**Previous months:**

2007 - 0703(3) - 0706(2)

2008 - 0807(1) - 0809(1) - 0810(1) - 0812(2)

2009 - 0901(2) - 0904(2) - 0907(2) - 0908(4) - 0909(1) - 0910(2) - 0911(1) - 0912(1)

2010 - 1001(3) - 1002(1) - 1003(55) - 1004(50) - 1005(36) - 1006(7) - 1007(11) - 1008(16) - 1009(21) - 1010(8) - 1011(7) - 1012(13)

2011 - 1101(14) - 1102(7) - 1103(13) - 1104(3) - 1105(1) - 1106(2) - 1107(1) - 1108(2) - 1109(2) - 1110(5) - 1111(4) - 1112(4)

2012 - 1201(2) - 1202(7) - 1203(6) - 1204(6) - 1205(7) - 1206(6) - 1207(5) - 1208(5) - 1209(11) - 1210(14) - 1211(10) - 1212(4)

2013 - 1301(5) - 1302(9) - 1303(16) - 1304(15) - 1305(12) - 1306(12) - 1307(25) - 1308(11) - 1309(8) - 1310(13) - 1311(15) - 1312(21)

2014 - 1401(20) - 1402(10) - 1403(26) - 1404(10) - 1405(17) - 1406(20) - 1407(33) - 1408(51) - 1409(47) - 1410(16) - 1411(16) - 1412(18)

2015 - 1501(14) - 1502(14) - 1503(33) - 1504(23) - 1505(18) - 1506(12) - 1507(15) - 1508(14) - 1509(14) - 1510(11) - 1511(9) - 1512(25)

2016 - 1601(14) - 1602(17) - 1603(77) - 1604(54) - 1605(28) - 1606(17) - 1607(19) - 1608(16) - 1609(22) - 1610(22) - 1611(12) - 1612(19)

2017 - 1701(19) - 1702(24) - 1703(25) - 1704(32) - 1705(25) - 1706(25) - 1707(21) - 1708(26) - 1709(17) - 1710(26) - 1711(25) - 1712(34)

2018 - 1801(34) - 1802(23) - 1803(21)

Any replacements are listed farther down

[1727] **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

[1726] **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

[1725] **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

[1724] **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

[1723] **viXra:1803.0317 [pdf]**
*submitted on 2018-03-19 20:47:26*

**Authors:** John Atwell Moody

**Comments:** 2 Pages.

Conjecture: d/dc of the magnitude of the integral of e^{(c-1+iw)t} times log(\lambda/q)dt is <0 when c\in (0,1/2) and w>0.

Theorem: The conjecture implies Riemann's hypothesis.

**Category:** Number Theory

[1722] **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

[1721] **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

[1720] **viXra:1803.0265 [pdf]**
*submitted on 2018-03-19 06:45:58*

**Authors:** Yuri Heymann

**Comments:** 20 Pages.

In the present study we use the Dirichlet eta function as an extension of the Riemann zeta function in the strip Re(s) in ]0, 1[. We then determine the domain of admissible complex zeros of the Riemann zeta function in this strip using the symmetries of the function and minimal constraints. We also check for zeros outside this strip. We nd 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

[1719] **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

[1718] **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

[1717] **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

[1716] **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

[1715] **viXra:1803.0178 [pdf]**
*submitted on 2018-03-12 18:15:11*

**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

[1714] **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

[1713] **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

[1712] **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

[1711] **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

[1710] **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

[1709] **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

[1708] **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

[1707] **viXra:1803.0017 [pdf]**
*submitted on 2018-03-01 10:12:24*

**Authors:** Pablo Hernan Pereyra

**Comments:** 3 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

[1706] **viXra:1802.0433 [pdf]**
*submitted on 2018-02-28 20:58:26*

**Authors:** Clive Jones

**Comments:** 2 Pages.

Featuring the PF5 Function

**Category:** Number Theory

[1705] **viXra:1802.0427 [pdf]**
*submitted on 2018-02-28 07:01:31*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

This note presents some formulas related with Dottie number.

**Category:** Number Theory

[1704] **viXra:1802.0395 [pdf]**
*submitted on 2018-02-26 20:37:20*

**Authors:** Radomir Majkic

**Comments:** 15 Pages.

Abstract: Goldbach's conjectures are inseparable and both of them stem from an underlying fundamental structure of the natural numbers.Thus,
one of them must consistently imply the other one, and the Goldbach's weak conjecture must imply the Goldbach's strong conjecture. Finally, all natural
numbers are the Goldbach's numbers.

**Category:** Number Theory

[1703] **viXra:1802.0363 [pdf]**
*submitted on 2018-02-26 10:46:52*

**Authors:** Zeolla Gabriel martin

**Comments:** 8 Pages.

This paper develops the divisibility of the so-called Simple Primes numbers-13, the discovery of a pattern to infinity, the demonstration of the inharmonics that are 2,3,5,7,11,13 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-13 and simple composite number-13
The simple prime numbers-13 are known as the 17-rough numbers.

**Category:** Number Theory

[1702] **viXra:1802.0353 [pdf]**
*submitted on 2018-02-25 04:25:14*

**Authors:** Dave Ryan T. Cariño

**Comments:** 12 Pages.

Function and method for solving the distribution of prime numbers accurately using the combination of step functions, polynomial functions, inverse functions and continuous functions. Equation 〖lim┬(n→∞) p(n)〗〖={2(n+x_p )+3|x_p=x_3+x_5+x_7+x_11+⋯x_p }〗 is true for all integer where for the distribution and generation of exact values of prime numbers without exception. This formula is efficient by means of modern supercomputers for the task of adding new expression for .

**Category:** Number Theory

[1701] **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

[1700] **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

[1699] **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

[1698] **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

[1697] **viXra:1802.0268 [pdf]**
*submitted on 2018-02-19 17:08:03*

**Authors:** Phil A. Bloom

**Comments:** Pages.

For x ^ n + y ^ n = z ^ n with positive co-prime x, y, z and positive integral n, we take Fermat's last theorem (FLT) as if still unproven. For some value of n (n = 1,2, at minimum) there exist positive co-prime r, s, t for which r ^ n + s ^ n = t ^ n, our algebraic identity, holds. These two equations directly imply other true statements : (r s)/t determines uniquely (r, s, t); (x y)/z determines uniquely (x, y, z); {(r s)/t} = {(x y)/z}; So, {r, s, t}={x, y, z}. We show, for n > 2, that there exists no co-prime (r, s, t). Hence, for n > 2, no co-prime (x, y, z) exists. Thus, for n > 2, no integral (x, y, z) exists.

**Category:** Number Theory

[1696] **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

[1695] **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

[1694] **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

[1693] **viXra:1802.0198 [pdf]**
*submitted on 2018-02-15 14:55:26*

**Authors:** John Yuk Ching Ting

**Comments:** 67 Pages. This research paper contains 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

[1692] **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

[1691] **viXra:1802.0154 [pdf]**
*submitted on 2018-02-13 22:51:29*

**Authors:** Réjean Labrie

**Comments:** 6 Pages.

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

[1690] **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

[1689] **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

[1688] **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

[1687] **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

[1686] **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

[1685] **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

[1684] **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

[1683] **viXra:1801.0416 [pdf]**
*submitted on 2018-01-30 10:29:06*

**Authors:** Timothy W. Jones

**Comments:** 3 Pages. This suggests an entirely different angle on traditional number theory.

A phenomenon is described for analytic number theory. The purpose is to coordinate number theory and to give it a specific goal of modeling the phenomenon.

**Category:** Number Theory

[1682] **viXra:1801.0381 [pdf]**
*submitted on 2018-01-27 16:15:50*

**Authors:** Zeolla Gabriel martin

**Comments:** 5 Pages.

This article develops a formula for calculating the simple prime numbers-7 and the simple composite numbers-7 of the Golden Pattern.

**Category:** Number Theory

[1681] **viXra:1801.0365 [pdf]**
*submitted on 2018-01-26 05:18:33*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I present seven sequences of Poulet numbers selected by some properties of their digits product: (1) - (5) Poulet numbers for which the product of their digits is equal to (1) q^2 – 1, where q prime; (2) q^2 – 9, where q prime; (3) 9*q^2 – 9, where q prime; (4) 2^n, where n natural; (5) Q – 1, where Q is also a Poulet number and (6) – (7) Poulet numbers divisible by 5 for which the product of their digits taken without the last one is equal to (6) q^2 – 1, where q prime; (7) Q – 1, where Q is also a Poulet number. Finally, I conjecture that all these seven sequences have an infinity of terms.

