ϕ, ζ(2) and various parameters of String Theory and Particle Physics.
Category: Number Theory

How Truly Random is the Brownian Motion Applied Especially in Quantitative Finance ? the Hidden Connection with Riemann Zetafunction & Riemann Hypothesis! is Randomness Truly Random or the Existence of Duality ?

Authors: Pankaj Mani
Comments: 10 Pages.

Random Numbers and random processes to model the dynamics of financial assets typically involve Brownian motion/Wiener process. In this article we’ll explore how random Brownian motion truly is and is it really compatible and capable to model randomness in financial variables ? The logic behind all these goes to the patterns hidden inside the Prime Numbers distribution and dynamics in pure mathematics from where they originate.Thus it's linked to Riemann Hypothesis ! Hence,it will possibly lead forward to revise foundationally how far Brownian motion is really capable to model random processes especially in quantitative finance or otherwise.
Category: Number Theory

How Truly Random is the Brownian Motion Applied Especially in Quantitative Finance? the Hidden Connection with Riemann Zetafunction & Riemann Hypothesis! is Randomness Truly Random or the Existence of Duality ?

Authors: Pankaj Mani
Comments: 10 Pages.

Random Numbers and random processes used to model the dynamics of financial assets which typically involve Brownian motion,Wiener process. In this article I’ll explore how random Brownian motion truly is and is it really compatible and capable to model randomness in financial variables ? The logic behind all these goes to the existence of Prime Numbers and their dynamics in pure mathematics from where they originate. This could possibly make the way forward to the powerful revision of Brownian motion application as the process of randomness especially in quantitative finance.
Category: Number Theory

An Equivalent Contour Integral to Riemann Hypothesis

Authors: Shekhar suman
Comments: 4 Pages.

Integration
Category: Number Theory

On Some Ramanujan Equations: Mathematical Connections with Various Topics Concerning Number Theory, ϕ, ζ(2) and Several Parameters of Particle Physics. IV

Authors: Michele Nardelli, Antonio Nardelli
Comments: 81 Pages.

In this paper we have described and analyzed some Ramanujan equations. We have obtained several mathematical connections between some topics concerning Number Theory, ϕ, ζ(2) and various parameters of Particle Physics.
Category: Number Theory

On Some Ramanujan Equations: Mathematical Connections with Various Topics Concerning Prime Numbers Theory, ϕ, ζ(2) and Several Parameters of Particle Physics. III

Authors: Michele Nardelli, Antonio Nardelli
Comments: 64 Pages.

In this paper we have described and analyzed some Ramanujan equations. We have obtained several mathematical connections between some topics concerning Prime Numbers Theory, ϕ, ζ(2) and various parameters of Particle Physics.
Category: Number Theory

Analyzing Some Ramanujan Equations: Mathematical Connections with Prime Numbers Theory, ϕ, ζ(2) and Various Parameters of Particle Physics. II

Authors: Michele Nardelli, Antonio Nardelli
Comments: 85 Pages.

In this paper we have described and analyzed some Ramanujan equations. We have obtained several mathematical connections between Prime Numbers Theory, ϕ, ζ(2) and various parameters of Particle Physics.
Category: Number Theory

Analyzing Various Ramanujan Equations: Mathematical Connections with Some Prime Numbers Linked to the Supersingular Elliptic Curves, ϕ, ζ(2) and to the Mass of Candidate Glueball F0(1710) Scalar Meson.

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

In this paper we have described and analyzed various Ramanujan equations. We have obtained several mathematical connections between some Prime Numbers linked to the Supersingular Elliptic Curves, ϕ, ζ(2) and to the mass of candidate glueball f0(1710) scalar meson.
Category: Number Theory

On Some Ramanujan Equations: Mathematical Connections with Prime Number Theorem, ϕ, ζ(2) and Various Parameters of Particle Physics.

Authors: Michele Nardelli, Antonio Nardelli
Comments: 90 Pages.

