[25] **viXra:1803.0715 [pdf]**
*submitted on 2018-03-30 04:02:45*

**Authors:** Andrey B. Skrypnik

**Comments:** 13 Pages.

Complete solution of Queens Puzzle

**Category:** Number Theory

[24] **viXra:1803.0689 [pdf]**
*submitted on 2018-03-28 06:44:40*

**Authors:** BERKOUK Mohamed

**Comments:** 27 Pages.

et si nous essayons d'extraire les nombres composés de l'ensemble des entiers naturels , à commencer par trouver la formule qui génère tous les entiers sans les multiple de 2 et 3 ( 1er polynôme ) puis de générer les entiers sans les multiples de 2,3,5 (2eme polynôme ) le but est de trouver l’équation , la formule ou le polynôme qui ne génèrera que les NOMBRES PREMIERS...

**Category:** Number Theory

[23] **viXra:1803.0668 [pdf]**
*replaced on 2018-04-18 10:58:01*

**Authors:** Haofeng Zhang

**Comments:** 19 Pages.

In this paper the author gives an elementary mathematics method to solve Fermat's
Last Theorem (FLT), in which let this equation become an one unknown number equation, in order to solve this equation the author invented a method called “Order reducing method for equations”, where the second order root compares to one order root, and with some necessary techniques the author successfully proved when x^(n-1)+y^(n-1)- z^(n-1) <= x^(n-2)+y^(n-2)-z^(n-2) there are no positive solutions for this equation, and also proves with the increasing of x there are still no positive integer solutions for this equation when x^(n-1)+y^(n-1)- z^(n-1)<=x^(n-2)+y^(n-2)- z^(n-2) is not satisfied.

**Category:** Number Theory

[22] **viXra:1803.0654 [pdf]**
*submitted on 2018-03-25 19:15:54*

**Authors:** Zeolla Gabriel Martín

**Comments:** 5 Pages.

This paper develops the formula that calculates the sum of simple composite numbers by golden patterns.

**Category:** Number Theory

[21] **viXra:1803.0635 [pdf]**
*submitted on 2018-03-23 20:55:27*

**Authors:** Waldemar Puszkarz

**Comments:** 2 Pages.

This note presents some properties of a quadratic polynomial 13n^2 + 53n + 41. One of them is unique, while others are shared with other prime-generating quadratics. The main purpose of this note is to emphasize certain common features of such quadratics that may not have been noted before.

**Category:** Number Theory

[20] **viXra:1803.0546 [pdf]**
*submitted on 2018-03-23 10:15:03*

**Authors:** Waldemar Puszkarz

**Comments:** 3 Pages.

This note lists all the known prime-generating quadratics with at most two-digit positive coefficients that generate at least 20 primes in a row. The Euler polynomial is the best-known member of this class of six.

**Category:** Number Theory

[19] **viXra:1803.0493 [pdf]**
*submitted on 2018-03-22 22:28:25*

**Authors:** Elizabeth Gatton-Robey

**Comments:** 22 Pages.

The current mathematical consensus is that Prime numbers, those integers only divisible by one and themselves, follow no standard predictable pattern.
This body of work provides the first formula to predict prime numbers. In doing so, this proves that prime numbers follow a pattern, and proves Goldbach’s Conjecture to be true.
This is done by forming an algorithm that considers all even integers, systematically eliminates some, and the resulting subset of even integers produces all prime numbers once three is subtracted from each.

**Category:** Number Theory

[18] **viXra:1803.0362 [pdf]**
*submitted on 2018-03-21 07:57:40*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

This note presents some formulas related with pi.

**Category:** Number Theory

[17] **viXra:1803.0317 [pdf]**
*replaced on 2018-04-09 06:07:22*

**Authors:** John Atwell Moody

**Comments:** 8 Pages.

Let p(c,r,v)=e^{(c-1)(r+2v)} log({{\lambda(r+v)}\over{q(r+v)}}) log({{\lambda(v)}\over{q(v)}}),
f(c,r)=\int_{-\infty}^\infty p(c,r,v)+p(c,-r,v) dv. Let c be a real number such that 0

f(c,r)<0 and {{\partial}\over{\partial r}}f(c,r)>0 for all $r\ge 0$
while {{\partial}\over{\partial c}}f(c,r)<0 and {{\partial^2}\over {\partial c \partial r}}f(c,r)>0 for all r>0.

Then \zeta(c+i\omega) \ne 0 for all \omega.