**Category:** Number Theory

[1680] **viXra:1801.0341 [pdf]**
*submitted on 2018-01-26 02:58:03*

**Authors:** Predrag Terzic

**Comments:** 8 Pages.

Theorems and conjectures about prime numbers .

**Category:** Number Theory

[1679] **viXra:1801.0311 [pdf]**
*submitted on 2018-01-24 11:27:10*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In a previous paper I presented seven sequences of numbers of the form 2*k*P – (30 + 290*n)*k – 315, where P is Poulet number, and I conjectured that two of them have all the terms odd abundant numbers and the other five have an infinity of terms odd abundant numbers. Because it is known that all the smaller numbers of amicable pairs are abundant numbers (see A002025 in OEIS), in this paper I revert the relation from above and I conjecture that all Poulet numbers P divisible by 5 can be written as P = (A + 315 + (30 + 290*n)*k)/(2*k), where A is a smaller of an amicable pair and n and k naturals. For example: 645 = (12285 + 315 + 30*10)/(2*10); also 1105 = (12285 + 315 + 2060*84)/(2*84) or 1105 = (69615 + 315 + 320*37)/(2*37). Note that for the first 17 such Poulet numbers there exist at least a combination [n, k] for A = 12285, the first smaller of an amicable pair divisible by 5!

**Category:** Number Theory

[1678] **viXra:1801.0310 [pdf]**
*submitted on 2018-01-23 17:13:34*

**Authors:** Jean BENICHOU

**Comments:** 1 Page. jean.benichou@icloud.com

All curves defined by x^n + y^n = z^n should intersect the circle x^2 + y^2 but are contained in it and no common point exists when x, y, z, n are integers. This contradiction forbid x^n + y^n = z^n for n>2.

**Category:** Number Theory

[1677] **viXra:1801.0297 [pdf]**
*submitted on 2018-01-23 08:51:22*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I present seven sequences of numbers of the form 2*k*P – (30 + 290*n)*k – 315, where P is Poulet number and n and k naturals; I conjecture that two of them have all the terms odd abundant numbers (corresponding to [P, n] = [645, 0] and [1105, 1]) and the other five (corresponding to [P, n] = [11305, 4], [16705, 13], [11305, 25], [10585, 28] and [16705, 34]) have an infinity of terms odd abundant numbers.

**Category:** Number Theory

[1676] **viXra:1801.0295 [pdf]**
*submitted on 2018-01-23 09:57:45*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I present three sequences of numbers of the form (4*k + 2)*P + n*(2002*k + 1001), where P is Poulet number, k natural and n integer (corresponding to [P, n] = [41041, -1], [101101, 5] and [401401, 35]); I conjecture that they have all the terms odd abundant numbers.

**Category:** Number Theory

[1675] **viXra:1801.0257 [pdf]**
*submitted on 2018-01-20 10:14:22*

**Authors:** Timothy W. Jones

**Comments:** 3 Pages.

In this article we derive the values of zeta(2) and zeta(2n) using Euler's original insights.

**Category:** Number Theory

[1674] **viXra:1801.0256 [pdf]**
*submitted on 2018-01-20 12:23:10*

**Authors:** Ilija Barukčić

**Comments:** 14 pages. Open Letter To Professor Saburou Saitoh. Copyright © 2017 by Ilija Barukčić, Jever, Germany. Published by:

Abstract
Objective: Accumulating evidence indicates that zero divided by zero equal one. Still it is not clear what number theory is saying about this.
Methods: To explore relationship between the problem of the division of zero by zero and number theory, a systematic approach is used while analyzing the relationship between number theory and independence.
Result: The theorems developed in this publication support the thesis that zero divided by zero equals one. It is possible to define the law of independence under conditions of number theory.
Conclusion: The findings of this study suggest that zero divided by zero equals one.
Keywords
Zero, One, Zero divide by zero, Independence, Number theory

**Category:** Number Theory

[1673] **viXra:1801.0219 [pdf]**
*submitted on 2018-01-17 16:29:46*

**Authors:** Haofeng Zhang

**Comments:** 6 Pages.

This paper proves that equation 4/(x+y+z)=1/x+1/y+1/z has no positive integer solutions
using the method of solving third order equation.

**Category:** Number Theory

[1672] **viXra:1801.0193 [pdf]**
*submitted on 2018-01-16 07:00:17*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

In this note we briefly explore the Ramanujan's cubic continued fraction.

**Category:** Number Theory

[1671] **viXra:1801.0190 [pdf]**
*submitted on 2018-01-16 07:44:06*

**Authors:** Ryujin Choe

**Comments:** 13 Pages.

Proof of Goldbach's conjecture and twin prime conjecture

**Category:** Number Theory

[1670] **viXra:1801.0187 [pdf]**
*submitted on 2018-01-16 14:33:02*

**Authors:** Juan Moreno Borrallo

**Comments:** 6 Pages.

In this paper it is proposed a conjecture of existence of prime numbers on a particular arithmetic progression, and demonstrated a particular case.

**Category:** Number Theory

[1669] **viXra:1801.0182 [pdf]**
*submitted on 2018-01-17 06:15:54*

**Authors:** Haofeng Zhang

**Comments:** 30 Pages.

In this paper for equation Ax^m+By^n=Cz^k , where m,n,k > 2, x,y,z > 1, A,B,C≥1 and
gcd(Ax,By,Cz)=1, the author proved there are no positive integer solutions for this equation using“Order reducing method for equations” that the author invented for solving high order equations,in which let the equation become two equations, through comparing the two roots to prove there are no positive integer solutions for this equation.

**Category:** Number Theory

[1668] **viXra:1801.0165 [pdf]**
*submitted on 2018-01-15 00:42:03*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following three conjectures: (I) All numbers of the form 2*k*645 – (345 + 30*(k – 1)), where k natural, are odd abundant numbers; the sequence of these numbers is 945, 2205, 3465, 4725, 5985, 7245, 8505, 9765...(II) All numbers of the form 2*k*1905 – (345 + 30*(k – 1)), where k natural, are odd abundant numbers; the sequence of these numbers is 3465, 7245, 11025, 14805, 18585, 22365, 26145, 29925...(III) There exist an infinity of Poulet numbers P such that all the numbers 2*k*P – (345 + 30*(k – 1)), where k natural, are odd abundant numbers.

**Category:** Number Theory

[1667] **viXra:1801.0164 [pdf]**
*submitted on 2018-01-15 03:28:23*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following four conjectures: (I) All numbers of the form 2*k*41041 – 1001*k, where k odd, are odd abundant numbers; the sequence of these numbers is 81081, 243243, 405405, 567567, 729729, 891891, 1054053, 1216215...(II) All numbers of the form 2*k*101101 + 5005*k, where k odd, are odd abundant numbers; the sequence of these numbers is 207207, 621621, 1036035, 1450449, 1864863, 2279277, 2693691, 3108105...(III) There exist an infinity of Poulet numbers P such that all the numbers 2*k*P – 1001*k, where k odd, are odd abundant numbers; (IV) There exist an infinity of Poulet numbers P such that all the numbers 2*k*P + 5005*k, where k odd, are odd abundant numbers.

**Category:** Number Theory

[1666] **viXra:1801.0161 [pdf]**
*submitted on 2018-01-14 05:43:59*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following observation: there exist palindromic abundant numbers P such that n = P – q^2 + 1 is an abundant number for any q prime, q ≥ 5 (of course, for q^2 < P + 1). The first such P is the first palindromic abundant number 66 (with corresponding [q, n] = [5, 42], [7, 18]. Another such palindromic abundant numbers are 222, 252, 282, 414, 444, 474, 606, 636, 666. Up to 666, the palindromic abundant numbers 88, 272, 464, 616 don’t have this property. Questions: are there infinite many such palindromic abundant numbers? What other sets of integers have this property beside palindromic abundant numbers?

**Category:** Number Theory

[1665] **viXra:1801.0153 [pdf]**
*submitted on 2018-01-14 03:08:57*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following observation: there exist Poulet numbers P such that n = P – q^2 is an abundant number for any q prime, q ≥ 5 (of course, for q^2 < P). The first such P is 1105 (with corresponding [q, n] = [5, 1080], [7, 1056], [11, 984], [13, 936], [17, 816], [19, 744], [23, 576], [29, 264], [31, 144]). Another such Poulet numbers are 1387, 1729, 2047, 2701, 2821. Up to 2821, the Poulet numbers 341, 561, 645, 1905, 2465 don’t have this property. Questions: are there infinite many such Poulet numbers? What other sets of integers have this property beside Poulet numbers?

