[19] **viXra:1802.0321 [pdf]**
*submitted on 2018-02-22 07:29:28*

**Authors:** Andrea Prunotto

**Comments:** 2 Pages.

The condition of equiprobability among two events involving independent extractions of elements from a finite set is shown to coincide with Fermat's Diophantine equation. The problem of the division of the stakes, related to the events, is also discussed

**Category:** Number Theory

[18] **viXra:1802.0309 [pdf]**
*submitted on 2018-02-21 19:03:59*

**Authors:** David Stacha

**Comments:** Pages.

In this article I will provide the solution of Brocard`s problem n!+1=x^2 and I will prove the existence of the finite amount of Brown numbers, where the largest Brown number is (7,71), which represents the equation 7!+1=71^2. Brocard`s problem represents one of the open problem in mathematics from the field of number theory, which has been formulated by Henri Brocard in 1876 and represents the solutions of the following Diophantine equation n!+1=x^2.

**Category:** Number Theory

[17] **viXra:1802.0303 [pdf]**
*submitted on 2018-02-22 00:31:26*

**Authors:** Pedro Caceres

**Comments:** 56 Pages.

The function x(j,k)=δ+ω(α+βj)^φk in C→C is a generalization of the power function y(α)=α^k in R→R and the exponential function y(k)=α^k in R→R. In this paper we are going to calculate the values of infinite and partial sums and products involving elements of the matrix Xjk=[x(j,k)]∈C
As a result, several new representations will be made for some infinite series, including the Riemann Zeta Function in C.

**Category:** Number Theory

[16] **viXra:1802.0269 [pdf]**
*submitted on 2018-02-19 17:14:55*

**Authors:** Ayal Sharon

**Comments:** 11 Pages.

The Law of the Excluded Middle holds that either a statement "X" or its opposite "not X" is true. In Boolean algebra form, Y = X XOR (not X). Riemann's analytic continuation of Zeta(s) contradicts the Law of the Excluded Middle, because the Dirichlet series Zeta(s) is proven divergent in the half-plane Re(s)<=1. Further inspection of the derivation of Riemann's analytic continuation of $\zeta(s)$ shows that it is wrongly based on the Cauchy integral theorem, and thus false.

**Category:** Number Theory

[15] **viXra:1802.0268 [pdf]**
*replaced on 2018-02-20 16:56:49*

**Authors:** Phil A. Bloom

**Comments:** Pages.

For x ^ n + y ^ n = z ^ n with positive co-prime values of x, y, z and positive integral values of n, we take Fermat's last theorem (FLT) as if it were still unproven. For some value of n (n = 1, 2, at minimum) there exist positive co-prime values of r, s, t for which r ^ n + s ^ n = t ^ n, our algebraic identity, holds. These two equations directly imply other true statements, which are A : It is true that {(r s)/t} = {(x y)/z}; B : Values of (r s)/t determine uniquely values of (r, s, t) ; C : Values of (x y)/z determine uniquely values of (x, y, z) ; Hence, D : It is true that {r, s, t} = {x, y, z}. We show E : For n > 2, no positive co-prime values of (r, s, t) exist; therefore, E : For n > 2, no positive integral values of (x, y, z) exist.

**Category:** Number Theory

[14] **viXra:1802.0236 [pdf]**
*submitted on 2018-02-18 17:27:55*

**Authors:** Zeolla Gabriel martin

**Comments:** 14 Pages.

This paper develops the divisibility of the so-called Simple Primes numbers-11 (1 to 11), the discovery of a pattern to infinity, the demonstration of the Inharmonics that are 2,3,5,7 and 11 and the harmony of 1. The discovery of infinite harmony represented in fractal numbers and patterns. This is a family before the prime numbers.
The simple prime numbers-11 are known as the 13-rough numbers.

**Category:** Number Theory

[13] **viXra:1802.0213 [pdf]**
*submitted on 2018-02-17 10:38:14*

**Authors:** ANIRILASY Méleste

**Comments:** 2 Pages.

We suggest that there exists, at least, one prime number in four intervals between n² and (n+1)² for any integer n 2 such that :
all intervals are half-open;
the excluded endpoints are multiples of n;
the number of elements in each interval is equal to the least even upper bound for the biggest prime number strictly less than n.
This conjecture is a strong form of Oppermann’s one.

