**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(3)

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(3) - 1110(5) - 1111(4) - 1112(4)

2012 - 1201(2) - 1202(10) - 1203(6) - 1204(8) - 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(34) - 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(12) - 1511(9) - 1512(25)

2016 - 1601(14) - 1602(17) - 1603(77) - 1604(54) - 1605(28) - 1606(17) - 1607(20) - 1608(17) - 1609(22) - 1610(24) - 1611(12) - 1612(20)

2017 - 1701(19) - 1702(26) - 1703(29) - 1704(32) - 1705(27) - 1706(24)

Any replacements are listed further down

[1522] **viXra:1706.0457 [pdf]**
*submitted on 2017-06-23 13:24:49*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

In this note we present an integral for the constant pi:pi=3.1415926535...

**Category:** Number Theory

[1521] **viXra:1706.0421 [pdf]**
*submitted on 2017-06-22 03:34:26*

**Authors:** Andrea Ossicini

**Comments:** 10 pages in English and 10 pages in Italian

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

**Category:** Number Theory

[1520] **viXra:1706.0414 [pdf]**
*submitted on 2017-06-21 05:00:47*

**Authors:** Marius Coman

**Comments:** 20 Pages.

A selection of forty sequences regarding primes and Fermat pseudoprimes from my yet unpublished papers, presented in "OEIS style", with definition of the terms of a sequence, examples, few first terms, notes and conjectures.

**Category:** Number Theory

[1519] **viXra:1706.0410 [pdf]**
*submitted on 2017-06-21 00:28:19*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: There exist an infinity of primes p having the property that concatenating s(p) – d(1) with s(p) – d(2) and repeatedly up to s(p) – d(k), where s(p) is the sum of digits of p and d(1),...,d(k) are the digits of p, is obtained a prime q. Example: such prime p is 127 because concatenating 9 (= 10 – 1) with 8 (= 10 – 2) and with 3 (= 10 – 7) is obtained a prime q = 983.

**Category:** Number Theory

[1518] **viXra:1706.0408 [pdf]**
*submitted on 2017-06-21 02:28:55*

**Authors:** Choe Ryong Gil

**Comments:** 12 pages, 2 tables

The aim of this paper is to show a new sufficient condition (NSC) by the Euler function for the Riemann hypothesis and its possibility. We build the NSC for any natural numbers ≥ 2 from well-known Robin theorem, and prove that the NSC holds for all odd and some even numbers while, the NSC holds for any even numbers under a certain condition, which would be called the condition (d).

**Category:** Number Theory

[1517] **viXra:1706.0407 [pdf]**
*submitted on 2017-06-21 02:30:07*

**Authors:** Choe Ryong Gil

**Comments:** 27 pages, 6 tables

In this paper, it is obtained a new estimate for the error term E(t) of the Mertens' formula sum_{p≤t}{p^{-1}}=loglogt+b+E(t), where t>1 is a real number, p is the prime number and b is the well-known Mertens' constant. We , first, provide an upper bound, not a lower bound, of E(p) for any prime number p≥3 and, next, give one in the form as E(t)<logt/√t for any real number t≥3. This is an essential improvement of already known results. Such estimate is very effective in the study of the distribution of the prime numbers.

**Category:** Number Theory

[1516] **viXra:1706.0381 [pdf]**
*submitted on 2017-06-18 22:43:41*

**Authors:** Lahcen Aghray

**Comments:** 5 Pages.

We obtain a parameterization of a Diophantine equation of degree 4

**Category:** Number Theory

[1515] **viXra:1706.0380 [pdf]**
*submitted on 2017-06-18 23:13:05*

**Authors:** Lahcen Aghray

**Comments:** 2 Pages.

The resolution of a Diophantine equation by calculating the intersection of a curve of degree 3 with a line

**Category:** Number Theory

[1514] **viXra:1706.0288 [pdf]**
*submitted on 2017-06-15 07:46:58*

**Authors:** Gang Li

**Comments:** 14 Pages.

An attempt of using elementary approach to prove Fermat's last theorem (FLT)
is given. For infinitely many prime numbers, Case I of the FLT can be proved
using this approach. Furthermore, if a conjecture proposed in this paper is
true (k-3 conjecture), then case I of the FLT is proved for all prime numbers.
For case II of the FLT, a constraint for possible solutions is obtained.

**Category:** Number Theory

[1513] **viXra:1706.0206 [pdf]**
*submitted on 2017-06-13 13:41:27*

**Authors:** Edgar Valdebenito

**Comments:** 16 Pages.

In this note we recall some formulas related with continued fractions , numbers , sequences and the constant pi.

**Category:** Number Theory

[1512] **viXra:1706.0205 [pdf]**
*submitted on 2017-06-13 13:45:55*

**Authors:** Edgar Valdebenito

**Comments:** 10 Pages.

In this note we briefly explore the equation: z^5+z^4-1=0

**Category:** Number Theory

[1511] **viXra:1706.0197 [pdf]**
*submitted on 2017-06-14 09:53:22*

**Authors:** Ryan Zielinski

**Comments:** 1 Page. This work is licensed under the CC BY 4.0, a Creative Commons Attribution License.

In this note we will use Faulhaber's Formula to explain why the odd Bernoulli numbers are equal to zero.

**Category:** Number Theory

[1510] **viXra:1706.0196 [pdf]**
*submitted on 2017-06-14 15:11:32*

**Authors:** Mendzina Essomba Francois

**Comments:** 2 Pages.

J present
two algorithms for calculating the natural logarithm of any real number. The first is an algorithm obtained by the
method of Archimedes for the calculation of pi and the second the product of a succession of rad
icals.

**Category:** Number Theory

[1509] **viXra:1706.0192 [pdf]**
*submitted on 2017-06-15 02:06:54*

**Authors:** Leszek W. Guła

**Comments:** 5 Pages. Certainly no scientist was working under such conditions. Nobody will ever announce to the world my creative proposals.

1. The truly marvellous proof of The Fermat's Last Theorem (FLT).
2. The proof of the theorem - For all n∈{3,5,7,…} and for all z∈{3,7,11,…} and for all natural numbers u,υ: z^n≠u^2+υ^2.