**Category:** Number Theory

[1664] **viXra:1801.0151 [pdf]**
*submitted on 2018-01-13 08:02:40*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following two conjectures: (I) There exist an infinity of Poulet numbers P such that D + R(D), where R(D) is the number obtained reversing the digits of D which is the number obtained concatenating the prime factors of P, is a palindromic number (example: such a Poulet number is P = 12801; the prime factors of 12801 are 3, 17 and 251, then D = 317251 and D + R(D) = 317251 + 152713 = 469964, a palindromic number); (II) There is no a number obtained concatenating the prime factors of a Poulet number to be a Lychrel number.

**Category:** Number Theory

[1663] **viXra:1801.0140 [pdf]**
*submitted on 2018-01-12 09:11:07*

**Authors:** Timothy W. Jones

**Comments:** 8 Pages. This does generalize to all zeta(n>1), but the necessary inequality for the general case is difficult.

We prove that a partial sum of zeta(2)-1=z is not given by any single decimal in a number base given by a denominator of its terms. This result, applied to all partials, shows that there are an infinite number of partial sums in one interval of the form [.(x-1),.x] where .x is single decimal in a number base of the denominators of the terms of z. We show that z is contained in an open interval inside [.(x-1),.x]. As all possible rational values of z are in these intervals, z must be irrational.

**Category:** Number Theory

[1662] **viXra:1801.0138 [pdf]**
*submitted on 2018-01-12 11:09:28*

**Authors:** Preininger Helmut

**Comments:** 11 Pages.

This paper is an appendix of Natural Squarefree Numbers: Statistical Properties [PR04]. In this appendix we calculate the probability of c is squarefree, where c=a*b, a is an element of the set X and b is an element of the set Y.

**Category:** Number Theory

[1661] **viXra:1801.0118 [pdf]**
*submitted on 2018-01-10 11:13:58*

**Authors:** MENDZINA ESSOMBA François

**Comments:** 03 Pages.

a new continuous fractions...

**Category:** Number Theory

[1660] **viXra:1801.0093 [pdf]**
*submitted on 2018-01-08 09:25:55*

**Authors:** Zeolla Gabriel Martin

**Comments:** 10 Pages.

This paper develops a modified an old and well-known expression for calculating and obtaining all prime numbers greater than three, composite numbers and all twin prime numbers greater than three. The conditioning (n) will be the key to make the formula work.

**Category:** Number Theory

[1659] **viXra:1801.0087 [pdf]**
*submitted on 2018-01-07 17:15:09*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following observation: the number n = p – q, where p and q are Poulet numbers, needs very few iterations of “reverse and add” to reach a palindrome. For instance, taking q = 1729 and p = 999986341201, it can be seen that only 3 iterations are needed to reach a palindrome: n = 999986341201 – 1729 = 999986339472 and we have: 999986339472 + 274933689999 = 1274920029471; 1274920029471 + 1749200294721 = 3024120324192 and 3024120324192 + 2914230214203 = 5938350538395, a palindromic number. So, relying on this, I conjecture that there exist an infinity of n, even considering q and p successive, that need just one such iteration to reach a palindrome (see sequence A015976 in OEIS for these numbers) and I also conjecture that there is no a difference between two Poulet numbers to be a Lychrel number.

**Category:** Number Theory

[1658] **viXra:1801.0082 [pdf]**
*submitted on 2018-01-08 02:20:51*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following observation: the number n = p^2 – q^2, where p and q are primes, needs very few iterations of “reverse and add” to reach a palindrome. For instance, taking q = 563 and p = 104723, it can be seen that only 3 iterations are needed to reach a palindrome: n = 104723^2 – 563^6 = 10966589760 and we have: 10966589760 + 6798566901 = 17765156661; 17765156661 + 16665156771 = 34430313432 and 34430313432 + 23431303443 = 57861616875, a palindromic number. So, relying on this, I conjecture that there exist an infinity of n, even considering q and p successive, that need just one such iteration to reach a palindrome (see sequence A015976 in OEIS for these numbers) and I also conjecture that there is no a difference between two squares of primes to be a Lychrel number.

**Category:** Number Theory

[1657] **viXra:1801.0080 [pdf]**
*submitted on 2018-01-07 05:12:33*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following two conjectures: (I) There exist an infinity of Poulet numbers P such that (P + 4*196) + R(P + 4*196), where R(n) is the number obtained reversing the digits of n, is a palindromic number; note that I wrote 4*196 instead 784 because 196 is a number known to be related with palindromes: is the first Lychrel number, which gives the name to the “196-algorithm”; (II) For every Poulet number P there exist an infinity of primes q such that the number (P + 16*q^2) + R(P + 16*q^2) is a palindrome. The three sequences (presumed infinite by the conjectures above) mentioned in title of the paper are: (1) Palindromes of the form (P + 4*196) + R(P + 4*196), where P is a Poulet number; (2) Palindromes of the form (P + 16*q^2) + R(P + 16*q^2), where P is a Poulet number and q the least prime for which is obtained such a palindrome; (3) Palindromes of the form (1729 + 16*q^2) + R(1729 + 16*q^2), where q is prime (1729 is a well known Poulet number).

**Category:** Number Theory

[1656] **viXra:1801.0078 [pdf]**
*submitted on 2018-01-07 07:13:35*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following two conjectures: (I) There exist an infinity of squares of primes p^2 such that (p^2 + 4*196) + R(p^2 + 4*196), where R(n) is the number obtained reversing the digits of n, is a palindromic number; note that I wrote 4*196 instead 784 because 196 is a number known to be related with palindromes: is the first Lychrel number, which gives the name to the “196-algorithm”; (II) For every square of odd prime p^2 there exist an infinity of primes q such that the number (p^2 + 16*q^2) + R(p^2 + 16*q^2) is a palindrome. The three sequences (presumed infinite by the conjectures above) mentioned in title of the paper are: (1) Palindromes of the form (p^2 + 4*196) + R(p^2 + 4*196), where p^2 is a square of prime; (2) Palindromes of the form (p^2 + 16*q^2) + R(p^2 + 16*q^2), where p^2 is a square of prime and q the least prime for which is obtained such a palindrome; (3) Palindromes of the form (13^2 + 16*q^2) + R(13^2 + 16*q^2), where q is prime.

**Category:** Number Theory

[1655] **viXra:1801.0070 [pdf]**
*submitted on 2018-01-06 07:15:45*

**Authors:** Timothy W. Jones

**Comments:** 2 Pages. It might help to read "Visualizing Zeta(n>1) and Proving Its Irrationality" by the same author.

In a universe with meteorites on concentric circles equally spaced, spaceships can avoid collisions by every smaller increments of their trajectories. Using this idea, a story conveys the sense that Zeta increments avoid all meteorites and thus converge to an irrational number.

**Category:** Number Theory

[1654] **viXra:1801.0068 [pdf]**
*submitted on 2018-01-06 09:26:17*

**Authors:** Haofeng Zhang

**Comments:** 13 Pages.

Abstract
: In this paper the author gives a simplest elementary mathematics method to solve the
famous Fermat's Last Theorem(FLT), in which let
this equation become a 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 su
ccessfully proved that there are
no positive integer solutions for x^n+y^n=z^n
which means FLT has been proved by elementary mathematics.

**Category:** Number Theory

[1653] **viXra:1801.0065 [pdf]**
*submitted on 2018-01-05 06:35:44*

**Authors:** Zeolla Gabriel martin

**Comments:** 27 Pages.

This paper develops a modified an old and well-known expression for calculating and obtaining all prime numbers greater than three and composite numbers divisible by numbers greater than three. This paper develops formulas to break down the prime numbers and the composite numbers in their reductions, these formulas based on equalities allow to regroup them according to congruence characteristics.

**Category:** Number Theory

[1652] **viXra:1801.0064 [pdf]**
*submitted on 2018-01-05 06:38:16*

**Authors:** Zeolla Gabriel martin

**Comments:** 7 Pages.

This paper develops the divisibility of the so-called Simple Primes numbers (1 to 9), the discovery of a pattern to infinity, the demonstration of the Inharmonics that are 2,3,5,7 and the harmony of 1. The discovery of infinite harmony represented in fractal numbers and patterns. This is a family before the prime numbers.