In this paper we have described and analyzed some Ramanujan equations. We have obtained several mathematical connections between Prime Number Theorem, ϕ, ζ(2) and various parameters of Particle Physics.
Category: Number Theory

On Some Ramanujan Integrals Concerning Riemann’s Functions ξ(s) and Ξ(t): Mathematical Connections with ϕ, ζ(2) and Various Parameters of Particle Physics. II

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

In this paper we have described and analyzed some Ramanujan integrals concerning Riemann’s functions ξ(s) and Ξ(t). Furthermore, we have obtained several mathematical connections between ϕ, ζ(2) and various parameters of Particle Physics.
Category: Number Theory

Riemann Hypothesis Using Integration Equivalent Condition

Authors: Shekhar Suman
Comments: 5 Pages.

Riemann zeta
Category: Number Theory

On Some Ramanujan Definite Integrals: Mathematical Connections with ϕ, ζ(2) and Various Parameters of Particle Physics

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

In this paper we have described and analyzed some Ramanujan definite integrals. Furthermore, we have obtained several mathematical connections between ϕ, ζ(2) and various parameters of Particle Physics.
Category: Number Theory

On the Mathematical Connections Between ϕ, ζ(2), Some Ramanujan Equations and Various Parameters of String Theory Mathematics and Particle Physics.

Authors: Michele Nardelli, Antonio Nardelli
Comments: 60 Pages.

In this paper we have described and analyzed some Ramanujan equations. Furthermore, we have obtained several mathematical connections between ϕ, ζ(2), and various parameters of String Theory Mathematics and Particle Physics.
Category: Number Theory

On Some Ramanujan Equations: Mathematical Connections Between ϕ, ζ(2), Mock Theta Functions and Various Parameters of Particle Physics.

Authors: Michele Nardelli, Antonio Nardelli
Comments: 55 Pages.

In this paper we have described and analyzed some Ramanujan equations. Furthermore, we have obtained several mathematical connections between ϕ, ζ(2), Mock theta functions and various parameters of Particle Physics.
Category: Number Theory

The Shortest Proof that Eulers-Mascheroni’s Constant is Transcendental

Authors: Noah White
Comments: 1 Page. It's very short

The shortest proof n/k is not algebraic thus this constant is not algebraic
Category: Number Theory

On Some Ramanujan Equations: Mathematical Connections with ϕ, ζ(2), Monstrous Moonshine and Various Parameters of Particle Physics. II

Authors: Michele Nardelli, Antonio Nardelli
Comments: 62 Pages.

In this paper we have described and analyzed some Ramanujan equations. Furthermore, we have obtained several mathematical connections with ϕ, ζ(2), Monstrous Moonshine and various parameters of Particle Physics.
Category: Number Theory

On Some Ramanujan Equations: Mathematical Connections with ϕ, ζ(2), Monstrous Moonshine and Various Parameters of Particle Physics.

Authors: Michele Nardelli, Antonio Nardelli
Comments: 64 Pages.

In this paper we have described and analyzed some Ramanujan equations. Furthermore, we have obtained several mathematical connections with ϕ, ζ(2), Monstrous Moonshine and various parameters of Particle Physics.
Category: Number Theory

Riemann Hypothesis Using Integral Equation

Authors: Shekhar Suman
Comments: 4 Pages.

Integration
Category: Number Theory

Numbers Are 3 Dimensional

Authors: Surajit Ghosh
Comments: 99 Pages.

Riemann hypothesis stands proved in three different ways of three different level of complexity.To prove Riemann hypothesis from the functional equation concept of Delta function and periodic harmonic conjugate of both Gamma and Delta functions are introduced similar to Gamma and Pi function. Other two proofs are derived using Eulers formula and elementary algebra. Analytically continuing zeta function to an extended domain, poles and zeros of zeta values are redefined. 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 three dimensional 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 and periodic harmonic conjugate of both Gamma and Delta functions.
Category: Number Theory

Quantom Test of Prime Numbers

Authors: Abdallah Chaib Edderraa
Comments: 5 Pages.

we give a new method to for testing whether any positive integer is prime or not using real experiment by throwing neutron into a Plutonium.
Category: Number Theory

On Some Ramanujan Equations: Mathematical Connections with ϕ, ζ(2), and Various Parameters of Cosmology and Particle Physics. II

Authors: Michele Nardelli, Antonio Nardelli
Comments: 68 Pages.