**Category:** Number Theory

[1508] **viXra:1706.0134 [pdf]**
*submitted on 2017-06-09 07:24:09*

**Authors:** Kolosov Petro

**Comments:** 12 pages, arXiv:1603.02468, MSC 2010: 40C15, 32A05

This paper describes a method of natural exponented power's $y=x^n, \ \forall(x,n) \in {\mathbb{N}}$ to the numerical series. The most widely used methods to solve this problem are Newton’s Binomial Theorem and Fundamental Theorem of Calculus (that is, derivative and integral are inverse operators). The paper provides the other kind of solution, based on induction, except above described theorems.
Keywords: power, power function, monomial, polynomial, power series, third power, series, finite difference, divided difference, high order finite difference, derivative, binomial coefficient, binomial theorem, Newton's binomial theorem, binomial expansion, n-th difference of n-th power, number theory, cubic number, cube, Euler number, exponential function, Pascal triangle, Pascal’s triangle, mathematics, math, maths, algebra, science, arxiv, preprint, series representation, series expansion, open scicence, calculus

**Category:** Number Theory

[1507] **viXra:1706.0112 [pdf]**
*submitted on 2017-06-07 14:51:48*

**Authors:** Kolosov Petro

**Comments:** 12 pages, 6 figures, arXiv:1705.02516

Calculating the value of $C^{k\in\{1,\infty\}}$ class of smoothness real-valued function's derivative in point of $\mathbb{R}^+$ in radius of convergence of its Taylor polynomial (or series), applying an analog of Newton's binomial theorem and $q$-difference operator. $(P,q)$-power difference introduced in section 5. Additionally, by means of Newton's interpolation formula, the discrete analog of Taylor series, interpolation using $q$-difference and $p,q$-power difference is shown.
Keywords: derivative, differential calculus, differentiation, Taylor's theorem, Taylor's formula, Taylor's series, Taylor's polynomial, power function, Binomial theorem, smooth function, real calculus, Newton's interpolation formula, finite difference, q-derivative, Jackson derivative, q-calculus, quantum calculus, (p,q)-derivative, (p,q)-Taylor formula, mathematics, math, maths, science, arxiv, preprint

**Category:** Number Theory

[1506] **viXra:1706.0111 [pdf]**
*submitted on 2017-06-07 19:48:40*

**Authors:** Kolosov Petro

**Comments:** 12 pages, 1 figure, arXiv:1608.00801

The main aim of this paper to establish the relations between forward, backward and central finite (divided) differences (that is discrete analog of the derivative) and partial & ordinary high-order derivatives of the polynomials.
Keywords: finite difference, divided difference, high order finite difference, derivative, ode, pde, partial derivative, partial difference, power, power function, polynomial, monomial, power series, high order derivative, mathematics, differential calculus, math, maths, science, arxiv, preprint, algebra, calculus, open science, differential equations

**Category:** Number Theory

[1505] **viXra:1706.0102 [pdf]**
*submitted on 2017-06-06 11:10:53*

**Authors:** Marius Coman

**Comments:** 2 Pages.

This paper is inspired by one of my previous papers, namely “Large primes obtained concatenating the numbers P - d(k) where d(k) are the prime factors of the Poulet number P”, where I conjectured that there are an infinity of primes which can be obtained concatenating the numbers P - d(1); P - d(2); ...; P – d(k); P, where d(1), ..., d(k) are the prime factors of the Poulet number P. Because some of these Poulet numbers are 3-Poulet numbers of the form (6k + 1)*(6h + 1)*(6j + 1) I extend in this paper that idea conjecturing that for any prime p of the form 6k + 1 there exist an infinity of pairs of primes [q, r], of the form 6h + 1 and 6j + 1, such that the number obtained concatenating p*q*r – p with p*q*r – q with p*q*r – r then with p*q*r is prime.

**Category:** Number Theory

[1504] **viXra:1706.0097 [pdf]**
*submitted on 2017-06-06 04:10:22*

**Authors:** Marius Coman

**Comments:** 2 Pages.

This paper is inspired by one of my previous papers, namely “Large primes obtained concatenating the numbers P - d(k) where d(k) are the prime factors of the Poulet number P”, where I conjectured that there are an infinity of primes which can be obtained concatenating the numbers P - d(1); P - d(2); ...; P – d(k); P, where d(1), ..., d(k) are the prime factors of the Poulet number P. Because some of these Poulet numbers are 2-Poulet numbers of the form (6k + 1)*(6h + 1) I extend in this paper that idea conjecturing that for any prime p of the form 6k + 1 there exist an infinity of primes q of the form 6h + 1 such that the number obtained concatenating p*q – p with p*q – q then with p*q is prime.

**Category:** Number Theory

[1503] **viXra:1706.0037 [pdf]**
*submitted on 2017-06-05 05:55:08*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that there are an infinity of primes which can be obtained concatenating the numbers P - d(1); P - d(2); ...; P – d(k); P, where d(1), ..., d(k) are the prime factors of the Poulet number P. Example: using the sign “//” with the meaning “concatenated to”, for the Poulet number 129921 (= 3*11*31*127), the number (129921 – 3)//(129921 – 11)//(129921 – 31)//(129921 – 127)//129921 = 129918129910129890129794129921 is prime. Note that such primes are obtained for 10 from the first 90 Poulet numbers!

**Category:** Number Theory

[1502] **viXra:1706.0033 [pdf]**
*submitted on 2017-06-04 11:53:02*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following observation: for many squares of primes (I conjecture that for an infinity of them) the numbers obtained concatenating 30 – d(1), 30 – d(2),..., 30 – d(k), where d(1),..., d(k) are the digits of a square of a prime, are primes. Example: for 1369 (= 37^2) the number obtained concatenating 29 = 30 – 1 with 27 = 30 – 3 with 24 = 30 – 6 with 21 = 30 – 9, i.e. the number 29272421, is prime. Note that for 35 from the first 200 squares of primes the numbers obtained this way are primes!

**Category:** Number Theory

[1501] **viXra:1706.0032 [pdf]**
*submitted on 2017-06-04 12:51:12*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I make the following observation: for many Poulet numbers (I conjecture that for an infinity of them) the numbers obtained concatenating 30 – d(1), 30 – d(2),..., 30 – d(n), where d(1),..., d(n) are the digits of a n-digits Poulet number, are primes. Example: for 8911 the number obtained concatenating 22 = 30 – 8 with 21 = 30 – 9 with 29 = 30 – 1 with 29 = 30 – 1, i.e. the number 22212929, is prime.

**Category:** Number Theory

[1500] **viXra:1706.0029 [pdf]**
*submitted on 2017-06-04 02:09:22*

**Authors:** Mendzina Essomba Francois, Essomba Essomba Dieudonne Gael

**Comments:** 24 Pages.

Introduction of new trigonometric functions and mathematical constants.
The same mathematical equation connects the circle to the square, the sphere to the cube, the hyper-sphere to the hyper-cube, another also connects the ellipse to the rectangle, the ellipsoid to a rectangular parallelepiped, the hyper-ellipsoid To the rectangular hyper-parallelepiped.

**Category:** Number Theory

[1499] **viXra:1706.0022 [pdf]**
*submitted on 2017-06-02 18:39:11*

**Authors:** T.Nakashima

**Comments:** 2 Pages.

This paper is the answer of collatz's Problem.

**Category:** Number Theory

[1498] **viXra:1705.0471 [pdf]**
*submitted on 2017-05-31 07:58:56*

**Authors:** V.I.Saenko

**Comments:** 2 Pages. This is a Russian preprint. The English version is sent to the peer-reviewed journal

It is proved that a perfect cuboid, i.e., a rectangular parallelepiped having integer edges, integer face diagonals, and integer space diagonal, is not possible

**Category:** Number Theory

[1497] **viXra:1705.0461 [pdf]**
*submitted on 2017-05-29 12:33:16*

**Authors:** Edgar Valdebenito

**Comments:** 12 Pages.

This note presents some formulas and fractals related with the equation : x^3+x^2+1=0.