**Category:** Number Theory

[1651] **viXra:1801.0063 [pdf]**
*submitted on 2018-01-05 06:43:09*

**Authors:** Zeolla Gabriel martin

**Comments:** 8 Pages.

This paper develops a modified an old and well-known expression for calculating and obtaining all prime numbers greater than three and composite numbers divisible by numbers greater than 3. The key for this formula to work correctly is in the equalities and inequalities. These equalities and inequalities are created from the uncovering of the patterns of the composite numbers. The composite numbers follow very clear and determining patterns, making it possible to find them through a formula.

**Category:** Number Theory

[1650] **viXra:1801.0006 [pdf]**
*submitted on 2018-01-01 20:57:30*

**Authors:** A. Polorovskii

**Comments:** 6 Pages.

Let |l| ⊂ ℵ0. In [19], the authors extended manifolds. We show that F ≤ M. In this context, the
results of [10] are highly relevant. It is not yet known whether there exists a Fourier additive polytope,
although [10] does address the issue of uniqueness.

**Category:** Number Theory

[1649] **viXra:1801.0001 [pdf]**
*submitted on 2018-01-01 12:02:19*

**Authors:** Leonhard Schuster

**Comments:** 13 Pages.

We investigate Riemann's Zeta function, as $(s-1)\zeta(s)$, under the M{\"o}bius transformation $s = \frac{1}{1-z}$ which maps the half plane right to the critical strip to the unit disk. Application of a generalized Poisson-Jensen formula (due to Nevanlinna) shows that the investigated function has only a finite number of zeros in the interior of the unit disk. We show, that the Li coefficients $\lambda_n = \sum_\rho (1-(1-1/\rho)^n)$ are positive for $n> 10^{24}$, and discuss consequences.

**Category:** Number Theory

[1648] **viXra:1712.0679 [pdf]**
*submitted on 2017-12-31 06:29:29*

**Authors:** Julian Beauchamp

**Comments:** 1 Page.

In this paper, we reveal a new binomial formula that expresses the sum of, or difference between two powers, a^x \pm b^y, as a binomial expansion of a single power, z. Like the standard binomial formula it includes the normal binomial coefficients, factors and indices, but includes an additional non-standard factor. The new formula (with an upper index z) mimics a standard binomial formula (to the power z) without the value of the binomial expansion changing even when z itself changes. This has exciting implications for certain diophantine equations. This short paper simply highlights its existence.

**Category:** Number Theory

[1647] **viXra:1712.0669 [pdf]**
*submitted on 2017-12-30 17:43:18*

**Authors:** Arthur Shevenyonov

**Comments:** 17 Pages. ABC conjecture

ABC conjecture and beyond, with cross-operational linkage hinting at broader convergence

**Category:** Number Theory

[1646] **viXra:1712.0662 [pdf]**
*submitted on 2017-12-29 15:51:45*

**Authors:** Julian Beauchamp

**Comments:** 9 Pages.

In psychology, the Chameleon Effect describes how an animal's behaviour can adapt to, or mimic, its environment through non-conscious mimicry. In the first part of this paper, we show how $a^x - b^y$ can be expressed as a binomial expansion (with an upper index, $z$) that, like a chameleon, mimics a standard binomial formula (to the power $z$) without its own value changing even when $z$ itself changes. In the second part we will show how this leads to a proof for the Beal Conjecture. We finish by outlining how this method can be applied to a more generalised form of the equation.

**Category:** Number Theory

[1645] **viXra:1712.0660 [pdf]**
*submitted on 2017-12-29 05:27:48*

**Authors:** Victor Sorokine

**Comments:** 3 Pages.

The proof is based on studying digits in the endings of different numbers in Fermat's equation.

**Category:** Number Theory

[1644] **viXra:1712.0656 [pdf]**
*submitted on 2017-12-29 08:47:39*

**Authors:** Zhang Tianshu

**Comments:** 24 Pages.

Positive integers which are able to be operated to 1 by the leftwards operational rule and generating positive integers which start with 1 to operate by the rightwards operational rule are one-to-one correspondence and the same. So, we refer to the bunch of integers’ chains to apply the mathematical induction, next classify positive integers to get comparable results via operations, such that finally summarize out a proof at substep according to beforehand prepared two theorems as judgmental criteria.

**Category:** Number Theory

[1643] **viXra:1712.0653 [pdf]**
*submitted on 2017-12-29 12:07:03*

**Authors:** Victor Sorokine

**Comments:** 2 Pages. Russian version

We study the digits at the end of different numbers in the Fermat's Equality, and arrive to a contradiction.

**Category:** Number Theory

[1642] **viXra:1712.0641 [pdf]**
*submitted on 2017-12-28 10:19:41*

**Authors:** Timothy W. Jones

**Comments:** 2 Pages. You may need to read Visualizing Zeta(n>1) and Proving its Irrationality by the same author.

This is an alternative proof that zeta(n>1) is irrational. It uses nested intervals and Cantor's Nested Interval Theorem. It is a follow up for the article Visualizing Zeta(n>1) and Proving its Irrationality.

**Category:** Number Theory

[1641] **viXra:1712.0588 [pdf]**
*submitted on 2017-12-24 11:13:39*

**Authors:** Ricardo Gil

**Comments:** 5 Pages. I will attempt any problem in any subject, just provide some background.

The objective of this paper is to show people that I am now for hire($). While many scientist and mathematicians are bound by the laws of nature and physics,I am able to look beyond the laws of nature and physics and come up with solutions for virtually every problem(See my papers).While my degrees are in education I have had a hobby of submitting unsolicited solutions to the CIA for the last 20 years for free. Whether they use the solutions of not is not relevant or if they deny it or confirm it. What is relevant is that if you have a problem and are willing to pay for a solution via paypal Im willing to solve it. Submit it to Ricardo.gil@sbcglobal.net. I ask for a fair price for a viable solution. "& Ye shall know the truth and any project is achievable in 18 mos or less."

**Category:** Number Theory

[1640] **viXra:1712.0572 [pdf]**
*submitted on 2017-12-22 12:57:33*

**Authors:** J. Mitchell

**Comments:** 35 Pages.

Multi-dimensional identity refers to the many labels which describe any ‘Thing’ as it exists, meaning both its describable states of existence and whatever processes generate, connect and count across those states. It develops through and out of a base2 pattern that becomes a multi-layered function which generates, relates and counts base10 numbers.

**Category:** Number Theory

[1639] **viXra:1712.0565 [pdf]**
*submitted on 2017-12-23 01:17:56*

**Authors:** Divyendu Priyadarshi

**Comments:** 1 Page. i am not a professional mathematician, so if there is some silly mistake or misconceptions , please point out.

In this small paper, I have argued very simply that "Twin Prime Conjecture" is quite obvious and there is nothing to prove. In fact, it reduces to the hypothesis that prime numbers are infinite in number if we accept the quite random pattern of occurrences of of prime numbers on number line.

**Category:** Number Theory

[1638] **viXra:1712.0554 [pdf]**
*submitted on 2017-12-21 12:46:22*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following two conjectures: (I) There exist an infinity of squares of odd numbers n^2 such that n^2 + R(n^2), where R(n^2) is the number obtained reversing the digits of n^2, is a palindromic number; (II) There is no a square of an odd number to be as well Lychrel number. Note that a Lychrel number is a natural number that cannot form a palindrome through the iterative process of repeatedly reversing its digits and adding the resulting numbers (process sometimes called the 196-algorithm, 196 being the smallest such number) – see the sequence A023108 in OEIS.

**Category:** Number Theory

[1637] **viXra:1712.0543 [pdf]**
*submitted on 2017-12-21 08:23:56*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following two conjectures: (I) There exist an infinity of Poulet numbers P such that P + R(P), where R(P) is the number obtained reversing the digits of P, is a palindromic number; (II) There is no a Poulet number to be as well Lychrel number. Note that a Lychrel number is a natural number that cannot form a palindrome through the iterative process of repeatedly reversing its digits and adding the resulting numbers (process sometimes called the 196-algorithm, 196 being the smallest such number) – see the sequence A023108 in OEIS.

**Category:** Number Theory

[1636] **viXra:1712.0532 [pdf]**
*submitted on 2017-12-20 16:05:37*

**Authors:** Ricardo Gil

**Comments:** 1 Page. So let it be written so let it be done by CERN or any other MADD Scientist Club !!!

The objective of this paper is to suggest that a photon can be cooled to -273 Kelvin and the photon can be slowed down around 10 m/s**2 and then a positive charge can be added to the minimally charged electron of the photon to create a positron which would be dark matter or antimatter, No Que No Carnal???

