[25] **viXra:1904.0592 [pdf]**
*submitted on 2019-04-30 08:39:11*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

In this note we recall a formula for Pi.

**Category:** Number Theory

[24] **viXra:1904.0561 [pdf]**
*replaced on 2019-07-14 19:53:00*

**Authors:** Kouji Takaki

**Comments:** 14 Pages.

We have obtained the conclusion that there are no odd perfect numbers.

**Category:** Number Theory

[23] **viXra:1904.0517 [pdf]**
*submitted on 2019-04-26 09:22:45*

**Authors:** Afmika, AF. Michael

**Comments:** 4 Pages.

This is a simple proof of the Legendre's conjecture.
afmichael73@gmail.com
afmichael.san@gmail.com

**Category:** Number Theory

[22] **viXra:1904.0507 [pdf]**
*replaced on 2019-05-02 07:08:03*

**Authors:** Algirdas Antano Maknickas

**Comments:** 2 Pages.

This remarks prove, that Riemann zeta function has infinitesimal amount of zeros.

**Category:** Number Theory

[21] **viXra:1904.0489 [pdf]**
*submitted on 2019-04-26 01:20:05*

**Authors:** Kunle Adegoke

**Comments:** 5 Pages.

We evaluate the sums $\sum_{j=0}^k{u_{rj+s}^{2n}\,z^j}$, $\sum_{j=0}^k{u_{rj+s}^{2n-1}\,z^j}$ and $\sum_{j=0}^k{v_{rj+s}^{n}\,z^j}$, where $r$, $s$ and $k$ are any integers, $n$ is any nonnegative integer, $z$ is arbitrary and $(u_n)$ and $(v_n)$ are the Lucas sequences of the first kind and of the second kind, respectively. As natural consequences we obtain explicit forms of the generating functions for the powers of the terms of Lucas sequences with indices in arithmetic progression. This paper therefore extends the results of P.~Sta\u nic\u a who evaluated $\sum_{j=0}^k{u_{j}^{2n}\,z^j}$ and $\sum_{j=0}^k{u_{j}^{2n-1}\,z^j}$; and those of B. S. Popov who obtained generating functions for the powers of these sequences.

**Category:** Number Theory

[20] **viXra:1904.0454 [pdf]**
*submitted on 2019-04-23 08:38:55*

**Authors:** Edgar Valdebenito

**Comments:** 5 Pages.

In this note we give some formulas related with the number: alpha=0.5*arccos(0.5*arccos(0.5*arccos(0.5*...))).

**Category:** Number Theory

[19] **viXra:1904.0446 [pdf]**
*submitted on 2019-04-23 18:28:40*

**Authors:** Zeolla Gabriel Martín

**Comments:** 7 Pages.

This document develops and demonstrates the discovery of a new square potentiation algorithm that works absolutely with all the numbers using the formula of the square of a binomial.

**Category:** Number Theory

[18] **viXra:1904.0428 [pdf]**
*replaced on 2019-05-03 18:29:29*

**Authors:** Ralf Wüsthofen

**Comments:** 2 Pages. Proof of the Goldbach conjecture on http://vixra.org/abs/1702.0300

Based on a strengthened form of the strong Goldbach conjecture, this paper presents an antinomy within the Peano arithmetic (PA). We derive two contradictory statements by using the same main instrument as in the proof of the conjecture, i.e. a set that is a structuring of the natural numbers starting from 3.

**Category:** Number Theory

[17] **viXra:1904.0422 [pdf]**
*replaced on 2019-05-02 04:00:00*

**Authors:** Hajime Mashima

**Comments:** 2 Pages.

The three sides of the right triangle are rational numbers, and those
with natural numbers are congruent numbers.

**Category:** Number Theory

[16] **viXra:1904.0410 [pdf]**
*submitted on 2019-04-21 15:17:55*

**Authors:** Yuly Shipilevsky

**Comments:** 3 Pages.

We consider generalized Fermat equation for hypercomplex numbers, in
order to stimulate research and development of those generalization

**Category:** Number Theory

[15] **viXra:1904.0386 [pdf]**
*submitted on 2019-04-19 11:38:30*

**Authors:** Divyendu Priyadarshi

**Comments:** 1 Page.

In this short paper, I have tried to give a physical meaning to irrational numbers.