**Category:** Number Theory

[1496] **viXra:1705.0460 [pdf]**
*submitted on 2017-05-29 07:06:59*

**Authors:** John Yuk Ching Ting

**Comments:** 66 Pages. Rigorous proofs for Riemann hypothesis, Polignac's conjecture and Twin prime conjecture

L-functions form an integral part of the 'L-functions and Modular Forms Database' which is associated with far-reaching applications and implications. In perspective, Riemann zeta function is the simplest example of an L-function. Riemann hypothesis refers to the 1859 proposal by German mathematician Bernhard Riemann whereby all nontrivial zeros of Riemann zeta function are conjectured to be located on the critical line. This proposal is equivalently stated in this research paper as all nontrivial zeros are conjectured to exactly match the 'Origin' intercepts of Riemann zeta function. Deeply entrenched in number theory, prime number theorem involves analysis of the prime counting function for prime numbers. Solving Riemann hypothesis would result in a crucial primary by-product whereby absolute and full delineation of this important prime number theorem will occur. Involving the study of prime numbers [which are Incompletely Predictable entities], Twin prime conjecture involves the analysis of prime gap = 2 [representing all twin primes] and is thus a subset of Polignac's conjecture which involves the analysis of all even number prime gaps = 2, 4, 6,... [representing prime numbers in totality except for the first prime number '2']. Nontrivial zeros of Riemann zeta function are also Incompletely Predictable entities. With the common presence of Incompletely Predictable entities and with this helpful presence considered a major asset; the task to solve the above mentioned intractable open problems of Riemann hypothesis, Polignac's and Twin prime conjectures is conveniently amalgamated together in this paper. We employ our novel Virtual Container Research Method which acts essentially as foundation for the mathematical framework enabling successful completion of this monumental task.

**Category:** Number Theory

[1495] **viXra:1705.0395 [pdf]**
*submitted on 2017-05-28 03:10:36*

**Authors:** Oleg Cherepanov

**Comments:** 6 Pages. http://www.trinitas.ru/rus/doc/0016/001d/2254-chr.pdf

The discovered algorithm for extracting prime numbers from the natural series is alternative to both the Eratosthenes lattice and Sundaram and Atkin's sentences. The distribution of prime numbers does not have a formula, but if the number is one less than the prime number is an exponent of the integers, then there are no two scalar scalars whose sum is equal to the third integer in the same degree. This is the sound of P. Fermat's Great Theorem, the proof of which he could begin by using the Minor theorem known to him. The first part of the proof is here restored. But how did P. Fermat finish it?

**Category:** Number Theory

[1494] **viXra:1705.0393 [pdf]**
*submitted on 2017-05-27 18:13:53*

**Authors:** Caitherine Gormaund

**Comments:** 2 Pages.

Herein we introduce the subject of the Gormaund numbers, and prove a fundamental property thereof.

**Category:** Number Theory

[1493] **viXra:1705.0392 [pdf]**
*submitted on 2017-05-27 10:56:20*

**Authors:** M. A. Thomas

**Comments:** 71 Pages. A PDF copy of a slide presentation containing 71 slides

A PDF copy of a Slide Presentation. Relationship of number theory to physics forms are established using a 'wholly trinity' consisting of the first odd Riemann Zeta function, the Euler-Mascheroni constant and the imaginary part of the first non-trivial Zeta zero. Quark flavor changes occur under extremal gravities with only slight asymmetric changes to dimensionless ratios. Symmetric invariance is maintained throughout changes.

**Category:** Number Theory

[1492] **viXra:1705.0390 [pdf]**
*submitted on 2017-05-27 07:41:40*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I present a method to obtain from a given prime p1 larger primes, namely inserting before of a digit of p1 a power of 3, and, once a prime p2 is obtained, repeating the operation on p2 and so on. By this method I obtained from a prime with 9 digits a prime with 36 digits (the steps are showed in this paper) using just the numbers 3, 9(3^2), 27(3^3) and 243(3^5).

**Category:** Number Theory

[1491] **viXra:1705.0379 [pdf]**
*submitted on 2017-05-26 06:15:45*

**Authors:** Ricardo.gil

**Comments:** 1 Page. Email solutions or suggestions to Ricardo.gil@sbcglobal.net

In math the the 7 Clay Math unsolved problems? Another problem is the question if there is a God(s)? In my paper the purpose is to explain that in the end we all meet our maker and that man does not have the power to cheat death. Like the Riemann Zeta function that remains unsolved and when solved will give insight to distribution of the Primes, giving or solving this open-end problem will help me solve a problem. This is the only problem I have not been able to solve and I am open sourcing it.

**Category:** Number Theory

[1490] **viXra:1705.0360 [pdf]**
*submitted on 2017-05-25 05:19:12*

**Authors:** Maik Becker-Sievert

**Comments:** 1 Page. Identity Proof FLT

This Identity proofs direct Fermats Last Theorem

**Category:** Number Theory

[1489] **viXra:1705.0343 [pdf]**
*submitted on 2017-05-22 13:32:15*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

This note presents some formulas for pi constant

**Category:** Number Theory

[1488] **viXra:1705.0342 [pdf]**
*submitted on 2017-05-22 13:36:50*

**Authors:** Edgar Valdebenito

**Comments:** 16 Pages.

This note presents some formulas related with the number z=LambertW(i),where LambertW(x) is the Lambert function.

**Category:** Number Theory

[1487] **viXra:1705.0293 [pdf]**
*submitted on 2017-05-19 11:18:33*

**Authors:** Prashanth Rao

**Comments:** 1 Page.

If p is any odd prime number and c is any odd number less than p, then there must exist a positive number c’ less than p, such that cc’= -2modp

**Category:** Number Theory

[1486] **viXra:1705.0289 [pdf]**
*submitted on 2017-05-19 07:56:37*

**Authors:** Helmut Preininger

**Comments:** 14 Pages.

In this paper we take a closer look to the distribution of the residues of squarefree natural numbers and explain an algorithm to compute those distributions.
We also give some conjectures about the minimal number of cycles in the squarefree arithmetic progression and explain an algorithm to compute this minimal numbers.

**Category:** Number Theory

[1485] **viXra:1705.0277 [pdf]**
*submitted on 2017-05-19 01:01:55*

**Authors:** Shaban A. Omondi Aura

**Comments:** 30 Pages. Preferably for journals, academies and conferences

This paper is concerned with formulation and demonstration of new versions of equations that can help us resolve problems concerning maximal gaps between consecutive prime numbers, the number of prime numbers at a given magnitude and the location of nth prime number. There is also a mathematical argument on why prime numbers as elementary identities on their own respect behave the way they do. Given that the equations have already been formulated, there are worked out examples on numbers that represent different cohorts. This paper has therefore attempted to formulate an equation that approximates the number of prime numbers at a given magnitude, from N=3 to N=〖10〗^25. Concerning the location of an nth prime number, the paper has devised a method that can help us locate a given prime number within specified bounds. Nonetheless, the paper has formulated an equation that can help us determine extremely bounded gaps. Lastly, using trans-algebraic number theory method, the paper has shown that unpredictable behaviors of prime numbers are due to their identity nature.

