[21] **viXra:1607.0569 [pdf]**
*submitted on 2016-07-31 17:30:32*

**Authors:** Hervé G.

**Comments:** 19 Pages.

It is presented an elementary proof that Beta(3)=Pi^3/32. Beta is the Dirichlet Beta function.

**Category:** Number Theory

[20] **viXra:1607.0557 [pdf]**
*submitted on 2016-07-30 15:05:24*

**Authors:** Simon Plouffe

**Comments:** 3 Pages.

An extension of a known result of Ramanujan is used to produce sums with exponential terms that gives a representation of many prime numbers.

**Category:** Number Theory

[19] **viXra:1607.0551 [pdf]**
*submitted on 2016-07-29 12:19:22*

**Authors:** Richard Broxley Omeston

**Comments:** 1 Page.

In this paper I show how the equivalence of the summation of the Móbius function with the Zeta function allows for proof of the Riemann Hypothesis.

**Category:** Number Theory

[18] **viXra:1607.0536 [pdf]**
*submitted on 2016-07-28 15:35:29*

**Authors:** José de Jesús Camacho Medina

**Comments:** 1 Page.

The present article shows an unpublished formula to evaluate twin primes, the formula is based on the theorem of Wilson and contains mathematical functions such that greatest common divisor, factorial and floor function.

**Category:** Number Theory

[17] **viXra:1607.0522 [pdf]**
*submitted on 2016-07-27 10:29:42*

**Authors:** Matilda Walter

**Comments:** 3 Pages.

Lemoine - Levy Conjecture, probably the least known of the 'Goldbach Conjectures', states that
every positive odd integer > 5 is a sum of a prime and double of a prime. We present a simple sieve
procedure for finding all existing solutions to the problem for any given odd number > 5.

**Category:** Number Theory

[16] **viXra:1607.0468 [pdf]**
*submitted on 2016-07-25 01:02:22*

**Authors:** Dhananjay P. Mehendale

**Comments:** 3 Pages

This paper proposes a generalised ABC conjecture and assuming its validity settles a generalised version of Fermat’s last theorem.

**Category:** Number Theory

[15] **viXra:1607.0437 [pdf]**
*submitted on 2016-07-23 12:06:52*

**Authors:** Kunle Adegoke

**Comments:** 15 Pages.

Using a straightforward elementary approach, we derive numerous infinite arctangent summation formulas involving Fibonacci and Lucas numbers. While most of the results obtained are new, a couple of celebrated results appear as particular cases of the more general formulas derived here.

**Category:** Number Theory

[14] **viXra:1607.0434 [pdf]**
*submitted on 2016-07-23 12:24:01*

**Authors:** Terubumi Honjou

**Comments:** 7 Pages.

Chapter12. Challenge "proof of the Lehman expectation".
A mathematics difficult problem biggest in history.
[1] With the mathematics difficult problem "proof of the Lehman expectation" biggest in history.
[2] I challenge the difficult problem Lehman expectation that rejected the geniuses challenge for 150 years.
[3] It is challenged the mystery of the prime number, a mathematics difficult problem biggest in history, proof of the Lehman expectation.
[4] Neology of the Lehman expectation. A point of intersection that all 0 points are straight.
[5] An elementary particle pulsation principle founds a door of the Lehman expected proof.

**Category:** Number Theory

[13] **viXra:1607.0400 [pdf]**
*submitted on 2016-07-21 22:44:27*

**Authors:** Quang Nguyen Van

**Comments:** 3 Pages.

We give an illogical point in Dirichlet's proof, therefore the used infininite descent is not
powered in his proof

**Category:** Number Theory

[12] **viXra:1607.0381 [pdf]**
*submitted on 2016-07-20 11:59:45*

**Authors:** Peter Bissonnet

**Comments:** 10 Pages.

Prime products are analyzed from various points of view, with an emphasis on graphical representation and analysis. A prime product N is determined to have two integer coordinates D and m. These coordinates are related to the solutions of a parabola, as well as to right triangles, in what the author calls a ‘backbone - rib’ representation. A prime number or a prime product fall on three dimensional helices, which can be represented in two dimensions as sets of parallel lines. If a prime or a prime product can be represented by 6s - 1, then helix 1 or H1 is designated; if a prime or a prime product can be represented by 6s + 1, then helix 2 or H2 is designated. The integer s is really a composite number, which can be represented as s = r + n, where r is the row number and n is the grouping number called the complex number, both determined from the two dimensional representation of the double helices.
It is also discovered that, due to the mathematical form relating N to D and m, that there must be Lorentz - like transformations between N, D, and m and a new set Nʹ, Dʹ and mʹ; however, the concept of velocity and the speed of light seem out of place in this instance. Nevertheless, the question is asked as to whether or not prime products can be considered to be away to unite relativity and quantum mechanics, which also depends upon integers in a large measure.

**Category:** Number Theory

[11] **viXra:1607.0360 [pdf]**
*replaced on 2016-10-25 13:27:33*

**Authors:** Reuven Tint

**Comments:** 7 Pages.

A variant of the solution with the help of Bill hypothesis direct evidence "Great" Fermat's theorem elementary methods rows. New are "invariant identity" (keyword) and obtained by us in the text, the identity of the work, which allowed directly to solve the FLT, and several others.