**Category:** Number Theory

[14] **viXra:1904.0378 [pdf]**
*submitted on 2019-04-19 21:33:45*

**Authors:** Pedro Caceres

**Comments:** 27 Pages.

The Riemann Zeta function or Euler–Riemann Zeta function, ζ(s), is a function of a complex variable z that analytically continues the sum of the Dirichlet series:
() = ∑ ^(-z) from k=1,∞
The Riemann zeta function is a meromorphic function on the whole complex z-plane, which is holomorphic everywhere except for a simple pole at z = 1 with residue 1. One of the most important advance in the study of Prime numbers was the paper by Bernhard Riemann in November 1859 called “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse” (On the number of primes less than a given quantity). In this paper, Riemann gave a formula for the number of primes less than x in terms the integral of 1/log(x), and also provided insights into the roots (zeros) of the zeta function, formulating a conjecture about the location of the zeros of () in the critical line Re(z)=1/2.
The Riemann Zeta function is one of the most studied and well known mathematical functions in history. In this paper, we will formulate nine new propositions to advance in the knowledge of the Riemann Zeta function

**Category:** Number Theory

[13] **viXra:1904.0376 [pdf]**
*replaced on 2019-05-14 02:42:38*

**Authors:** Surajit Ghosh

**Comments:** 32 Pages.

Based on Euler ’s formula a concept of duality unit or dunit circle is discovered. Continuing with Riemann hypothesis is proved from diﬀerent angles, zeta values are renormalised to remove the poles of zeta function and discover relationships between numbers and primes. Other unsolved prime conjectures are also proved with the help of theorems of numbers and number theory. Imaginary number i can be deﬁned such a way that it eases the complex logarithm and accounts for the scale diﬀerence between very big and very small. Pi can also be a base to natural logarithm and complement the scale gap. 96 complex constants derived from complex logarithm can explain everything in the universe.

**Category:** Number Theory

[12] **viXra:1904.0235 [pdf]**
*submitted on 2019-04-12 17:45:46*

**Authors:** Arthur Shevenyonov

**Comments:** 8 Pages. Trilinear, IIIVNII

A set of distinct and elementary approaches, all embarking on the Euler-Riemann equivalence representing the zeta at zero, invariably point to a consistent solution structure. The Riemann Hypothesis as regards Re=1/2 gains full support as a core solution, albeit one amounting to a special nontrivial case warranting extensions and qualifications.

**Category:** Number Theory

[11] **viXra:1904.0227 [pdf]**
*replaced on 2019-05-26 08:48:53*

**Authors:** Elizabeth Gatton-Robey

**Comments:** 5 Pages.

I created an algorithm capable of proving Goldbach’s Conjecture. This is not a claim to have proven the
conjecture. The algorithm and all work contained in this document is original, so no outside sources have
been used. This paper explains the algorithm then applies the algorithm with examples. The final section
of the paper contains information to accompany my thoughts on why I believe Goldbach’s Conjecture can
be proven with the use of my algorithm.

**Category:** Number Theory

[10] **viXra:1904.0219 [pdf]**
*submitted on 2019-04-11 18:49:36*

**Authors:** Yuly Shipilevsky

**Comments:** 2 Pages.

We consider a new conjecture regarding powers of integer numbers and
more specifically, we are interesting in existence and finding pairs of integers:
n ≥ 2 and m ≥ 2, such that n^m = m^n.

**Category:** Number Theory

[9] **viXra:1904.0214 [pdf]**
*replaced on 2019-05-27 18:47:01*

**Authors:** John Yuk Ching Ting

**Comments:** 18 Pages. Rigorous proofs for Polignac's and Twin prime conjectures.

Prime numbers are Incompletely Predictable numbers calculated using complex algorithm Sieve of Eratosthenes. Involving proposals that prime gaps and associated sets of prime numbers are infinite in magnitude, Twin prime conjecture deals with even prime gap 2 and is a subset of Polignac's conjecture which deals with all even prime gaps 2, 4, 6, 8, 10,.... Treated as Incompletely Predictable problems, we solve these conjectures with research method Information-Complexity conservation to get Plus Gap 2 Composite Number Continuous Law and Plus-Minus Gap 2 Composite Number Alternating Law.

**Category:** Number Theory

[8] **viXra:1904.0146 [pdf]**
*submitted on 2019-04-07 14:40:11*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 10 Pages. Submitted to the journal Research In Number Theory. Comments welcome.

In this paper, we consider the $abc$ conjecture in the case $c=a+1$. Firstly, we give the proof of the first conjecture that $c