**Category:** Number Theory

[1484] **viXra:1705.0242 [pdf]**
*submitted on 2017-05-16 03:47:59*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that for any pair of twin primes [p, q], p ≥ 11, there exist a number n having the sum of its digits equal to 12 such that inserting n after the first digit of p respectively q are obtained two primes (almost always twins, as in the case [1481, 1483] where n = 48 is inserted in [11, 13], beside the case that the first digit of twins is different, as in the case [5669, 6661] where n = 66 is inserted in [59, 61]).

**Category:** Number Theory

[1483] **viXra:1705.0224 [pdf]**
*submitted on 2017-05-15 02:29:14*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that for any prime p, p ≥ 7, there exist a prime q obtained inserting a number n with the sum of digits equal to 12 before the last digit of p.

**Category:** Number Theory

[1482] **viXra:1705.0221 [pdf]**
*submitted on 2017-05-15 03:41:04*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that for any prime p, p ≥ 5, there exist a prime q obtained inserting a number n with the sum of digits equal to 12 after the first digit of p.

**Category:** Number Theory

[1481] **viXra:1705.0154 [pdf]**
*submitted on 2017-05-09 12:16:42*

**Authors:** Mesut Kavak

**Comments:** 1 Page.

This works aims to bring a simple solution to the Riemann Hypothesis over the Lagarias Transformation.

**Category:** Number Theory

[1480] **viXra:1705.0152 [pdf]**
*submitted on 2017-05-09 12:39:34*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

This note presents some formulas related with ahmed's integral.

**Category:** Number Theory

[1479] **viXra:1705.0151 [pdf]**
*submitted on 2017-05-09 12:43:55*

**Authors:** Edgar Valdebenito

**Comments:** 19 Pages.

This note presents formulas related with Euler-Mascheroni constant and fractals.

**Category:** Number Theory

[1478] **viXra:1705.0142 [pdf]**
*submitted on 2017-05-09 06:38:25*

**Authors:** Carlos Castro

**Comments:** 11 Pages. Submitted to Mod. Phys. Letts A.

An approach to solving the Riemann Hypothesis is revisited within the framework of the special properties of $\Theta$ (theta) functions, and the notion of $ {\cal C } { \cal T} $ invariance. The conjugation operation $ {\cal C }$ amounts to complex scaling transformations, and the $ {\cal T } $ operation
$ t \rightarrow ( 1/ t ) $ amounts to the reversal $ log (t) \rightarrow - log ( t ) $. A judicious scaling-like operator is constructed whose spectrum $E_s = s ( 1 - s ) $ is real-valued, leading to $ s = {1\over 2} + i \rho$,
and/or $ s $ = real. These values are the location of the non-trivial and trivial zeta zeros, respectively.
A thorough analysis of the one-to-one correspondence among the zeta zeros, and the orthogonality conditions among pairs of eigenfunctions,
reveals that $no$ zeros exist off the critical line. The role of the $ {\cal C }, {\cal T } $ transformations, and the properties of the Mellin transform of $ \Theta$ functions were essential in our construction.

**Category:** Number Theory

[1477] **viXra:1705.0130 [pdf]**
*submitted on 2017-05-07 08:49:48*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that there exist an infinity of Poulet numbers P such that concatenating P to the left with the number (s(P) – 1)/2, where s is the sum of digits of P, is obtained a prime; also I make the same conjecture for (s(P) – 1)/3 respectively for (s(P) – 1)/6.

**Category:** Number Theory

[1476] **viXra:1705.0129 [pdf]**
*submitted on 2017-05-07 08:51:48*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In a previous paper, “Primes obtained concatenating a Poulet number P with (s - 1)/n where s digits sum of P and n is 2, 3 or 6”, I noticed that in almost all the cases that I considered if a prime was obtained through this concatenation than the digits sum of P was a prime. That gave me the idea for this paper where I observe that for many primes p having an odd prime digit sum s there exist a prime obtained concatenating p to the left with a divisor of s – 1 (including 1 and s – 1).

**Category:** Number Theory

[1475] **viXra:1705.0126 [pdf]**
*submitted on 2017-05-07 11:22:44*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In a previous paper, “Primes obtained concatenating to the left a prime having an odd prime digit sum s with a divisor of s - 1”, I observed that for many primes p having an odd prime digit sum s there exist a prime obtained concatenating p to the left with a divisor of s – 1. In this paper I conjecture that for any prime p, p ≠ 5, having an odd prime digit sum s there exist an infinity of primes obtained concatenating to the left p with multiples of s – 1. Yet I conjecture that there exist at least a prime obtained concatenating n*(s – 1) with p such that n < sqr s.

**Category:** Number Theory

[1474] **viXra:1705.0117 [pdf]**
*submitted on 2017-05-06 08:50:03*

**Authors:** Jason Cole

**Comments:** 4 Pages.

There is exciting research trying to connect the nontrivial zeros of the Riemann Zeta function to Quantum mechanics as a breakthrough towards proving the 160-year-old Riemann Hypothesis. This research offers a radically new approach.
Most research up to this point have focused only on mapping the nontrivial zeros directly to eigenvalues. Those attempts have failed or didn’t yield any new breakthrough. This research takes a radically different approach by focusing on the quantum mechanical properties of the wave graph of Zeta as ζ(0.5+it) and not the nontrivial zeros directly. The conjecture is that the wave forms in the graph of the Riemann Zeta function ζ(0.5+it) is a wave function ψ. It is made of a Complex version of the Parity Operator wave function. The Riemann Zeta function consists of linked Even and Odd Parity Operator wave functions on the critical line. From this new approach, it shows the Complex version of the Parity Operator wave function is Hermitian and it eigenvalues matches the zeros of the Zeta function.

**Category:** Number Theory

[1473] **viXra:1705.0115 [pdf]**
*submitted on 2017-05-05 16:53:48*

**Authors:** Christopher Goddard

**Comments:** 23 Pages.

Using an extension of the idea of the radical of a number, as well as a few other ideas, it is indicated as to why one might expect the Oesterle-Masser conjecture to be true. Based on structural elements arising from this proof, a criterion is then developed and shown to be potentially sufficient to resolve two relatively deep conjectures about the structure of the prime numbers. A sketch is consequently provided as to how it might be possible to demonstrate this criterion, borrowing ideas from information theory and cybernetics.

**Category:** Number Theory

[1472] **viXra:1705.0100 [pdf]**
*submitted on 2017-05-03 13:05:47*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

This note presents some double integrals for Euler-Mascheroni constant and related fractals.