**Category:** Number Theory

[1635] **viXra:1712.0488 [pdf]**
*submitted on 2017-12-17 05:24:36*

**Authors:** Mesut Kavak

**Comments:** 5 Pages.

I published some solutions a time ago to Goldbach Conjecture, Collatz Problem and Twin Primes; but I noticed that there were some serious logic voids to explain the problems. After that I made some corrections in my another article; but still there were some mistakes. Even so, I can say it easily that here I brought exact solutions for them out by new methods back to the drawing board.

**Category:** Number Theory

[1634] **viXra:1712.0483 [pdf]**
*submitted on 2017-12-16 19:45:21*

**Authors:** Ricardo Gil

**Comments:** 2 Pages. This paper is about Human performance, mass reduction through repulsion and Mathematical Physics.

The purpose of this paper is to show how these electronic mass reduction repulsion boot(similar to magnetic field disruption TR3B (https://www.youtube.com/watch?v=au4hbUm4mMo) could be used on the Talos to reduce the weight of the suit.(https://www.youtube.com/watch?v=pFmFl5eE8vc).

**Category:** Number Theory

[1633] **viXra:1712.0482 [pdf]**
*submitted on 2017-12-17 02:19:30*

**Authors:** Ricardo Gil

**Comments:** 1 Page. To make a Tesla car, put one or more car batteries in the front seat and run a jumper cable to the cigarette lighter.

The purpose of this paper is to point out the Blue Rib Bridge or Highway A.K.A the Pinn Oak or Hausmann Stargate.

**Category:** Number Theory

[1632] **viXra:1712.0458 [pdf]**
*submitted on 2017-12-14 09:17:53*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following two conjectures: (1) There exist an infinity of primes obtained concatenating 9*p – 12 with p^2 where p is a prime (for example, such a prime is 208554289 obtained concatenating 9*233 – 12 = 2085 with 233^2 = 54289); (2) There exist an infinity of primes obtained concatenating 9*p – 12 with p^2 where p is a Poulet number (for example, such a prime is 155492989441 obtained concatenating 9*1729 – 12 = 15549 with 1729^2 = 2989441).

**Category:** Number Theory

[1631] **viXra:1712.0457 [pdf]**
*submitted on 2017-12-14 09:19:56*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I make the following conjecture: There exist an infinity of primes obtained concatenating 2*n + 4 with 2*n + 4 then with n where n = 3*p and p is a prime; for example, such primes are 19019093 obtained concatenating 190 = 2*(3*31) + 4 with 190 then with 93 = 3*31 or 12701270633 obtained concatenating 1270 = 2*(3*211) + 4 with 1270 then with 633 = 3*211. Note that for twenty-five from the first eighty primes p are obtained primes with this method.

**Category:** Number Theory

[1630] **viXra:1712.0452 [pdf]**
*submitted on 2017-12-15 03:07:11*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: For any a, b, c distinct numbers of the form 6*k – 1 there exist an infinity of numbers d of the form 6*h – 1 such that the number n = 2^a*2^b*2^c + d is prime. This is a formula that conducts often to primes and composites with very few prime factors; for instance, taking a = 5 and b = 11 are obtained seventeen primes for c and d both less than 100 (for c = 17, n is prime for six values of d up to 100: 17, 29, 35, 59, 71, 77)! Also note that for [a, b, c, d] = [59, 65, 71, 53] (all four less than or equal to 71) is obtained a prime with 59 digits!

**Category:** Number Theory

[1629] **viXra:1712.0451 [pdf]**
*submitted on 2017-12-15 03:09:04*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: For any a, b, c distinct numbers of the form 6*k + 1 there exist an infinity of numbers d of the form 6*h + 1 such that the number n = 2^a*2^b*2^c - d is prime. This is a formula that conducts often to primes and composites with very few prime factors; for instance, taking a = 7 and b = 13 are obtained eighteen primes for c and d both less than 100 (for c = 19, n is prime for four values of d up to 100: 7, 19, 67, 91)! Also note that for [a, b, c, d] = [49, 55, 61, 61] (all four less than or equal to 61) is obtained a prime with 50 digits!

**Category:** Number Theory

[1628] **viXra:1712.0441 [pdf]**
*submitted on 2017-12-13 11:00:55*

**Authors:** Helmut Preininger

**Comments:** 42 Pages.

n this paper we calculate for various sets X (some subsets of the natural numbers) the probability of an element a of X is also squarefree. Furthermore we calculate the probability of c is squarefree, where c=a+b, a is an element of the set X and b is an element of the set Y.

**Category:** Number Theory

[1627] **viXra:1712.0421 [pdf]**
*submitted on 2017-12-12 09:31:48*

**Authors:** Ricardo Gil

**Comments:** 2 Pages. Don't worry folks I have been there on foot and in a car. Remain Calm.Have fun with it.

The purpose of this paper is to explain a Stargate or Temporal anomaly on Pin Oak Road

**Category:** Number Theory

[1626] **viXra:1712.0397 [pdf]**
*submitted on 2017-12-10 06:30:48*

**Authors:** Ricardo Gil

**Comments:** 2 Pages. If your computer can handle it the sky's the limit with regards to digits.

The objective of this paper is to provide everyone with a program in Piethon to be able to print 250,000 digits and if your computer allow to be able to print > 299792458 digits.

**Category:** Number Theory

[1625] **viXra:1712.0396 [pdf]**
*submitted on 2017-12-10 10:16:41*

**Authors:** Ricardo Gil

**Comments:** 3 Pages. Mosses equals 500 in the Torah.

The objective of this paper is simplify frequency topology of encryption and lininear Graphs that can be represent in dimension 2 or greater.

**Category:** Number Theory

[1624] **viXra:1712.0384 [pdf]**
*submitted on 2017-12-10 12:27:40*

**Authors:** Timothy W. Jones

**Comments:** 11 Pages. This fixes a number of typos and adds two appendices to the 2010 article published by the MAA's Monthly.

Ivan Niven's proof of the irrationality of pi is often cited because it is brief and uses only calculus. However it is not well motivated. Using the concept that a quadratic function with the same symmetric properties as sine should when multiplied by sine and integrated obey upper and lower bounds for the integral, a contradiction is generated for rational candidate values of pi. This simplifying concept yields a more motivated proof of the irrationality of pi and pi squared.

**Category:** Number Theory

[1623] **viXra:1712.0366 [pdf]**
*submitted on 2017-12-10 01:30:48*

**Authors:** Barry Foster

**Comments:** 1 Page.

This treatment uses two simple facts and seems to confirm the Conjecture without providing an obvious method for discovering GP primes.

**Category:** Number Theory

[1622] **viXra:1712.0359 [pdf]**
*submitted on 2017-12-08 08:14:04*

[1621] **viXra:1712.0353 [pdf]**
*submitted on 2017-12-09 04:18:48*

**Authors:** Bado idriss olivier

**Comments:** 6 Pages.

In this paper, we are going to give the proof of the Goldbach conjecture by introducing the lemma which implies Goldbach conjecture. first of all we are going to prove that the lemma implies Goldbach conjecture and in the following we are going to prove the validity of the lemma by using Chébotarev-Artin theorem's, Mertens formula and the Principle of inclusion - exclusion of Moivre

**Category:** Number Theory

[1620] **viXra:1712.0352 [pdf]**
*submitted on 2017-12-09 04:21:53*

**Authors:** Bado idriss olivier

**Comments:** 5 Pages.

In this paper, we are going to give the proof of legendre conjecture by using the Chebotarev -Artin 's theorem ,Dirichlet arithmetic theorem and the principle inclusion-exclusion of Moivre

**Category:** Number Theory

[1619] **viXra:1712.0342 [pdf]**
*submitted on 2017-12-07 19:37:22*

**Authors:** F.L.B.Périat

**Comments:** 2 Pages.

Voici la démonstration que les nombres univers sont infiniment rare.

**Category:** Number Theory

[1618] **viXra:1712.0202 [pdf]**
*submitted on 2017-12-06 19:30:08*

**Authors:** Réjean Labrie

**Comments:** 1 Page.

542 Place Macquet

**Category:** Number Theory

[1617] **viXra:1712.0098 [pdf]**
*submitted on 2017-12-05 06:02:16*

**Authors:** Choe Ryujin

**Comments:** 6 Pages.

Proof of Goldbach's conjecture and twin prime conjecture

**Category:** Number Theory

[1616] **viXra:1712.0073 [pdf]**
*submitted on 2017-12-03 17:31:02*

**Authors:** Leszek W. Guła

**Comments:** 1 Page.

The Goldbach's Theorem

**Category:** Number Theory

[780] **viXra:1803.0317 [pdf]**
*replaced on 2018-03-21 12:07:10*

**Authors:** John Atwell Moody

**Comments:** 3 Pages.

Conjecture: d/dc of the magnitude of the integral of e^{(c-1+iw)t} log(\lambda/q)dt is <0 when c\in (0,1/2) and w>10.