**Category:** Number Theory

[12] **viXra:1802.0201 [pdf]**
*submitted on 2018-02-15 12:14:36*

**Authors:** Zeolla Gabriel martin

**Comments:** 9 Pages.

This paper develops the divisibility of the so-called Simple Primes numbers-5, the discovery of a pattern to infinity, the demonstration of the inharmonics that are 2,3,5 and the harmony of 1. The discovery of infinite harmony represented in fractal numbers and patterns. This is a family before the prime numbers. This paper develops a formula to get simple prime number-5 and simple composite number-5
The simple prime numbers-5 are known as the 7-rough numbers.

**Category:** Number Theory

[11] **viXra:1802.0198 [pdf]**
*replaced on 2018-02-18 05:08:06*

**Authors:** John Yuk Ching Ting

**Comments:** 65 Pages. Targeting the General Public - Rigorous proofs for Riemann hypothesis, Polignac's and Twin prime conjectures

L-functions form an integral part of the 'L-functions and Modular Forms Database' with far-reaching implications. In perspective, Riemann zeta function is the simplest example of an L-function. Riemann hypothesis refers to the 1859 proposal by Bernhard Riemann whereby all nontrivial zeros are [mathematically] conjectured to lie on the critical line of this function. This proposal is equivalently stated in this research paper as all nontrivial zeros are [geometrically] conjectured to exactly match the 'Origin' intercepts of this function. Deeply entrenched in number theory, prime number theorem entails analysis of prime counting function for prime numbers. Solving Riemann hypothesis would enable complete delineation of this important theorem. Involving proposals on the magnitude of prime gaps and their associated sets of prime numbers, Twin prime conjecture deals with prime gap = 2 (representing twin primes) and is thus a subset of Polignac's conjecture which deals with all even number prime gaps = 2, 4, 6,... (representing prime numbers in totality except for the first prime number '2'). Both nontrivial zeros and prime numbers are Incompletely Predictable entities allowing us to employ our novel Virtual Container Research Method to solve the associated hypothesis and conjectures.

**Category:** Number Theory

[10] **viXra:1802.0176 [pdf]**
*submitted on 2018-02-14 10:10:56*

**Authors:** Philip Gibbs

**Comments:** 11 Pages.

A Diophantine m-tuple is a set of m distinct non-zero integers such that the product of any two elements of the set is one less than a square. The definition can be generalised to any commutative ring. A computational search is undertaken to find Diophantine 5-tuples (quintuples) over the ring of quadratic integers Z[√D] for small positive and negative D. Examples are found for all positive square-free D up to 22, but none are found for the complex rings including the Gaussian integers.

**Category:** Number Theory

[9] **viXra:1802.0154 [pdf]**
*replaced on 2018-02-14 09:08:19*

**Authors:** Réjean Labrie

**Comments:** 6 Pages.

Abstract: Let N, n and k be integers larger than 1. Then for all N there exists a minimum threshold k such that for n>=N, if we cut the sequence of consecutive integers from 1 to n*(n+k) into n+k slices of length n, we always find at least a prime number in each slice.
It follows that π(n*(n+k)) > π(n*(n+k-1)) > π(n*(n+k-2)) > π(n*(n+k-3))> ...> π(2n)> π(n) where π(n) is the quantity of prime numbers smaller than or equal to n.

**Category:** Number Theory

[8] **viXra:1802.0141 [pdf]**
*submitted on 2018-02-12 14:37:55*

**Authors:** Marius Coman

**Comments:** 2 Pages.

The set of Poulet numbers having only odd digits is: 1333333, 1993537, 3911197, 5351537, 5977153, 7759937, 11777599...(22 from the first 7196 Poulet numbers belong to this set). Question: is this sequence infinite? Observations: the numbers n*P + R(P) – n respectively P + n*R(P) - n, where R(P) is the reversal of P and n positive integer, are often primes. Examples: for P = 1333333, the number 1333333 + 3333331 – 1 = 4666663, prime; also 3*1333333 + 3333331 – 3 = 7333327, prime; also 5*1333333 + 3333331 – 5 = 9999991, prime. For the same P, the number 1333333 + 2*3333331 - 2 = 7999993, prime; also the number 1333333 + 4*3333331 - 4 = 14666653, prime.