**Category:** Number Theory

[1471] **viXra:1704.0393 [pdf]**
*submitted on 2017-04-29 21:17:45*

**Authors:** A. A. Frempong

**Comments:** 2 Pages. Copyright © by A. A. Frempong

The above conjecture evolved when after proving the Beal conjecture algebraically (viXra:1702.0331), the author attempted to prove the same conjecture geometrically. The new conjecture states that if A^x + B^y = C^z, where A, B, C, x, y, z are positive integers, x, y, z > 2, and A ≠ B ≠ C ≠ 2, then A, B and C cannot be the lengths of the sides of a triangle. A proof of the above conjecture may shed some light on the relationships between similar equations and the lengths of the sides of polygons. Counterexamples could be added to the exceptions.

**Category:** Number Theory

[1470] **viXra:1704.0392 [pdf]**
*submitted on 2017-04-29 23:48:52*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following three conjectures: (I) The set of the primes which are the sum of three consecutive Poulet numbers is infinite; (II) The set of the primes which are partial sums of the sequence of Poulet numbers is infinite; (III) The set of the primes which are obtained concatenating four consecutive 2-Poulet numbers is infinite.

**Category:** Number Theory

[1469] **viXra:1704.0391 [pdf]**
*submitted on 2017-04-29 23:52:47*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following three conjectures: (I) The set of the primes which are obtained concatenating to the left a prime with its digital sum is infinite; (II) The set of the primes which are obtained concatenating to the left a prime with its digital root is infinite; (III) The set of the primes which are equal to the sum of a prime p with the number obtained concatenating to the left p with its digital sum and the number obtained concatenating to the left p with its digital root is infinite.

**Category:** Number Theory

[1468] **viXra:1704.0306 [pdf]**
*submitted on 2017-04-23 12:08:04*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following observation: Let d be a factor (not necessarily prime) of the Poulet number P such that d < sqr P and m the least number such that m*d*(d – 1) > (P – 1)/2. Let n be equal to P – m*d*(d – 1). Then often exist a set of Poulet numbers Q such that Q mod(m*d*(d – 1)) = n. For example, for P = 2047 = 23*89 and d = 23, where d < sqr 2047, the least m such that m*23*22 > (P – 1)/2 is equal to 3 (1518 > 1023, while, for 2, 1012 < 1023); so, n = 2047 – 3*23*22 = 2047 – 1518 = 529 and indeed there exist a set of Poulet numbers Q such that Q mod 1518 = 529; the formula 1518*x + 529 gives the Poulet numbers 2047, 6601, 15709, 30889 (...) for x = 1, 4, 10, 20 (...).

**Category:** Number Theory

[1467] **viXra:1704.0296 [pdf]**
*submitted on 2017-04-22 23:35:39*

**Authors:** Marius Coman

**Comments:** 1 Page.

In a previous paper, “Poulet numbers in Smarandache prime partial digital sequence and a possible infinite set of primes” I conjectured that there exist an infinity of Poulet numbers which admit a deconcatenation in prime numbers. In this paper I conjecture that there exist an infinity of Poulet numbers which admit a deconcatenation in two prime numbers p and q where q = p + 30*k, where k integer.

**Category:** Number Theory

[1466] **viXra:1704.0295 [pdf]**
*submitted on 2017-04-22 23:37:50*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I conjecture that there exist an infinity of primes obtained concatenating four consecutive numbers, the largest one from them being a Poulet number. For example, 1726172717281729 is such a prime, obtained concatenating the numbers 1726, 1727, 1728 and 1729, where 1729 is a Poulet number (see the sequence A030471 in OEIS for primes which are concatenation of four consecutive numbers).

**Category:** Number Theory

[1465] **viXra:1704.0293 [pdf]**
*submitted on 2017-04-23 04:09:15*

**Authors:** Marius Coman

**Comments:** 2 Pages.

It is well known the story of the Hardy-Ramanujan number, 1729 (also a Poulet number), which is the smallest number expressible as the sum of two cubes in two different ways, but I have not met yet, not even in OEIS, the sequence of the Poulet numbers which can be written as x^3±y^3, sequence that I conjecture in this paper that is infinite. I also conjecture that there are infinite Poulet numbers which are centered cube numbers (equal to 2*n^3 + 3*n^2 + 3*n + 1), also which are centered hexagonal numbers (equal to 3*n^2 + 3*n + 1).

**Category:** Number Theory

[1464] **viXra:1704.0274 [pdf]**
*submitted on 2017-04-21 11:49:07*

**Authors:** François Mendzina Essomba

**Comments:** 1 Page. extreme fomulas

I present in this small article two algorithms of calculation of pi, they are characterized by two extremities, one is the most convergent and the other the slowest of the imaginable formulas.

**Category:** Number Theory

[1463] **viXra:1704.0260 [pdf]**
*submitted on 2017-04-20 07:53:36*

**Authors:** Edgar Valdebenito

**Comments:** 37 Pages.

This note presents a collection of fractals related with constant pi

**Category:** Number Theory

[1462] **viXra:1704.0259 [pdf]**
*submitted on 2017-04-20 07:57:06*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

This note presents some series for pi constant.

**Category:** Number Theory

[1461] **viXra:1704.0258 [pdf]**
*submitted on 2017-04-20 08:02:54*

**Authors:** Edgar Valdebenito

**Comments:** 7 Pages.

In this research , the autor has detailed about: Numerical evaluation of the Ramanujan-Göllnitz-Gordon continued fraction.

**Category:** Number Theory

[1460] **viXra:1704.0240 [pdf]**
*submitted on 2017-04-19 11:32:59*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

This note presents some formulas related with Gelfond constant:exp(pi)

**Category:** Number Theory

[1459] **viXra:1704.0225 [pdf]**
*submitted on 2017-04-17 17:28:46*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: For any n positive integer there exist an infinity of primes which can be deconcatenated in three numbers, i.e., from left to right, a, b and a + b + n. Examples: for n = 0, the least such prime is 101 (1 + 0 + 0 = 1); for n = 1, the least such prime is 113 (1 + 1 + 1 = 3); for n = 2, the least such prime is 103 (1 + 0 + 2 = 3); for n = 3, the least such prime is 137 (1 + 3 + 3 = 7); for n = 4, the least such prime is 127 (1 + 2 + 4 = 7); for n = 5, the least such prime is 139 (1 + 3 + 5 = 9); for n = 6, the least such prime is 107 (1 + 0 + 6 = 7); for n = 7, the least such prime is 3313 (3 + 3 + 7 = 13).

**Category:** Number Theory

[1458] **viXra:1704.0224 [pdf]**
*submitted on 2017-04-17 17:31:06*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: For any n even there exist an infinity of primes which can be deconcatenated in three numbers, i.e., from left to right, p, n and p + n, where p and p + n are primes. Examples: for n = 2, the least such prime is 11213 (11 + 2 = 13); for n = 4, the least such prime is 347 (3 + 4 = 7); for n = 6, the least such prime is 11617 (11 + 6 = 17); for n = 8, the least such prime is 5813 (5 + 8 = 13); for n = 10, the least such prime is 31013 (3 + 10 = 13); for n = 12, the least such prime is 51217 (5 + 12 = 17); for n = 14, the least such prime is 51419 (5 + 14 = 19); for n = 16, the least such prime is 431659 (43 + 16 = 59).