Theorem: The conjecture implies Riemann's hypothesis.

**Category:** Number Theory

[779] **viXra:1803.0317 [pdf]**
*replaced on 2018-03-21 05:39:58*

**Authors:** John Atwell Moody

**Comments:** 3 Pages.

Conjecture: d/dc of the magnitude of the integral of e^{(c-1+iw)t} log(\lambda/q)dt is <0 when c\in (0,1/2) and w>0.

Theorem: The conjecture implies Riemann's hypothesis.

**Category:** Number Theory

[778] **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

[777] **viXra:1803.0017 [pdf]**
*replaced on 2018-03-21 15:48:41*

**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

[776] **viXra:1803.0017 [pdf]**
*replaced on 2018-03-09 13:03:48*

**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

[775] **viXra:1803.0017 [pdf]**
*replaced on 2018-03-07 11:30:17*

**Authors:** Pablo Hernan Pereyra

**Comments:** 4 Pages.

**Category:** Number Theory

[774] **viXra:1803.0017 [pdf]**
*replaced on 2018-03-06 14:56:08*

**Authors:** Pablo Hernan Pereyra

**Comments:** 3 Pages.

**Category:** Number Theory

[773] **viXra:1802.0353 [pdf]**
*replaced on 2018-03-08 04:44:34*

**Authors:** Dave Ryan T. Cariño

**Comments:** 17 Pages.

Function and method for solving the distribution of prime numbers accurately using the combination of step functions, polynomial functions, inverse functions and continuous functions. Equation 〖lim┬(n→∞) p(n)〗〖={3+2(n+x_p )├|x_p=x_3+x_5+x_7+x_11+⋯x_p ┤}〗 is true for all integer where n>0 for the distribution and generation of exact values of prime numbers without exception. This formula is efficient by means of modern supercomputers for the task of expanding term x_p.

**Category:** Number Theory

[772] **viXra:1802.0353 [pdf]**
*replaced on 2018-02-28 01:16:15*

**Authors:** Dave Ryan T. Cariño

**Comments:** 17 Pages.

Function and method for solving the distribution of prime numbers accurately using the combination of step functions, polynomial functions, inverse functions and continuous functions. Equation 〖lim┬(n→∞) p(n)〗〖={3+2(n+x_p )├|x_p=x_3+x_5+x_7+x_11+⋯x_p ┤}〗 is true for all integer where n>0 for the distribution and generation of exact values of prime numbers without exception. This formula is efficient by means of modern supercomputers for the task of expanding term x_p.

**Category:** Number Theory

[771] **viXra:1802.0353 [pdf]**
*replaced on 2018-02-27 05:19:51*

**Authors:** Dave Ryan T. Cariño

**Comments:** 16 Pages.

Function and method for solving the distribution of prime numbers accurately using the combination of step functions, polynomial functions, inverse functions and continuous functions. Equation 〖lim┬(n→∞) p(n)〗〖={3+2(n+x_p )├|x_p=x_3+x_5+x_7+x_11+⋯x_p ┤}〗 is true for all integer where n>0 for the distribution and generation of exact values of prime numbers without exception. This formula is efficient by means of modern supercomputers for the task of expanding term x_p.

**Category:** Number Theory

[770] **viXra:1802.0353 [pdf]**
*replaced on 2018-02-27 00:43:09*

**Authors:** Dave Ryan T. Cariño

**Comments:** 16 Pages.

**Category:** Number Theory

[769] **viXra:1802.0353 [pdf]**
*replaced on 2018-02-25 22:33:51*

**Authors:** Dave Ryan T. Cariño

**Comments:** 12 Pages.

Function and method for solving the distribution of prime numbers accurately using the combination of step functions, polynomial functions, inverse functions and continuous functions. Equation 〖lim┬(n→∞) p(n)〗〖={3+2(n+x_p )├|x_p=x_3+x_5+x_7+x_11+⋯x_p ┤}〗 is true for all integer where n>0 for the distribution and generation of exact values of prime numbers without exception. This formula is efficient by means of modern supercomputers for the task of adding new expression for x_p.

**Category:** Number Theory

[768] **viXra:1802.0309 [pdf]**
*replaced on 2018-02-22 14:33:51*

**Authors:** David Stacha

**Comments:** 9 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

[767] **viXra:1802.0268 [pdf]**
*replaced on 2018-03-22 18:24:25*

**Authors:** Phil A. Bloom

**Comments:** 3 Pages.

Let F be positive co-prime{r, s, t} for applicable positive integral n, at least n = 1, 2, for which r^n + s^n = t^n, our algebraic identity, holds. Let J be positive co-prime {x, y, z} for applicable positive integral n, at least n = 1, 2, for which x^n + y^n = z^n holds. Let L be rational (r s)\t. Let M be rational {(x y)/z}. Satisfied equations r^n + s^n = t^n and x^n + y^n = z^$ directly imply statements: (x y)/z determines uniquely (x, y, z); (r s)/t determines uniquely (r, s ,t); {(r s)/t} = {(x y)/z}; thus, {r, s, t} = {x, y, z}. We show that for n > 2, no (r, s, t) exists. Hence, it is true that for n > 2, no integral (x, y, z) exists.

**Category:** Number Theory

[766] **viXra:1802.0268 [pdf]**
*replaced on 2018-03-17 23:33:25*

**Authors:** Phil A. Bloom

**Comments:** 2 Pages.

Let F be positive co-prime{r, s, t} for applicable positive integral n (at least n = 1,2) for which r^n + s^n = t^n, our algebraic identity, holds. Let J be positive co-prime x, y, z for applicable positive integral n (at least n = 1, 2) for which x^n + y^n = z^n holds. Let L be a non-standard {(r s)/t} such that t is not a divisor. Let M be a non-standard {(x y)/z} such that z is not a divisor. Satisfied equations r^n + s^n = t^n and x^n + y^n = z^n directly imply statements: {(r s)/t} = {(x y)/z}; (x y)/z determines uniquely (x, y, z)$; (r s)/t determines uniquely (r, s, t); so, {r, s, t}={x, y, z}. We show the truth of the statement: For n > 2, no (r, s, t) exists. Hence, for n > 2, no integral (x, y, z) exists.

**Category:** Number Theory

[765] **viXra:1802.0268 [pdf]**
*replaced on 2018-03-14 16:20:38*

**Authors:** Phil A. Bloom

**Comments:** 2 Pages.

Let J be the set of all positive co-prime triples (x, y, z) for which x^n + y^n = z^n holds for applicable positive integral n (at least n = 1, 2). Let L be the set of all positive co-prime triples (r, s, t) for which r^n + s^n = t^n, our algebraic identity, holds. Let K be the set of all (x y)/z; let M be the set of all (r s)/t. Satisfied equations r^n + s^n = t^n and x^n+y^n=z^n directly imply statements: {(r s)/t} = {(x y)/z}; (x y)/z determines uniquely (x, y, z); (r s)/t determines uniquely (r, s, t); hence, {r, s, t}={x, y, z}. We show the truth of statement : For n > 2, no (r,s,t) exists. Hence, for n > 2 , no integral (x, y, z) exists.

**Category:** Number Theory

[764] **viXra:1802.0268 [pdf]**
*replaced on 2018-03-06 13:03:19*

**Authors:** Phil A. Bloom

**Comments:** 2 Pages.

Let J be the set of all positive co-prime triples (x, y, z) for which x^n + y^n = z^n holds for applicable positive integral values of n (at least n = 1, 2). Let K be the set of all positive co-prime triples (r, s, t) for which r^n + s^n = t^n, our devised, algebraic identity, holds. Let L be the set of all (x y)/z; let M be the set of all (r s)/t. Satisfied equations r^n + s^n = t^n and x^n + y^n = z^n directly imply statements A : {(r s)/t} = {(x y)/z}; B : (x y)/z determines uniquely (x, y, z); C : (r s)/t determines uniquely (r, s, t); hence, D : {r, s, t} = {x, y, z}. We show the truth of E : For n > 2 no (r, s, t) exists. So, F : For n > 2, no integral (x, y, z) exists.