**Category:** Number Theory

[7] **viXra:1802.0135 [pdf]**
*submitted on 2018-02-13 02:20:28*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In a previous paper I noticed that the numbers n*P + R(P) – n respectively P + n*R(P) - n, where P are Poulet numbers having only odd digits, R(P) the reversals of P and n positive integer, are often primes. In this paper I notice that the same is true for primes having only odd digits (see A030096 in OEIS for a list of such primes). Taken thirteen randomly chosen consecutive primes P having nine (odd) digits (from 971111137 to 971111993) I see that for all of them there exist at least a value of n smaller than 15 for which the number n*P + R(P) – n is prime (for 971111591, for instance, there exist four such values of n: 9, 11, 14, 15; for 971111137 three: 2, 4, 7; for 971111551 also three: 1, 2, 6; for 971111959 also three: 1, 9, 10; for 971111993 also three: 5, 6, 14).

**Category:** Number Theory

[6] **viXra:1802.0134 [pdf]**
*submitted on 2018-02-11 06:47:29*

**Authors:** Ricardo Gil

**Comments:** 1 Page. There are alot of collected papers at Bexar County which I submitted to the Government.

Bexar County Detention Papers
(Topological Number Theory Formula/Equation/Algorithm)
By Ricard.gil@sbcglobal.net
January 9,2017 to Pretrail (Court February 26,2018 CCC4)
The objective of this paper is to show how one can take Gigori Pereleman complex arXiv 39 page paper and make it into a simple topological formula. I am revealing the ide I had in Bexar County Detention on viXra. I want to dedicate the Bexar County Detention papers to my Ex-Wife, Eddie Gil and Ashleigh Gil. (See attached Photo) & I can always be found at 3607 Ticonderoga, San Antonio Texas, where I am a permanent guest/resident.
I. The Topological Formula/Equation/Algorithm
1D=2D=3D/1=1/1D=2D=3D

**Category:** Number Theory

[5] **viXra:1802.0097 [pdf]**
*submitted on 2018-02-08 06:40:23*

**Authors:** Jesús Álvarez Lobo

**Comments:** 3 Pages. Revista Escolar de la Olimpiada Iberoamericana de Matemática. Volume 34.

In this paper is proved an inequality involving a function of Fibonacci numbers in generic form and its limit at infinity is calculated using the asymptotic relationship given by Barr and Schooling in "The Field" (December 14, 1912).

**Category:** Number Theory

[4] **viXra:1802.0095 [pdf]**
*submitted on 2018-02-08 06:59:49*

**Authors:** Jesús Álvarez Lobo

**Comments:** 2 Pages. Spanish.

Solution to the problem PMO33.2. Problem of Mathematical Duel 08 (Olomouc, Chorzow, Graz).
Determine all triples (x, y, z) of positive integers verifying the following equation:
3 + x + y + z = xyz

**Category:** Number Theory

[3] **viXra:1802.0093 [pdf]**
*submitted on 2018-02-08 07:18:35*

**Authors:** Jesús Álvarez Lobo

**Comments:** 1 Page. Revista Escolar de la Olimpiada Iberoamericana de Matemática. Volume 21. Spanish.

Lobo's theorem for heronian triangles:
"Exists at least one heronian triangle such that two sides are consecutive natural numbers and its area is equal to n times the perimeter, for n = 1, 2, 3".
Teorema de Lobo para triángulos heronianos: Existe al menos un triángulo heroniano tal que dos de sus lados son números naturales consecutivos y su área es igual a n veces su perímetro, para n = 1, 2, 3.

**Category:** Number Theory

[2] **viXra:1802.0056 [pdf]**
*submitted on 2018-02-05 15:45:25*

**Authors:** Igor Hrnčić

**Comments:** 6 Pages.

This paper disproves the Riemann hypothesis by analyzing the integral representation of the Riemann zeta function that converges absolutely in the root-free region.

**Category:** Number Theory

[1] **viXra:1802.0039 [pdf]**
*submitted on 2018-02-05 04:50:15*

**Authors:** Andrea Prunotto

**Comments:** 5 Pages.

The condition of equiprobability among two events involving independent extractions of elements from a finite set is shown to be related to the solutions of a Diophantine equation.

**Category:** Number Theory