**Category:** Number Theory

[1457] **viXra:1704.0210 [pdf]**
*submitted on 2017-04-16 19:36:57*

**Authors:** Zhang Tianshu

**Comments:** 13 Pages.

Due to exist forevermore uncorrelated limits of values of real number ε≥0, enable ABC conjecture to be able to be both proved and negated. In this article, we find a representative equality 1+2N(2N-2)=(2N-1)2 satisfying (2N-1)2>[Rad(1, 2N(2N-2), (2N-1)2)]1+ε, then both prove the ABC conjecture and negate the ABC conjecture according to two limits of values of ε.

**Category:** Number Theory

[1456] **viXra:1704.0196 [pdf]**
*submitted on 2017-04-15 06:48:55*

**Authors:** François Mendzina Essomba, Gael Dieudonné Essomba Essomba

**Comments:** 7 Pages. algorithm, convergence and approximation

We give algorithms for the calculation of pi. These algorithms
can be easily developed in a linear manner and allows the calculation
of pi with an infinite degree of convergence. Of course, the calculation
of the second term passes through the first one, and it is necessary, as
this type of algorithms, for a larger memory for calculations contrary
to the formula BBP [1] whose execution corresponds to the order of
the desired number.
The advantage of our formulas in spite of the dificulty associated with extracting sin(x) lies in their degree of convergence, which is infinite, they prove the Borweins brothers hypothesis on the construction of algorithms At any speed as symbolized in our generic formula (8) of this paper.
These formulas for the most part are totally new :
We had found several other formulas of pi l

**Category:** Number Theory

[1455] **viXra:1704.0146 [pdf]**
*submitted on 2017-04-11 14:33:49*

**Authors:** Jose Javier garcia Moreta

**Comments:** 4 Pages.

In this paper we define a new Mellin discrete convolution, which is related to Perron's formula. Also we introduce new explicit formulae for arithmetic function which generalize the explicit formulae of Weil.

**Category:** Number Theory

[1454] **viXra:1704.0129 [pdf]**
*submitted on 2017-04-10 22:54:58*

**Authors:** Barry Foster

**Comments:** 2 Pages.

This attempt uses Bertrand’s postulate.

**Category:** Number Theory

[1453] **viXra:1704.0121 [pdf]**
*submitted on 2017-04-10 07:46:25*

**Authors:** Shi-YuanDong

**Comments:** 2 Pages.

The prime partition of n!, On the Goldbach prime partition, and the algebraic sum of elements of prime.

**Category:** Number Theory

[1452] **viXra:1704.0114 [pdf]**
*submitted on 2017-04-09 11:23:01*

**Authors:** Abdelghaffar Slimane

**Comments:** 3 Pages. Academic use only

We give a condition that an odd number in the neighborhood of a successive collatz numbers
set must verify to be a non-collatz number, and we use the result for odd numbers of the form 6k−1 at the boundary of a successive collatz numbers set.

**Category:** Number Theory

[1451] **viXra:1704.0110 [pdf]**
*submitted on 2017-04-09 10:15:18*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 10 Pages.

In this research investigation, the author has presented a novel scheme of Universal Evolution Model. This model can be also successfully used for forecasting purposes.

**Category:** Number Theory

[1450] **viXra:1704.0102 [pdf]**
*submitted on 2017-04-09 00:26:53*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I make the following conjecture: there exist an infinity of Poulet numbers which can be written as a sum of two successive primes plus one (for the numbers that are the sum of two successive primes see the sequence A001043 in OEIS).

**Category:** Number Theory

[1449] **viXra:1704.0101 [pdf]**
*submitted on 2017-04-09 02:46:21*

**Authors:** Chongxi Yu

**Comments:** 18 Pages.

Prime numbers are the basic numbers and are crucial important. There are many conjectures concerning primes are challenging mathematicians for hundreds of years. Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics. A kaleidoscope can produce an endless variety of colorful patterns and it looks like a magic, but when you open it, it contains only very simple, loose, colored objects such as beads or pebbles and bits of glass. Human is very easily cheated by 2 words, infinite and anything, because we never see infinite and anything, so we always make simple thing complex. Goldbach’s conjecture is about all very simple numbers, the pattern of prime numbers likes a “kaleidoscope” of numbers, we divided any even numbers into 5 groups and primes into 4 groups, Goldbach’s conjecture becomes much simpler. Here we give a clear proof for Goldbach's conjecture based on the fundamental theorem of arithmetic, the prime number theorem, and Euclid's proof that the set of prime numbers is endless.
Key words: Goldbach's conjecture, fundamental theorem of arithmetic, Euclid's proof of infinite primes, the prime number theorem

**Category:** Number Theory

[1448] **viXra:1704.0098 [pdf]**
*submitted on 2017-04-08 09:49:00*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following conjecture: let P be a Poulet number and n the integer for which the number (P – 1)/2^n is odd; then there exist an infinity of Poulet numbers for which the number q = (P – 1)/2^n – 2^n is prime.

**Category:** Number Theory

[1447] **viXra:1704.0097 [pdf]**
*submitted on 2017-04-08 10:06:47*

**Authors:** Aaron Chau

**Comments:** 2 Pages.

或许在友善的下午茶叙上，笔者清心直说，既然代数无法正确筛选任一质数，这说明代数的
缺点是难免会把非质数来充当质数。不言而喻，数学毕竟不鼓励凭修饰把非质数来充当质数。
所以，虽然代数的工业用途广泛，但针对解决孪生质数猜想，算术才是一把能够开锁的钥匙。

**Category:** Number Theory

[1446] **viXra:1704.0093 [pdf]**
*submitted on 2017-04-08 03:00:47*

**Authors:** Bing He

**Comments:** 16 Pages.

In this paper we introduce a finite field analogue for the Appell series F_{3} and
give some reduction formulae and certain generating functions for this function over finite fields.

**Category:** Number Theory

[1445] **viXra:1704.0087 [pdf]**
*submitted on 2017-04-07 09:13:10*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that there are an infinity of primes which can be written as sqr ((p – q – 1)*p – q – 1), where p and q are successive primes, p > q.

**Category:** Number Theory

[1444] **viXra:1704.0079 [pdf]**
*submitted on 2017-04-06 23:53:45*

**Authors:** C Sloane

**Comments:** 29 Pages. This paper is with several institutions and this submission is a time-stamp of authorship.

We discovered a beautiful symmetry to the equation x^n+y^n± z^n , first studied by Fermat, in a dependent variable t = x+y-z and the product (xyz) if we introduce a term we call the symmetric r = x^2+yz-xt-t^2. Once x^n+y^n± z^n is written in terms of powers of t, r and (xyz) we looked at the coefficient vs. exponent abstract space and found Lucas, Fibonacci and Convoluted Fibonacci sequences among other corollaries. We also found that 3 cases of a prime decomposition factor q of x^2+yz gave certain results for Fermat’s Last Theorem which could be eliminated if a forth case could also be solved. Intrigued by this, we then introduce partial congruence representations modulo a prime for this much harder forth case to find the ‘form’ of the solutions modulo q. The form of the solutions leads us to a cubic congruence method that solves the special and general cases. There are several pages and stages of the proofs where computer verification of the results is possible.