**Category:** Number Theory

[7] **viXra:1904.0105 [pdf]**
*submitted on 2019-04-06 00:57:16*

**Authors:** Idriss Olivier Bado

**Comments:** 7 Pages.

In this paper we give a proof for Beal's conjecture . Since the discovery of the proof of Fermat's last theorem by Andre Wiles, several questions arise on the correctness of Beal's conjecture. By using a very rigorous method we come to the proof. Let $ \mathbb{G}=\{(x,y,z)\in \mathbb{N}^{3}: \min(x,y,z)\geq 3\}$
$\Omega_{n}=\{ p\in \mathbb{P}: p\mid n , p \nmid z^{y}-y^{z}\}$ ,
$$\mathbb{T}=\{(x,y,z)\in \mathbb{N}^{3}: x\geq 3,y\geq 3,z\geq 3\}$$
$\forall(x,y,z) \in \mathbb{T}$ consider the function $f_{x,y,z}$ be the function defined as :
$$\begin{array}{ccccc}
f_{x,y,z} & : \mathbb{N}^{3}& &\to & \mathbb{Z}\\
& & (X,Y,Z) & \mapsto & X^{x}+Y^{y}-Z^{z}\\
\end{array}$$
Denote by $$\mathbb{E}^{x,y,z}=\{(X,Y,Z)\in \mathbb{N}^{3}:f_{x,y,z}(X,Y,Z)=0\}$$
and $\mathbb{U}=\{(X,Y,Z)\in \mathbb{N}^{3}: \gcd(X,Y)\geq2,\gcd(X,Z)\geq2,\gcd(Y,Z)\geq2\}$
Let $ x=\min(x,y,z)$ . The obtained result show that :if $ A^{x}+B^{y}=C^{z}$ has a solution and $ \Omega_{A}\not=\emptyset$, $\forall p \in \Omega_{A}$ ,
$$ Q(B,C)=\sum_{j=1}^{x-1}[\binom{y}{j}B^{j}-\binom{z}{j}C^{j}]$$ has no solution in $(\frac{\mathbb{Z}}{p^{x}\mathbb{Z}})^{2}\setminus\{(\overline{0},\overline{0})\} $ Using this result we show that Beal's conjecture is true since $$ \bigcup_{(x,y,z)\in\mathbb{T}}\mathbb{E}^{x,y,z}\cap \mathbb{U}\not=\emptyset$$ Then $\exists (\alpha,\beta,\gamma)\in \mathbb{N}^{3}$ such that $\min(\alpha,\beta,\gamma)\leq 2$ and $\mathbb{E}^{\alpha,\beta,\gamma}\cap \mathbb{U}=\emptyset$
The novel techniques use for the proof can be use to solve the variety of Diophantine equations . We provide also the solution to Beal's equation . Our proof can provide an algorithm to generate solution to Beal's equation

**Category:** Number Theory

[6] **viXra:1904.0070 [pdf]**
*submitted on 2019-04-03 09:55:42*

**Authors:** Stephen Marshall

**Comments:** 8 Pages.

The Polignac prime conjecture, was made by Alphonse de Polignac in 1849. Alphonse de Polignac (1826 – 1863) was a French mathematician whose father, Jules de Polignac (1780-1847) was prime minister of Charles X until the Bourbon dynasty was overthrown in1830. Polignac attended the École Polytechnique (commonly known as Polytechnique) a French public institution of higher education and research, located in Palaiseau near Paris. In 1849, the year Alphonse de Polignac was admitted to Polytechnique, he made what's known as Polignac's conjecture:
For every positive integer k, there are infinitely many prime gaps of size 2k.
Alphonse de Polignac made other significant contributions to number theory, including the de Polignac's formula, which gives the prime factorization of n!, the factorial of n, where n ≥ 1 is a positive integer.
This paper presents a complete and exhaustive proof of the Polignac Prime Conjecture. The approach to this proof uses same logic that Euclid used to prove there are an infinite number of prime numbers.

**Category:** Number Theory

[5] **viXra:1904.0035 [pdf]**
*submitted on 2019-04-02 14:51:02*

**Authors:** Stephen Marshall

**Comments:** 10 Pages.

Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century.
The exponents n which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... (sequence A000043) and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ...
If n is a composite number then so is 2n − 1. More generally, numbers of the form Mn = 2n − 1 without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that n be prime. The smallest composite Mersenne number with prime exponent n is 211 − 1 = 2047 = 23 × 89.
Mersenne primes Mp are also noteworthy due to their connection to perfect numbers.
A new Mersenne prime was found in December 2017. As of January 2018, 50 Mersenne primes are now known. The largest known prime number 277,232,917 − 1 is a Mersenne prime. Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether the set of Mersenne primes is finite or infinite. Ever since M521 was proven prime in 1952, the largest known prime has always been Mersenne primes, which shows that Mersenne primes become large quickly. Since the prime numbers are infinite, and since all large primes discovered since 1952 have been Mersenne primes, this seems to be evidence indicating the infinitude of Mersenne primes since there has to continually be an infinite number of large primes, even if we don’t find them. Additional evidence, is that since prime numbers are infinite, there exist an infinite number of Mersenne numbers of form 2p – 1, meaning there exist an infinite number of Mersenne numbers that are candidates for Mersenne primes. However, as with 211 – 1, we know not all Mersenne numbers of form 2p – 1 are primes. All of this evidence makes it reasonable to conjecture that there exist an infinite number of Mersenne primes. First we will provide additional evidence indicating an infinite number of Mersenne primes. Then we will provide the proof.