**Category:** Number Theory

[763] **viXra:1802.0268 [pdf]**
*replaced on 2018-02-22 20:24:40*

**Authors:** Phil A. Bloom

**Comments:** 2 Pages.

For applicable positive integral n (at least n = 1, 2), let an always positive co-prime x, y, z satisfy only x^n + y^n = z^n. For same {n}, let an always positive co-prime r, s, t satisfy only r^n + s^n = t^n, our algebraic identity. These equations directly imply true statements, which are A : It is true that {(r s)/t} = {(x y)/z}; B : Values of (x y)/z determine uniquely values of (x, y, z); C : Values of (r s)/t determine uniquely values of (r, s, t); ; 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, F : Since every non-co-prime value of (x y, z) is a non-unity integral multiple of a co-prime (x, y, z), for n > 2, no positive integral value of (x, y, z) exists.

**Category:** Number Theory

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

**Authors:** Phil A. Bloom

**Comments:** Pages.

With x ^ n + y ^ n = z ^ n, we take Fermat's last theorem as true for some positive integral values of n (n = 1, 2, at least) and for positive co-prime values of x, y, z . With identical {n}, 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 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, F : For n > 2, no positive integral values of (x, y, z) exist.

**Category:** Number Theory

[761] **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

[760] **viXra:1802.0198 [pdf]**
*replaced on 2018-02-17 12:33:19*

**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

[759] **viXra:1802.0198 [pdf]**
*replaced on 2018-02-16 19:15:04*

**Authors:** John Yuk Ching Ting

**Comments:** 66 Pages. This research paper contains rigorous proofs for Riemann hypothesis, Polignac's and Twin prime conjectures

**Category:** Number Theory

[758] **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

[757] **viXra:1801.0341 [pdf]**
*replaced on 2018-02-21 05:21:44*

**Authors:** Predrag Terzic

**Comments:** 9 Pages.

Theorems and conjectures about prime numbers .

**Category:** Number Theory

[756] **viXra:1801.0341 [pdf]**
*replaced on 2018-01-26 05:43:55*

**Authors:** Predrag Terzic

**Comments:** 8 Pages.

Theorems and conjectures about prime numbers .

**Category:** Number Theory

[755] **viXra:1801.0257 [pdf]**
*replaced on 2018-01-26 16:59:16*

**Authors:** Timothy W. Jones

**Comments:** 7 Pages. A few edits.

In this article we derive the values of zeta(2) and zeta(2n) using Euler's original insights.

**Category:** Number Theory

[754] **viXra:1801.0187 [pdf]**
*replaced on 2018-01-20 09:13:54*

**Authors:** Juan Moreno Borrallo

**Comments:** 6 Pages.

In this paper it is proposed a conjecture of existence of prime numbers on a particular arithmetic progression, and demonstrated a particular case.

**Category:** Number Theory

[753] **viXra:1801.0182 [pdf]**
*replaced on 2018-03-17 12:52:43*

**Authors:** Haofeng Zhang

**Comments:** 30 Pages.

In this paper for equation Axm+Byn=Czk , where m,n,k > 2, x,y,z > 2, A,B,C≥1 and gcd(Ax,By,Cz)=1, the author proves there are no positive integer solutions for this equation using “Order reducing method for equations” that the author invented for solving high order equations, in which let the equation become two equations, through comparing the two roots to prove there are no positive integer solutions for this equation under the assumption of no positive integer solutions for Ax^3+By^3=Cz^3 when Ax^m-i+By^n-i>Cz^k-i.

**Category:** Number Theory

[752] **viXra:1801.0182 [pdf]**
*replaced on 2018-03-09 05:47:15*

**Authors:** Haofeng Zhang

**Comments:** 31 Pages.

In this paper for equation Axm+Byn=Czk , where m,n,k > 2, x,y,z > 2, A,B,C≥1 and
gcd(Ax,By,Cz)=1, the author proves there are no positive integer solutions for this equation using “Order reducing method for equations” that the author invented for solving high order equations, in which let the equation become two equations, through comparing the two roots to prove there are no positive integer solutions for this equation under the assumption of no positive integer solutions for Axm+Byn=C*3k when Axm-i+Byn-i>Czk-i .

**Category:** Number Theory

[751] **viXra:1801.0182 [pdf]**
*replaced on 2018-01-28 11:08:53*

**Authors:** Haofeng Zhang

**Comments:** 30 Pages.

In this paper for equation Ax^m+By^n=Cz^k , where m,n,k > 2, x,y,z > 1, A,B,C≥1 and gcd(Ax,By,Cz)=1, the author proved there are no positive integer solutions for this equation using“Order reducing method for equations” that the author invented for solving high order equations,in which let the equation become two equations, through comparing the two roots to prove there are no positive integer solutions for this equation.

**Category:** Number Theory

[750] **viXra:1801.0140 [pdf]**
*replaced on 2018-02-13 07:03:38*

**Authors:** Timothy W. Jones

**Comments:** 5 Pages. Proofs are added for two results used in earlier drafts.

We prove that partial sums of zeta(n>=2) are not given by any single decimal in a number base given by a denominator of their terms. This result, applied to all partials, shows that partials are excluded from an ever greater number of rational values. The limit of the partials is zeta(n) and the limit of the exclusions leaves only irrational numbers.

**Category:** Number Theory

[749] **viXra:1801.0140 [pdf]**
*replaced on 2018-01-18 15:58:09*

**Authors:** Timothy W. Jones

**Comments:** 6 Pages. One typo corrected.

We prove that partial sums of zeta(2) are not given by any single decimal in a number base given by a denominator of their terms. This result, applied to all partials, shows that partials are excluded from an ever greater number of rational values. The limit of the partials is zeta(2) and the limit of the exclusions leaves only irrational numbers.

**Category:** Number Theory

[748] **viXra:1801.0140 [pdf]**
*replaced on 2018-01-15 09:33:09*

**Authors:** Timothy W. Jones

**Comments:** 6 Pages. Gives a counter to the geometric counter example.

We prove that partial sums of $\zeta(2)-1=z_2$ are not given by any single decimal in a number base given by a denominator of their terms. This result, applied to all partials, shows that partials are excluded from an ever greater number of rational values. The limit of the partials is $z_2$ and the limit of the exclusions leaves only irrational numbers.

**Category:** Number Theory

[747] **viXra:1801.0140 [pdf]**
*replaced on 2018-01-14 03:44:54*

**Authors:** Timothy W. Jones

**Comments:** 6 Pages. An easier set theoretical proof has been added.

We prove that a partial sum of $\zeta(2)-1=z_2$ is not given by any single decimal in a number base given by a denominator of its terms. This result, applied to all partials, shows that partials are excluded from an ever greater number of rational values. The limit of the partials is $z_2$ and the limit of the exclusions leaves only irrational numbers. This is a set theoretical proof. We also give a topological proof using nested intervals and Cantor's intersection theorem.

**Category:** Number Theory

[746] **viXra:1801.0140 [pdf]**
*replaced on 2018-01-13 09:51:26*

**Authors:** Timothy W. Jones

**Comments:** 6 Pages. Correction of upper bound.

We prove that a partial sum of $\zeta(2)-1=z_2$ is not given by any single decimal in a number base given by a denominator of its terms. This result, applied to all partials, shows that there are an infinite number of partial sums in one interval of the form $X_{k^2}=[.(x-1),.x]$ where $.x$ is a single, non-zero decimal in a number base of the denominators of the terms of $z_2$, here $k^2$. Using this property we show that $z_2$ is contained in an open interval inside $X_{k^2}$. As all possible rational values of $z_2$ are the endpoints of these $X_k$ intervals, $z_2$ must be irrational.

**Category:** Number Theory

[745] **viXra:1801.0068 [pdf]**
*replaced on 2018-03-19 09:17:05*

**Authors:** Haofeng Zhang

**Comments:** 23 Pages.

In this paper the author gives a simplest elementary mathematics method to solve the famous Fermat's Last Theorem (FLT), in which let this equation become a 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 Fermat's Last Theorem.

**Category:** Number Theory

[744] **viXra:1801.0068 [pdf]**
*replaced on 2018-03-17 12:54:51*

**Authors:** Haofeng Zhang

**Comments:** 13 Pages.

In this paper the author gives a simplest elementary mathematics method to solve the famous Fermat's Last Theorem (FLT), in which let this equation become a 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 Fermat's Last Theorem.

**Category:** Number Theory

[743] **viXra:1801.0068 [pdf]**
*replaced on 2018-03-09 05:45:22*

**Authors:** Haofeng Zhang

**Comments:** 21 Pages.

In this paper the author gives a simplest elementary mathematics method to solve the famous Fermat's Last Theorem (FLT), in which let this equation become a 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 Fermat's Last Theorem.