In this paper we have described and analyzed some Ramanujan equations. Furthermore, we have obtained various mathematical connections with ϕ, ζ(2), and several parameters of Cosmology and Particle Physics.
Category: Number Theory

On the Distribution of Addition Chains

Authors: Theophilus Agama
Comments: 7 Pages.

In this paper we study the theory of addition chains producing any given number $n\geq 3$. With the goal of estimating the partial sums of an additive chain, we introduce the notion of the determiners and the regulators of an addition chain and prove the following identities\begin{align}\sum \limits_{j=2}^{\delta(n)+1}s_j=2(n-1)+(\delta(n)-1)+\kappa(a_{\delta(n)})-\varrho(r_{\delta(n)+1})+\int \limits_{2}^{\delta(n)-1}\sum \limits_{2\leq j\leq t}\varrho(r_j)dt\nonumber \end{align}where \begin{align}2,s_3=\kappa(a_3)+\varrho(r_3),\ldots,s_{k-1}=\kappa(a_{k-1})+\varrho(r_{k-1}),s_{k}=\kappa(a_{k})+\varrho(r_{k})=n\nonumber \end{align}are the associated generators of the chain $1,2,\ldots,s_{k-1},s_{k}=n$ of length $\delta(n)$. Also we obtain the identity\begin{align}\sum \limits_{j=2}^{\delta(n)+1}\kappa(a_j)=(n-1)+(\delta(n)-1)+\kappa(a_{\delta(n)})-\varrho(r_{\delta(n)+1})+\int \limits_{2}^{\delta(n)-1}\sum \limits_{2\leq j\leq t}\varrho(r_j)dt.\nonumber \end{align}
Category: Number Theory

Prime Numbers and Composite Numbers Congruent to 1 (Mod 9)

Authors: Zeolla Gabriel Martín
Comments: 7 Pages.

This paper develops a modified an old and well-known expression for calculating and obtaining all prime numbers greater than three and composite numbers divisible by numbers greater than three. This paper develops formulas to break down the prime numbers and the composite numbers in their reductions, these formulas based on equalities allow to regroup them according to congruence characteristics.
Category: Number Theory

On Some Ramanujan’s Approximations to π: Mathematical Connections with ϕ, ζ(2), and Various Parameters of Particle Physics.

Authors: Michele Nardelli, Antonio Nardelli
Comments: 77 Pages.

In this paper we have described and analyzed some Ramanujan’s Approximations to π. Furthermore, we have obtained various mathematical connections with ϕ, ζ(2), and several parameters of Particle Physics.
Category: Number Theory

On Some Equations Concerning a New Possible Method for the Calculation of the Prime Numbers: Mathematical Connections with Various Expressions of Some Sectors of String Theory and Number Theory

Authors: Roberto Servi, Michele Nardelli
Comments: 22 Pages.

In this paper, in Sections 1 and 2, we have described some equations and theorems concerning and linked to the Riemann zeta function. In the Section 3, we have showed the fundamental equation of the Riemann zeta function and the some equations concerning a new possible method for the calculation of the prime numbers. In conclusion, in the Section 4 we show the possible mathematical connections with various expressions of some sectors of String Theory and Number Theory and finally we suppose as the prime numbers can be identified as possible solutions to the some equations of the string theory (zeta string)
Category: Number Theory

On Some Equations Concerning the M Theory and Topological Strings and the Gopakumar Vafa Formula Applied in Some Sectors of String Theory and Number Theory

Authors: Roberto Servi, Michele Nardelli, Francesco Di Noto
Comments: 39 Pages.