**Category:** Number Theory

[10] **viXra:1607.0359 [pdf]**
*replaced on 2016-07-20 07:41:06*

**Authors:** Matilda Walter

**Comments:** 2 Pages.

We present a simple sieve algorithm for finding all existing solutions to the binary Goldbach
problem for a given even number 2N > 4.

**Category:** Number Theory

[9] **viXra:1607.0178 [pdf]**
*submitted on 2016-07-15 06:08:08*

**Authors:** Zhang Tianshu

**Comments:** 25 Pages.

In this article, we first classify A, B and C according to their respective odevity, and thereby get rid of two kinds from AX+BY=CZ. Then, affirmed AX+BY=CZ in which case A, B and C have at least a common prime factor by several concrete equalities. After that, proved AX+BY≠CZ in which case A, B and C have not any common prime factor by mathematical induction with the aid of the symmetric relations of positive odd numbers concerned after divide the inequality in four. Finally, reached a conclusion that the Beal’s conjecture holds water via the comparison between AX+BY=CZ and AX+BY≠CZ under the given requirements.

**Category:** Number Theory

[8] **viXra:1607.0159 [pdf]**
*submitted on 2016-07-13 11:11:18*

**Authors:** Maaninou Youssef

**Comments:** 1 Page.

by this theory you can chek any number if it is a prime or not
also you can generat a new prime number

**Category:** Number Theory

[7] **viXra:1607.0095 [pdf]**
*submitted on 2016-07-07 16:22:11*

**Authors:** Kolosov Petro

**Comments:** 2 Pages.

The basic and short description of proposed entry to Wolfram MathWorld, Finite Difference, starting from line ”An n-th power has a constant
nth finite difference. For example, take n=3 and make a difference table”.
The entry appears to to wide the properties of high order difference of
n-th power.

**Category:** Number Theory

[6] **viXra:1607.0094 [pdf]**
*submitted on 2016-07-07 16:24:21*

**Authors:** Kolosov Petro

**Comments:** 2 Pages.

The basic and short description of proposed entry to Wolfram MathWorld, ”Power”, properties section. The entry appears to connect the
high order finite difference and derivative of Power function.

**Category:** Number Theory

[5] **viXra:1607.0093 [pdf]**
*submitted on 2016-07-07 16:26:41*

**Authors:** Kolosov Petro

**Comments:** 12 Pages.

This paper presents the way to make expansion for the next form function: y =
x n , ∀(x,n) ∈ N to the numerical series. The most widely used methods to solve this
problem are Newton’s Binomial Theorem and Fundamental Theorem of Calculus (that
is, derivative and integral are inverse operators). The paper provides the other kind of
solution, except above described theorems.

**Category:** Number Theory

[4] **viXra:1607.0087 [pdf]**
*replaced on 2016-07-25 10:05:25*

**Authors:** Ryan Zielinski

**Comments:** 84 Pages. This work is licensed under the CC BY 4.0, a Creative Commons Attribution License.

What is a general expression for the sum of the first n integers, each raised to the mth power, where m is a positive integer? Answering this question will be the aim of the paper....We will take the unorthodox approach of presenting the material from the point of view of someone who is trying to solve the problem himself. Keywords: analogy, Johann Faulhaber, finite sums, heuristics, inductive reasoning, number theory, George Polya, problem solving, teaching of mathematics.

**Category:** Number Theory

[3] **viXra:1607.0072 [pdf]**
*replaced on 2016-07-15 07:54:41*

**Authors:** Korn Rakpradit

**Comments:** 47 Pages.

The opinions of this work are revising, stalking and proving in details the derivation of Riemann Zeta Function and Riemann Hypothesis, which Riemann did roughly for more than 150 years ago without proof, and correcting all mistakes about the boundaries of the integrals that was found and those undefined (and/or multiplied by zero) functional equations which caused very big problems to this Riemann Hypothesis.
Proof or disproof of Riemann Hypothesis’s derivation will be very useful for many mathematicians and physicists nowadays because the Hypothesis is widely used in many subjects and works, unaware of risks, thought it is not officially proved right or wrong.

**Category:** Number Theory

[2] **viXra:1607.0029 [pdf]**
*submitted on 2016-07-03 05:30:52*

**Authors:** Andrea OSSICINI

**Comments:** 16 Pages.

It is shown that an appropriate use of so-called << double equations >> by Diophantus provides the origin of the Frey elliptic curve and from it we can deduce an elementary proof of Fermat’s Last Theorem

**Category:** Number Theory

[1] **viXra:1607.0003 [pdf]**
*submitted on 2016-07-01 03:27:48*

**Authors:** Pingyuan Zhou

**Comments:** 13 Pages. If so-called strong Goldbach number sequence introduced in this paper is acceptable then the existence of strong Goldbach number sequence will imply both Goldbach and twin prime conjectures.

Abstract: In this note, we present a new and direct appraoch to prove the Goldbach conjecture that if the existence of the limit of

**Category:** Number Theory