**Category:** Number Theory

[16] **viXra:1803.0298 [pdf]**
*submitted on 2018-03-20 21:42:49*

**Authors:** Zeolla Gabriel Martin

**Comments:** 3 Pages.

This paper develops the formula that calculates the quantity of simple composite numbers that exist by golden patterns.

**Category:** Number Theory

[15] **viXra:1803.0289 [pdf]**
*submitted on 2018-03-21 03:18:10*

**Authors:** Bado idriss olivier

**Comments:** 6 Pages.

In this paper we are going to give the proof of Goldbach conjecture by introducing a new lemma which implies Goldbach conjecture .By using Chebotarev-Artin theorem , Mertens formula and Poincare sieve we establish the lemma

**Category:** Number Theory

[14] **viXra:1803.0265 [pdf]**
*replaced on 2018-04-09 12:03:56*

**Authors:** Yuri Heymann

**Comments:** 24 Pages.

In the present study we used the Dirichlet eta function as an extension of the Riemann zeta function in the strip Re(s) in ]0, 1[. We then determined the domain of admissible complex zeros of the Riemann zeta function in this strip using minimal constraints and alternative series of power functions. While proving the uniqueness of the line Re(s) = 1/2 in the strip Re(s) in ]0, 1[, we obtained the value of the Dirichlet eta function evaluated at the point s = 1/2 which was fortuitous. We also checked for zeros outside this strip. We found that the admissible domain of complex zeros excluding the trivial zeros is the critical line given by Re(s) = 1/2 as stated in the Riemann hypothesis.

**Category:** Number Theory

[13] **viXra:1803.0225 [pdf]**
*submitted on 2018-03-15 20:17:04*

**Authors:** Zeolla Gabriel martin

**Comments:** 4 Pages.

This paper develops the formula that calculates the sum of simple prime numbers by golden pattern.

**Category:** Number Theory

[12] **viXra:1803.0219 [pdf]**
*submitted on 2018-03-16 05:37:57*

**Authors:** Huseyin Ozel

**Comments:** 44 Pages.

The existing definition of imaginary numbers is solely based on the fact that certain mathematical operation, square operation, would not yield certain type of outcome, negative numbers; hence such operational outcome could only be imagined to exist. Although complex numbers actually form the largest set of numbers, it appears that almost no thought has been given until now into the full extent of all possible types of imaginary numbers. A close look into what further non-existing numbers could be imagined help reveal that we could actually expand the set of imaginary numbers, redefine complex numbers, as well as define imaginary and complex mathematical objects other than merely numbers.

**Category:** Number Theory

[11] **viXra:1803.0192 [pdf]**
*submitted on 2018-03-14 02:45:45*

**Authors:** Andrea Prunotto

**Comments:** 4 Pages.

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

**Category:** Number Theory

[10] **viXra:1803.0179 [pdf]**
*submitted on 2018-03-12 18:18:40*

**Authors:** Morgan Osborne

**Comments:** 22 Pages. Keywords: Beal, Diophantine, Continuity (2010 MSC: 11D99, 11D41)

The Beal Conjecture considers positive integers A, B, and C having respective positive integer exponents X, Y, and Z all greater than 2, where bases A, B, and C must have a common prime factor. Taking the general form A^X + B^Y = C^Z, we explore a small opening in the conjecture through reformulation and substitution to create two new variables. One we call 'C dot' representing and replacing C and the other we call 'Z dot' representing and replacing Z. With this, we show that 'C dot' and 'Z dot' are separate continuous functions, with argument (A^X + B^Y), that achieve all positive integers during their continuous non-constant rates of infinite ascent. Possibilities for each base and exponent in the reformulated general equation A^X +B^Y = ('C dot')^('Z dot') are examined using a binary table along with analyzing user input restrictions and 'C dot' values relative to A and B. Lastly, an indirect proof is made, where conclusively we find the continuity theorem to hold over the conjecture.

**Category:** Number Theory

[9] **viXra:1803.0178 [pdf]**
*replaced on 2018-03-16 15:50:40*

**Authors:** Zeolla Gabriel Martin

**Comments:** 3 Pages.

This paper develops the formula that calculates the quantity of simple prime numbers that exist by golden patterns.

**Category:** Number Theory

[8] **viXra:1803.0171 [pdf]**
*submitted on 2018-03-12 09:55:34*

**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. The analysis is performed upon the well known inverse Mellin transform of zeta, that defines the Mertens function. The contour of integration is taken arbitrarily close to the nontrivial roots, and then only arbitrarily small parts of the integrand that are infinitely close to the nontrivial roots on such contour are analyzed. The convergence of the integral at hand then implies a result that a series over the derivative of zeta and over nontrivial roots closest to the roots free region converges. This result is in a contradiction with the well known result that the very same series, when taken over the critical line and under the truth of the Riemann hypothesis, diverges. This disproves the Riemann hypothesis.