**Category:** Number Theory

[4] **viXra:1904.0034 [pdf]**
*submitted on 2019-04-02 15:00:11*

**Authors:** Stephen Marshall

**Comments:** 8 Pages.

Fermat prime is a prime number that are a special case, given by the binomial number of the form:
Fn = 22n + 1, for n ≥ 0
They are named after Pierre de Fermat, a Frenchman of the 17th Century, Pierre de Fermat, effectively invented modern number theory virtually single-handedly, despite being a small-town amateur mathematician. Throughout his life he devised a wide range of conjectures and theorems. He is also given credit for early developments that led to modern calculus, and for early progress in probability theory.
The only known Fermat primes are:
F0 = 3
F1 = 5
F2 = 17
F3 = 257
F4 = 65,537
It has been conjectured that there are only a finite number of Fermat primes, however, we will use the same technique the author used to prove that the Mersenne primes are infinite, to prove the Fermat primes are infinite.

**Category:** Number Theory

[3] **viXra:1904.0033 [pdf]**
*submitted on 2019-04-02 15:07:04*

**Authors:** Stephen Marshall

**Comments:** 9 Pages.

The Wagstaff prime is a prime number q of the form:
q = (2^p- 1)/3
where, p is an odd prime. Wagstaff primes are named after the mathematician Samuel S. Wagstaff Jr. Wagstaff primes appear in the New Mersenne conjecture and have applications in cryptography.
The New Mersenne conjecture (Bateman et al. 1989) states that for any odd natural number p, if any two of the following conditions hold, then so does the third:
1. p = 2k ± 1 or p = 4k ± 3 for some natural number k.
2. 2p − 1 is prime (a Mersenne prime).
3. (2p + 1) / 3 is prime (a Wagstaff prime).
There is no simple primality test analogous to the Lucas-Lehmer test for Wagstaff primes, so all recent primality proofs of Wagstaff primes have used elliptic curve primality proving which is very time consuming.
A Wagstaff prime can also be interpreted as a repunit prime of base , as
if p is odd, as it must be for the above number to be prime.
The first three Wagstaff primes are 3, 11, and 43 because
The first few Wagstaff primes are:
3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, … (sequence A000979 in the OEIS)
As of October 2014, known exponents which produce Wagstaff primes or probable primes are:
3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, (all known Wagstaff primes)
95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399, …, 13347311, 13372531 (Wagstaff probable primes) (sequence A000978 in the OEIS)
In February 2010, Tony Reix discovered the Wagstaff probable prime:
which has 1,213,572 digits and was the 3rd biggest probable prime ever found at this date.
In September 2013, Ryan Propper announced the discovery of two additional Wagstaff probable primes:
and,
Each is a probable prime with slightly more than 4 million decimal digits. It is not currently known whether there are any exponents between 4031399 and 13347311 that produce Wagstaff probable primes.
Note that when p is a Wagstaff prime, need not to be prime, the first counterexample is p = 683, and it is conjectured that if p is a Wagstaff prime and p>43, then is composite.

**Category:** Number Theory

[2] **viXra:1904.0032 [pdf]**
*submitted on 2019-04-02 15:13:48*

**Authors:** Stephen Marshall

**Comments:** 7 Pages.

At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as "unattackable at the present state of mathematics" and are now known as Landau's problems. They are as follows:
1.Goldbach's conjecture: Can every even integer greater than 2 be written as the sum of two primes?
2.Twin prime conjecture: Are there infinitely many primes p such that p + 2 is prime?
3.Legendre's conjecture: Does there always exist at least one prime between consecutive perfect squares?
4.Are there infinitely many primes p such that p − 1 is a perfect square? In other words: Are there infinitely many primes of the form n2 + 1?
We will solve Landau’s fourth problem by proving there are infinitely many primes of the form n2 + 1.

**Category:** Number Theory

[1] **viXra:1904.0025 [pdf]**
*replaced on 2019-05-09 08:03:40*

**Authors:** BERKOUK Mohamed

**Comments:** 12 Pages.

en ce qui concerne la conjecture forte, chaque nombre pair n, à partir de 4 peut générer plusieurs couples dont les éléments a et b < n et que parmi ces couples, qui déjà répondent à la conjecture par la sommation (n=a+b).Le nombre ou le cardinal des couples premiers sera estimé par le théorème fondamentale des nombres premiers , en démontrant que ce cardinal > 0 c'est-à-dire ∀ N pair > 3, ∃ un couplet Goldbach premier (p, p’) généré par N / N= p + p’ En établissant l’inéquation de Goldbach qui exprime autrement la conjecture dédié à Mostafa , mon petit frère décédé d'une mort subite (R.A).

**Category:** Number Theory