**Category:** Number Theory

[1443] **viXra:1704.0058 [pdf]**
*submitted on 2017-04-05 08:41:51*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 9 Pages.

In this research investigation, the author has presented a novel scheme of Universal Evolution Model. This can also be used as a Universal Forecasting Model.

**Category:** Number Theory

[1442] **viXra:1704.0056 [pdf]**
*submitted on 2017-04-05 10:40:16*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 10 Pages.

In this research investigation, the author has presented a novel scheme of Universal Evolution Model. Also, this model can be used as a Universal Forecasting Model.

**Category:** Number Theory

[1441] **viXra:1704.0029 [pdf]**
*submitted on 2017-04-03 11:20:53*

**Authors:** A. Zaganidis

**Comments:** 17 Pages. I have chosen the category Mathematics-Number Theory since most of the consequences of the present article are inside the number theory

In this work, we introduce the $n$-formal sequents and the formal numbers defined with the help of the second order logic. We give many concrete examples of formal numbers and $n$-formal sequents with the Peano's axioms and the axioms of the real numbers. Shortly, a sequent is $n$-formal iff the sequent is composed by some closed hypotheses and a $n$-formal formula (an explicit sub-formula with one free variable which is only true with the unique natural number $n$), and no sub-sequent are composed by some closed sub-hypotheses and a $m$-formal explicit sub-formula with $m>1$. The definition is motivated by the intuition that the ``nature's hypotheses'' do not carry natural numbers or "hidden natural numbers" except for the numbers $0$ and $1$, i.e., they can be used in a $n$-formal sequent. Moreover, we postulate at second order of logic that the ``nature's hypotheses'' are not chosen randomly: the ``nature's hypotheses'' are the only hypotheses which give the largest formal number $N_Z=1497$. The Goldbach's conjecture, the Dubner's conjecture, the Polignac's conjecture, the Firoozbakht's conjecture, the Oppermann's conjecture, the Agoh-Giuga conjecture, the generalized Fermat's conjecture and the Schinzel's hypothesis H are reviewed with this new (second order logic) formal axiom. Finally, three open questions remain: Can we prove that a natural number is not formal? If a formal number $n$ is found with a function symbol $f$ where its outputs values are only $0$ and $1$, can we always replace the function symbol $f$ by a another function symbol $\tilde{f}$ such that $\tilde{f}=1-f$ and the new sequent is still $n$-formal? Does a sequent exist to make a difference between the definition of the $n$-formal sequents and the following variant of that definition: we look at the explicit sub-formulas of $\phi$ which induce the $m$-formal formulas instead of looking at the explicit sub-formulas of $\phi_{n-formal}$ which are $m$-formal formulas?

**Category:** Number Theory

[1440] **viXra:1704.0012 [pdf]**
*submitted on 2017-04-02 07:11:13*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 9 Pages.

In this research investigation, the author has presented a novel scheme of Universal Evolution Model.

**Category:** Number Theory

[1439] **viXra:1703.0309 [pdf]**
*submitted on 2017-03-31 21:43:31*

**Authors:** Clive Jones

**Comments:** 6 Pages.

The Escape-Condition is a power of 2

**Category:** Number Theory

[1438] **viXra:1703.0308 [pdf]**
*submitted on 2017-03-31 21:46:05*

**Authors:** Clive Jones

**Comments:** 6 Pages.

Primes & Squares in Pyramids, Blocks & Triangles

**Category:** Number Theory

[1437] **viXra:1703.0304 [pdf]**
*submitted on 2017-03-31 11:38:22*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 9 Pages. In French. Submitted to the International Journal of Number Theory. Comments welcome.

In 1898, Riemann had announced the following conjecture : the nontrivial roots (zeros) $s=\sigma+it$ of the zeta function, defined by:
$$\zeta(s) = \sum_{n=1}^{+\infty}\frac{1}{n^s},\,\mbox{for}\quad \Re(s)>1$$
have real part $\sigma= \ds \frac{1}{2}$. We give a proof that $\sigma= \ds \frac{1}{2}$ using an equivalent statement of Riemann Hypothesis.

**Category:** Number Theory

[1436] **viXra:1703.0297 [pdf]**
*submitted on 2017-03-31 05:36:04*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I note two sequences of Poulet numbers: the terms of the first sequence are the Poulet numbers which can be written as P*2 – d; the terms of the second sequence are the Poulet numbers which can be written as (P*2 – d)*2 - d, where P is another Poulet number and d one of the prime factors of P. I also conjecture that the both sequences are infinite and I observe that the recurrent relation ((((P*2 – d)*2 – d)*2 – d)...) conducts sometimes to more than one Poulet number (for instance, starting with P = 4369 and d = 257, the first, the second and the third numbers obtained are 8481, 16705 and 33153, all three Poulet numbers).

**Category:** Number Theory

[1435] **viXra:1703.0243 [pdf]**
*submitted on 2017-03-25 15:57:12*

**Authors:** Wes Hansen

**Comments:** 7 Pages.

In the following we define a set of hyper-naturals on N x N with the lexicographic ordering and a novel definition of the arithmetical operation, multiplication. These hyper-naturals are isomorphic to ω2 yet have recursive arithmetical operations defined on them, demonstrating a counter-example to Tennenbaum’s Theorem.

**Category:** Number Theory

[1434] **viXra:1703.0241 [pdf]**
*submitted on 2017-03-26 03:17:22*

**Authors:** Barry Foster

**Comments:** 2 Pages.

This attempt is essentially an inductive approach.

**Category:** Number Theory

[1433] **viXra:1703.0237 [pdf]**
*submitted on 2017-03-25 08:25:21*

**Authors:** Ricardo Gil

**Comments:** 4 Pages.

The purpose of this papers is to share an encryption system based on a modified Riemann Zeta function which relates to prime
numbers.

**Category:** Number Theory

[1432] **viXra:1703.0226 [pdf]**
*submitted on 2017-03-23 22:58:58*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 4 Pages.

In this research investigation, the author has detailed about the Scheme of construction of Natural metric for any given positive Integer. Natural Metric can be used for Natural Scaling of any Set optimally. Natural Metric also forms the Universal Basis for the Universal Correspondence Principle between Quantum mechanics and Newtonian Mechanics. Furthermore, Natural Metric finds great use in the Science of Forecasting Engineering.

**Category:** Number Theory

[1431] **viXra:1703.0220 [pdf]**
*submitted on 2017-03-23 01:28:34*

**Authors:** Pedro Caceres

**Comments:** 23 Pages.

A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.
The crucial importance of prime numbers to number theory and mathematics in general stems from the fundamental theorem of arithmetic, which states that every integer larger than 1 can be written as a product of one or more primes in a way that is unique except for the order of the prime factors. Primes can thus be considered the “basic building blocks”, the atoms, of the natural numbers.
There are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no known simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, the statistical behavior of primes in the large, can be modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or to the logarithm of n.
The way to build the sequence of prime numbers uses sieves, an algorithm yielding all primes up to a given limit, using only trial division method which consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime.
This paper introduces a new way to approach prime numbers, called the DNA-prime structure because of its intertwined nature, and a new process to create the sequence of primes without direct division or multiplication, which will allow us to introduce a new primality test, and a new factorization algorithm.
As a consequence of the DNA-prime structure, we will be able to provide a potential proof of Golbach’s conjecture.