**Category:** Number Theory

[7] **viXra:1803.0150 [pdf]**
*submitted on 2018-03-10 16:37:39*

**Authors:** Pedro Caceres

**Comments:** 21 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:
ζ(z)=∑_(k=1)^∞ k^(-z)
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 ζ(z) 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 new propositions to advance in the knowledge of the Riemann Zeta function.
a) A constant C that can be used to express ζ(2n+1)≡a/b*C^(2n+1)
b) An approximation to the values of ζ(s) in R given by ζ(s)=1/(1-π^(-s)-2^(-s))
c) A theorem that states that the infinite sums ∑_(j=1)^∞[ζ(u*k±n)-ζ(v*k±m)] converge
to a value in the interval (-1,1) for all u≥1,v≥1,n,m ∈N such that (u*k±n)>1 and
(v*k±m)>1 for all j∈N
d) A new set of constants CZ_(u,n,v,m)calculated from infinite sums involving ζ(z)
e) A function in C2(x,a,b)= 2*x^(-a)*(∑_(j=1)^(x-1) [j^(-a)*cos(b*(ln(x/j)))]) in R with zeros in(a,b) with a=1/2 and b=Im(z*), with z*=non-trivial zero of ζ(z).
f) A C-transformation that allows for a decomposition of ζ(z) that can be used to study
the Riemann Hypothesis.
g) Linearization of the Harmonic function using Non-Trivial zeros of ζ(z).
h) An expression that links any two Non-Trivial zeros of ζ(z).

**Category:** Number Theory

[6] **viXra:1803.0121 [pdf]**
*submitted on 2018-03-09 10:43:27*

**Authors:** Zeolla Gabriel martin

**Comments:** 5 Pages.

This paper develops the construction of the Golden Patterns for different prime divisors, the discovery of patterns towards infinity. The discovery of infinite harmony represented in fractal numbers and patterns. The golden pattern works with the simple prime numbers that are known as rough numbers and simple composite number.

**Category:** Number Theory

[5] **viXra:1803.0110 [pdf]**
*submitted on 2018-03-08 06:44:51*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

This note presents some trigonometric formulas that involving nested radicals.

**Category:** Number Theory

[4] **viXra:1803.0108 [pdf]**
*submitted on 2018-03-08 08:14:15*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2017-2018 by Colin James III All rights reserved.

This is the briefest known such proof, and in mathematical logic.

**Category:** Number Theory

[3] **viXra:1803.0105 [pdf]**
*submitted on 2018-03-07 21:22:05*

**Authors:** Henry Göttler, Chantal Göttler, Heinrich Göttler, Thorsten Göttler, Pei-jung Wu

**Comments:** 7 Pages. Proof of Collatz Conjecture

Over 80 years ago, the German mathematician Lothar Collatz formulated an interesting mathematical problem, which he was afraid to publish, for the simple reason that he could not solve it. Since then the Collatz Conjecture has been around under several names and is still unsolved, keeping people addicted. Several famous mathematicians including Richard Guy stating “Dont try to solve this problem”. Paul Erd¨os even said ”Mathematics is not yet ready for such problems” and Shizuo Kakutani joked that the problem was a Cold War invention of the Russians meant to slow the progress of mathematics in the West. We might have ﬁnally freed people from this addiction.

**Category:** Number Theory

[2] **viXra:1803.0098 [pdf]**
*submitted on 2018-03-07 09:05:15*

**Authors:** Zeolla Gabriel martin

**Comments:** 6 Pages.

This paper develops the divisibility of the so-called Simple Primes numbers-3, the discovery of a pattern to infinity, the demonstration of the inharmonics that are 2,3, 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-3 and simple composite number-3
The simple prime numbers-3 is known as the 5-rough numbers.

**Category:** Number Theory

[1] **viXra:1803.0017 [pdf]**
*replaced on 2018-04-03 12:13:21*

**Authors:** Pablo Hernan Pereyra

**Comments:** 4 Pages.

A discrete condition for twin prime numbers is established by Wilson's theorem. By synchronization is obtained a linear diophantine equation that implies by Bertrand Chebyshev's theorem the existence of infinite twin prime numbers.

**Category:** Number Theory