In the present paper we have described in the Chapter 1 some equations concerning the M-Theory, the Topological strings and the Topological Gauge Theory, in the Chapter 2 some equations concerning the Gopakumar-Vafa formula in Type IIA compactification to four dimensions on a Calabi-Yau manifold in terms of a counting of BPS states in M-theory. Finally, in the Chapter 3, we have described some possible methods of factorization and their various possible mathematical connections concerning the solutions for some equations regarding the above sectors of string theory
Category: Number Theory

The $abc$ Conjecture: the Proof of $c

Authors: Abdelmajid Ben Hadj Salem
Comments: 9 Pages. Submitted to the journal Annals of Mathematics.

In this note, I present a very elementary proof of the conjecture $c<rad^2(abc)$ that constitutes the key to resolve the $abc$ conjecture. The method concerns the comparison of the number of primes of $c$ and $rad^2(abc)$ for large $a,b,c$ using the prime counting function $\pi(x)$ giving the number of primes $\leq x$. Some numerical examples are given.
Category: Number Theory

Composite Numbers

Authors: Yuji Masuda
Comments: 1 Page.

This formula of composite numbers shows √2.
Category: Number Theory

Proof of the Twin Prime Conjecture

Authors: K.H.K. Geerasee Wijesuriya
Comments: 13 Pages.

Twin prime numbers are two prime numbers which have the difference of 2 exactly. In other words, twin primes is a pair of prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes. Up to date there is no any valid verification for the twin prime conjecture. Through this research paper, my attempt is to provide a valid proof for twin prime conjecture.
Category: Number Theory

On Various Ramanujan’s Elliptic Integrals, Einstein Dilaton Gauss-Bonnet Gravity and Black Hole Physics Equations: Mathematical Connections with ϕ, ζ(2), and Some Parameters of High Energy Physics. Vii

Authors: Michele Nardelli, Antonio Nardelli
Comments: 54 Pages.

In this paper we have described several Ramanujan’s elliptic integrals, Einstein Dilaton Gauss-Bonnet Gravity and Black Hole Physics equations. Furthermore, we have obtained mathematical connections with ϕ, ζ(2), and some parameters of High Energy Physics.
Category: Number Theory

Riemann Hypothesis Using Simple Integration

Authors: Shekhar Suman
Comments: 8 Pages.

Analytic continuation
Category: Number Theory

On Some Ramanujan Equations: Various Mathematical Connections with ϕ, ζ(2), and Some Physical Parameters

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

In this paper we have described several Ramanujan equations and obtained various mathematical connections with ϕ, ζ(2), and some Physical parameters
Category: Number Theory

On the Ramanujan Taxicab Numbers: New Mathematical Connections with Some Equations of Einstein Dilaton Gauss-Bonnet Gravity

Authors: Michele Nardelli, Antonio Nardelli
Comments: 42 Pages.

In this paper we have described new mathematical connections between Ramanujan Taxicab numbers and some equations of Einstein Dilaton Gauss-Bonnet Gravity
Category: Number Theory

Riemann Hypothesis Using Simple Inequality

Authors: Shekhar Suman
Comments: 6 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

An Expression for π

Authors: Yuji Masuda
Comments: 1 Page.

This formula expresses π.
Category: Number Theory

Sum of Prime Numbers

Authors: Yuji Masuda
Comments: 1 Page.

This formula shows the sum of primes.
Category: Number Theory

On Some Mathematical Equations Concerning the Functions ζ (S) and ζ (S,w) and Some Ramanujan-Type Series for 1/π .Mathematical Connections with Some Equations Concerning the P-Adic Open String for the Scalar Tachyon Field and the Zeta Strings.

Authors: Michele Nardelli
Comments: 32 Pages.

In this paper we have described some mathematical equations concerning the functions ζ(s) and ζ(s,w) and some Ramanujan-type series for 1/π. Furthermore, we describe the mathematical connections with some equations concerning the p-adic open string for the scalar tachyon field and the zeta strings.
Category: Number Theory