Number Theory

Proof of All Out Primes

Authors: Yuji Masuda
Comments: 1 Page.

This is the proof of All Out Primes.
Category: Number Theory

All Out Primes

Authors: Yuji Masuda
Comments: 1 Page.

This is a break time story + β.
Category: Number Theory

Riemann Hypothesis Proof Using Infinite Product

Authors: Shekhar Suman
Comments: 11 Pages.

The proof involves analytic continuation of the Riemann Zeta
Category: Number Theory

Riemann Hypothesis and Non Trivial Zeroes of Riemann Zeta Elementary Proof

Authors: Shekhar Suman
Comments: 5 Pages.

The prooof involves analytic continuation of the Riemann zeta function to the whole of the complex plane except for a simple pole at s=1.
Category: Number Theory

Non - Prime Formula =???

Authors: Yuji Masuda
Comments: 5 Pages.

This is a break time story + α.
Category: Number Theory

Beal’s Conjecture is Tenable

Authors: Zhang Tianshu
Comments: 20 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, affirm that there are AX+BY=CZ in which case A, B and C have a common prime factor by several concrete equalities. After that, prove that there are AX+BY≠CZ where A, B and C have not a common prime factor by the mathematical induction with the aid of interrelations of 3 integers relating to symmetry after divide AX+BY=CZ in four. 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

Riemann Hypothesis a New Proof

Authors: Shekhar Suman
Comments: 10 Pages.

Riemann Xi function is defined and has the same zeroes as the non trivial zeroes of Riemann Zeta function
Category: Number Theory

New Clues on Arbitrary-Precision Calculation of the Riemann Zeta Function On The Critical Line

Authors: Alan Michael Gómez Calderón
Comments: 3 Pages. Attemp to solve Riemann Hypothesis

The Riemann Hypothesis, is considered by many mathematicians to be the most important unsolved problem, consist in the assertion that all of zeta's nontrivial zeros line up at the so called critical line, $\zeta(1/2+it)$. This paper presents an algorithm, based on a closed-form system of equations, that computes directly at $n^{th}$ decimal digit each non-trivial zeros of the Riemann Zeta Function.
Category: Number Theory

The Riemann Hypothesis and Zeroes of the Riemann Zeta Function

Authors: Shekhar Suman
Comments: 9 Pages.

The proof involves analytic continuation expressed as a Hadamard Product
Category: Number Theory

Riemann Hypothesis Proof Using Hadamard Product

Authors: Shekhar Suman
Comments: 9 Pages.

The proof involves analytic contination of Riemann zwta function as a Hadamard Product
Category: Number Theory

Riemann Hypothesis and Riemann Zeta Function

Authors: Shekhar Suman
Comments: 12 Pages.

the proof involves hadamard product representation of Riemann Xi function
Category: Number Theory

Approximation of Prime Graph

Authors: Yuji Masuda
Comments: 2 Pages.

This is a break time story Ⅲ.
Category: Number Theory

Riemann Hypothesis and the Zeroes of Riemann Zeta Function

Authors: Shekhar Suman
Comments: 12 Pages.

The proof involves analytic continuation of Riemann Zeta function. Further we work on the Hadamard product representation of Riemann Xi function to prove the Riemann Hypothesis
Category: Number Theory

Riemann Zeta Function and Riemann Hypothesis

Authors: Shekhar Suman
Comments: 9 Pages.

The proof involves analytic continuation of the Riemann zeta function as a hadamard product
Category: Number Theory

Proof of Ramanujan’s Identities for Q-Series and Theta Function

Authors: Omprakash Atale
Comments: 4 Pages.

This paper involves in providing proof of Ramanujan’s identities for q-hypergeometric series and Theta function, more specifically we are going to derive the q-analogue of gauss summation of ordinary hypergeometric series.
Category: Number Theory

Goedelian Encryption & Goldbach's Conjecture

Authors: Paris Rose Miles-Brenden
Comments: 8 Pages. Riemann Zeta Elimination Encryption Goedel Hypothesis Solution 2020

In qualitative terms the assumptive of mathematics is 'therefore'. To what would equivalently dissuade one from a problem; there is with immediacy a necessity to solution; to which is founded on a principle; or two... then to what we ponder of infinity and cognition; the realization of which is two. For then in balance and equipartition there is a modicum of folly to the arrow of balance; but of it's twin; it is suggestive that numbers may be thought or written; transcribed or copied. This principle stems bidirectionally the suggestion of twin's; and at it's base; founds a world free of theft; for of what is difference of accumen; the final truth of logic; that when assailed; a problem fall's way naturally when it will; but otherwise as alone only when properly expressed in the form of a solution; the joining; again therefore.
Category: Number Theory

Stairs of Prime Steps

Authors: Yuji Masuda
Comments: 1 Page.