**Category:** Number Theory

[742] **viXra:1801.0068 [pdf]**
*replaced on 2018-01-28 11:11:09*

**Authors:** Haofeng Zhang

**Comments:** 17 Pages.

**Category:** Number Theory

[741] **viXra:1801.0068 [pdf]**
*replaced on 2018-01-09 10:47:39*

**Authors:** Haofeng Zhang

**Comments:** 18 Pages.

**Category:** Number Theory

[740] **viXra:1801.0001 [pdf]**
*replaced on 2018-01-02 13:45:30*

**Authors:** Leonhard Schuster

**Comments:** 13 Pages.

In this paper, we prove the positivity of Li coefficients for n>10^24. We investigate the Riemann Zeta function, in the form (s-1)zeta(s), under the transformation s = 1/(1-z). We apply a generalised Poisson-Jensen formula to show that Riemann Zeta function has only a finite number of zeros not lying the critical line, and that the Li coefficients are positive for n>10^24. This implicitly proves the validity of Riemann Hypothesis.

**Category:** Number Theory

[739] **viXra:1712.0384 [pdf]**
*replaced on 2017-12-22 10:37:51*

**Authors:** Timothy W. Jones

**Comments:** 13 Pages. Additional clarifications and appendix added.

Ivan Niven's proof of the irrationality of pi is often cited because it is brief and uses only calculus. However it is not well motivated. Using the concept that a quadratic function with the same symmetric properties as sine should when multiplied by sine and integrated obey upper and lower bounds for the integral, a contradiction is generated for rational candidate values of pi. This simplifying concept yields a more motivated proof of the irrationality of pi and pi squared.

**Category:** Number Theory

[738] **viXra:1712.0359 [pdf]**
*replaced on 2017-12-12 02:10:44*

[737] **viXra:1712.0135 [pdf]**
*replaced on 2018-03-17 17:04:34*

**Authors:** Kamal Barghout

**Comments:** 35 Pages. The material in this article is copyrighted. Please obtain authorization to use from the author

We represent each term of Beal’s conjecture equation as a number in exponential form with unique prime base-unit as its building block and therefore unique discreteness. It will be proven that any two numbers in exponential form to be added together must have a common prime base-unit due to their discreteness property. We represent the solution of Beal’s conjecture equation as an identity that produces the sum of two monomials of common indeterminate. The monomial on the RHS of Beal’s equation can be built from the expression on the LHS. Upon factorization of the GCD of the two monomials on the LHS of Beal’s conjecture solution it must be combined with the sum of the two coefficients of the terms to yield the monomial on the RHS by the power rules based on the identity solution of the equation.

**Category:** Number Theory

[736] **viXra:1712.0135 [pdf]**
*replaced on 2018-03-13 05:47:37*

**Authors:** Kamal Barghout

**Comments:** 42 Pages. The material in this article is copyrighted. Please obtain authorization to use any part of the manuscipt from the author

In this article we prove Beal’s conjecture by deductive reasoning. The main assertion in the proof stems from that any solution to the Beal’s conjecture equation〖 a〗^x+b^y=c^z represents an identity equation and each term of the solution represents a “block-number”. By representing any number in exponential form of single power as unique and have a basis of base-unit made of a prime number that repeats to comprise the number of specific “size” that is made of those prime base-units, it will be proven that any number in exponential form to be added to it to yield a sum in exponential form of single power must have the same prime base-unit. The proof is mainly provided using Bezout’s Identity Theory and identifying Beal’s conjecture equation as an identity. Similar to taking a LCD when adding two fractions and convert the addition to multiplication we can take an exponential GCF of two exponential numbers to convert addition to multiplication. This way, the GCF of the two terms on the LHS of any solution of Beal’s conjecture equation can be factored out, and by the power rules, it must be combined with the sum of the two coefficients of the two terms to yield the term on the RHS of the equation, confirming the proposition that they must have a common modularity that evaluates to 0 and therefore a common distinct base-unit to successfully combine and build a single term based on the identity solution of Beal’s conjecture equation. Therefore, the process of adding the two numbers of exponential form together on the LHS of Beal’s solution is equivalent to increasing the size of one of them by the other by the same number of base-units of the other to produce the number on the RHS of Beal’s solution.

**Category:** Number Theory

[735] **viXra:1712.0135 [pdf]**
*replaced on 2018-02-20 11:06:15*

**Authors:** Kamal Barghout

**Comments:** 37 Pages. The material in this article is copyrighted. Please obtain authorization from the author before the use of any part of the manuscript

In this article we prove Beal’s conjecture by deductive reasoning by means of elementary algebraic methods. The main assertion in the proof stems from that any solution to the Beal’s equation += represents an identity equation and that the LHS of the equation represents the sum of two monomials of like terms with the value of their variables and their coefficients for each term of the equation combine by the power rules. Since the two monomials have like terms, we can factor out a common factor and add the coefficients to produce a product of two terms that can be combined by the power rules to yield the RHS of the equation. By representing a number in exponential form of single power as having a unique base-unit that repeats to comprise the number, it will be proven that any number in exponential form to be added to it to yield a sum in exponential form of single power must have the same base-unit by virtue of the two numbers having a “block-form” with a building block of their common base-unit. By conversion of the addition process of the two exponential numbers to multiplication, the GCF of the two terms on the LHS of any solution of Beal’s conjecture equation can be factored out, and by the power rules, it must be combined with the sum of the two coefficients of the two terms to yield the term on the RHS of the equation, confirming the proposition that they must have a common and distinct base-unit to successfully combine and build a single term based on the identity solution of Beal’s conjecture equation. Therefore, the process of adding the two numbers of exponential forms together on the LHS of Beal’s solution is equivalent to increasing the “size” of one of them by the other by the same number of base-units of the other to produce the number on the RHS of Beal’s solution

**Category:** Number Theory

[734] **viXra:1712.0135 [pdf]**
*replaced on 2018-02-05 10:31:56*

**Authors:** Kamal Barghout

**Comments:** 36 Pages. The material in this article is copyrighted. Please obtain authorization from the author before use of any part of the manuscript

In this article we prove Beal’s conjecture by deductive reasoning by means of elementary algebraic methods. The main assertion in the proof stems from that any solution to the Beal’s equation += represents an identity equation and that the LHS of the equation represents the sum of two monomials of like terms with the value of their variables and their coefficients for each term of the equation combine by the power rules. Since the two monomials have like terms, we can factor out a common factor and add the coefficients to produce a product of two terms that can be combined by the power rules to yield the RHS of the equation. By representing a number in exponential form of single power as having a unique base-unit that repeats to comprise the number, it will be proven that any number in exponential form to be added to it to yield a sum in exponential form of single power must have the same base-unit by virtue of the two numbers having a “block-form” with a building block of their common base-unit. By conversion of the addition process of the two exponential numbers to multiplication, the GCF of the two terms on the LHS of any solution of Beal’s equation can be factored out, and by the power rules, it must be combined with the sum of the two coefficients of the two terms to yield the term on the RHS of the equation, confirming the proposition that they must have a common and distinct base-unit to successfully combine and build a single term based on the identity solution of Beal’s equation. Therefore, the process of adding the two numbers of exponential forms on the LHS of Beal’s solution is equivalent to increasing the “size” of the number on the RHS of Beal’s solution.

**Category:** Number Theory

[733] **viXra:1712.0135 [pdf]**
*replaced on 2017-12-30 12:15:53*

**Authors:** Kamal Barghout

**Comments:** 21 Pages. The material in this article is copyrighted. Please obtain authorization to use any part of the manuscipt from the author

In this article we prove Beal’s conjecture by deductive reasoning by means of elementary algebraic methods. The main assertion in the proof stands upon that the LHS of Beal’s conjecture represents the sum of two monomials of like terms. The monomial on the RHS of Beal’s conjecture can be built by combining the two monomials on the LHS. By representing any number in exponential form of single power as having a unique base-unit it is to be proved that any number in exponential form to be added to it to yield a sum in exponential form of single power must have the same base-unit by virtue of the two numbers having a “block-form” with a building block of their common base-unit. By conversion of the addition process of the two exponential numbers to multiplication, the GCF of the two terms on the LHS of Beal’s conjecture can be factored. Upon factorization of the GCF and making use of power rules, it must be combined with the sum of the two coefficients of the two terms to yield the monomial on the RHS of the conjecture, confirming the proposition that they must have a common and unique base-unit to successfully combine.

**Category:** Number Theory