This is a break time story Ⅲ.
Category: Number Theory

Beal Conjecture & Equivalent Beal Conjecture Proved

Authors: A. A. Frempong
Comments: 10 Pages. Copyright © by A. A. Frempong

The author proves both the original Beal conjecture and the equivalent Beal conjecture. The original Beal conjecture states that if A^x + B^y = C^z, where A, B, C, x, y, z are positive integers and x, y, x >2 then A, B, and C have a common prime factor. The equivalent Beal conjecture states that if A, B, C, x, y, z are positive integers and A, B, and C are coprime , and x, y, z >2, then the equation if A^x + B^y = C^z, has no solutions. The principles applied in both proofs are based on the same properties of the factored Beal equation. However the proof of the equivalent conjecture is by contradiction. 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

Order in the Collatz: Fractal Symmetry Controlling Syracuse Function Trajectories

Authors: Derek Tucker
Comments: 11 Pages.

Consider the function C(n) = {█(3n+1, n≡1 mod 2.@n/2, n≡0 mod 2.)┤ The 3x + 1 problem or Collatz conjecture asks if all trajectories iterating recursively contain one. This holds empirically but eludes theoretical proof. Remove even entries to obtain the Syracuse function, T(u), mapping odd numbers u , T(u) = (3u +1)/2r, where r maximal for T(u) odd. Here we show by expressing u as 4x ±1, a fractal regularity given by the amplitude of a periodic function passing through the x axis in the curves defined y=N cos⁡(πx/2^N ) ≡ N mod((x+2^(N-1))/2^N ,1), with ℕ the natural numbers. The expression describes the length of ascending segments R and with translation, r the number of divisions in T(u). Algebraic analysis reveals the Syracuse function’s image neatly partitions onto arithmetic progressions 6k ± 1, as a function of r. This confirms that all natural trajectories in the 3x + 1 problem contain one..
Category: Number Theory

Assuming C

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

In this paper about the abc conjecture, we propose a new conjecture about an upper bound for c as c0, for a,b,c positive integers relatively prime with c=a+b, we have c< K(\epsilon).rad^{1+\epsilon}(abc). Some numerical examples are given.
Category: Number Theory

N.H.P=New Hypothesis of Primes (break time Ⅱ)

Authors: Yuji Masuda
Comments: 1 Page.

THis will be a new rule of primes.
Category: Number Theory

On the Links Between Some Ramanujan Formulas, the Golden Ratio and Various Equations of Several Sectors of Black Hole Physics

Authors: Michele Nardelli, Antonio Nardelli
Comments: 116 Pages.

The purpose of this paper is to show the links between some Ramanujan formulas, the golden ratio and the mathematical connections with various equations of several sectors of Black Hole Physics
Category: Number Theory

Analyzing Two Ramanujan Equations: Mathematical Connections with Various Parameters of Particle Physics and Cosmology II

Authors: Michele Nardelli, Antonio Nardelli
Comments: 75 Pages.

The purpose of this paper is to show how using certain mathematical values and / or constants from two Ramanujan equations, some important parameters of Particle Physics and Cosmology are obtained.
Category: Number Theory

Analyzing a Ramanujan Equation: Mathematical Connections with Various Parameters of Particle Physics and Cosmology

Authors: Michele Nardelli, Antonio Nardelli
Comments: 40 Pages.

The purpose of this paper is to show how using certain mathematical values and / or constants from a Ramanujan equation, some important parameters of Particle Physics and Cosmology are obtained.
Category: Number Theory

Goldbach Conjecture

Authors: Oluwatobi Samuel Bode-Odeyemi
Comments: 2 Pages.

The short document is a proof by induction of the Goldbach Conjecture
Category: Number Theory

Conjectures About Min/max Prime Divisor of the Natural Numbers and Prime Numbers

Authors: Edoardo Gueglio
Comments: 13 Pages.

Applying functions over prime divisors to the sequence of natural numbers and taking the differences of this sequence you can split this in subsequences by the sum equals length rule so that you can associate each subsequence to each gap between consecutive prime numbers
Category: Number Theory

On Some Ramanujan’s Equations (Hardy-Ramanujan Number and Mock Theta Functions) Linked to Various Parameters of Standard Model and Black Hole Physics: New Possible Mathematical Connections. III

Authors: Michele Nardelli, Antonio Nardelli
Comments: 116 Pages.

In this research thesis, we have described and deepened further Ramanujan equations (Hardy-Ramanujan number and mock theta functions) linked to various parameters of Standard Model and Black Hole Physics. We have therefore obtained further possible mathematical connections.
Category: Number Theory