**Category:** Number Theory

[1430] **viXra:1703.0211 [pdf]**
*submitted on 2017-03-22 01:05:49*

**Authors:** Simon Plouffe

**Comments:** 41 Pages. Conference is in French

Une conférence sur Pi, le jour de Pi : 14 mars 2017 au Lycée International Winston Churchill : Londres.
A conference on Pi on Pi Day, march 14 2017 at the Winston Churchill International College (Lycée ) London.

**Category:** Number Theory

[1429] **viXra:1703.0192 [pdf]**
*submitted on 2017-03-20 08:06:50*

**Authors:** Helmut Preininger

**Comments:** 10 Pages.

This paper introduces the notion of an S-Structure (short for Squarefree Structure.) After establishing a few simple properties of such S-Structures, we investigate the squarefree natural numbers as a primary example. In this subset of natural numbers we consider "arithmetic" sequences with varying initial elements. It turns out that these sequences are always periodic. We will give an upper bound for the minimal and maximal points of these periods.

**Category:** Number Theory

[1428] **viXra:1703.0180 [pdf]**
*submitted on 2017-03-19 02:37:08*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I conjecture that any number of the form 4*n^2 + 8*n + 3, where n is positive integer, is Fermat pseudoprime to base 2*n + 2.

**Category:** Number Theory

[1427] **viXra:1703.0177 [pdf]**
*submitted on 2017-03-18 07:50:11*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I conjecture that any Poulet number of the form (4^n + 1)/5 where n is prime is either 2-Poulet number either a product of primes p(1)*p(2)*...*p(k) such that all the semiprimes p(i)*p(j), where 1 ≤ i < j ≤ k, are 2-Poulet numbers.

**Category:** Number Theory

[1426] **viXra:1703.0174 [pdf]**
*submitted on 2017-03-18 04:21:30*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that any number of the form (4^n – 1)/3 where n is odd greater than 3 is divisible by a Poulet number (it is known that any number of this form is a Poulet number if n is prime greater than 3; such a number is called Cipolla pseudoprime to base 2, see the sequence A210454 in OEIS).

**Category:** Number Theory

[1425] **viXra:1703.0155 [pdf]**
*submitted on 2017-03-15 23:40:37*

**Authors:** T.Nakashima

**Comments:** 2 Pages.

Near m, the destance of primes is lower order than logm. This is
the key to solve the Legendre's conjecture.

**Category:** Number Theory

[1424] **viXra:1703.0147 [pdf]**
*submitted on 2017-03-14 15:20:02*

**Authors:** Philip A. Bloom

**Comments:** Pages. Useful bibliographic references do not exist for this study.

This two-page proof, by contraposition, of Fermat's last theorem, uses two analogous forms of a previously overlooked equation, each of which is equivalent to rational z^n-y^n = x^n. A relationship held in common by these two equations reveals the crucial n = 1, 2 limitation.

**Category:** Number Theory

[1423] **viXra:1703.0124 [pdf]**
*submitted on 2017-03-13 13:55:36*

**Authors:** Petr E. Pushkarev

**Comments:** 5 Pages. was published in the Global Journal of Pure and Applied Mathematics 13, no. 6 (2017): 1987-1992

In this article we are closely examining Riemann zeta function's non-trivial zeros. Especially, we examine real part of non-trivial zeros. Real part of Riemann zeta function's non-trivial zeros is considered in the light of constant quality of such zeros. We propose and prove a theorem of this quality. We also uncover a definition phenomenons of zeta and Riemann xi functions. In conclusion and as an conclusion we observe Riemann hypothesis in perspective of our researches.

**Category:** Number Theory

[1422] **viXra:1703.0115 [pdf]**
*submitted on 2017-03-13 02:26:30*

**Authors:** John Yuk Ching Ting

**Comments:** 18 Pages. This landmark research paper essentially contains the rigorous proofs for Polignac's and Twin prime conjectures.

Prime numbers and composite numbers are intimately related simply because the complementary set of composite numbers constitutes the set of natural numbers with the exact set of prime numbers excluded in its entirety. In this research paper, we predominantly use our 'Virtual container' method, which incorporates the novel mathematical tool coined Information-Complexity conservation with its core foundation based on this [complete] prime-composite number relationship, to solve the intractable open problem of whether prime gaps are infinite (arbitrarily large) in magnitude with each individual prime gap generating prime numbers which are again infinite in magnitude. This equates to solving Polignac's conjecture which involves analysis of all possible prime gaps = 2, 4, 6,... and [the subset] Twin prime conjecture which involves analysis of prime gap = 2 (for twin primes).

**Category:** Number Theory

[1421] **viXra:1703.0114 [pdf]**
*submitted on 2017-03-13 03:27:18*

**Authors:** John Yuk Ching Ting

**Comments:** 20 Pages. This research paper essentially contains the rigorous proof for Riemann hypothesis.

The triple countable infinite sets of (i) x-axis intercepts, (ii) y-axis intercepts, and (iii) both x- and y-axes [formally known as the 'Origin'] intercepts in Riemann zeta function are intimately related to each other simply because they all constitute complementary points of intersection arising from the single [exact same] countable infinite set of curves generated by this function. This [complete] relationship amongst all three sets of intercepts will enable us to simultaneously study important intrinsic properties derived from all those intercepts in a mathematically consistent manner which then provides the rigorous proof for Riemann hypothesis as well as fully explain x-axis intercepts (which is the usual traditionally-dubbed 'Gram points') and y-axis intercepts. Riemann hypothesis involves analysis of all nontrivial zeros of Riemann zeta function and refers to the celebrated proposal by famous German mathematician Bernhard Riemann in 1859 whereby all nontrivial zeros are conjectured to be located on the critical line [or equivalently stated as all nontrivial zeros are conjectured to exactly match the Origin intercepts]. Concepts from the Hybrid method of Integer Sequence classification, together with our key formulae coined Sigma-Power Laws, are some of the important mathematical tools employed in this paper to successfully achieve our proof. Not least in [again] using the same 'Virtual container' method in this current research paper, there are other additional deeply inseparable mathematical connections between the content of this paper and our recent publication [Ting, J. Y. C. (March 13, 2017), Unravelling the Dual Source of Prime Number Infiniteness from Prime Gaps using Information-Complexity Conservation, viXra, 1--18] on the dual source of prime number infiniteness.

**Category:** Number Theory

[1420] **viXra:1703.0104 [pdf]**
*submitted on 2017-03-11 10:45:33*

**Authors:** Pedro Caceres

**Comments:** 25 Pages.

PrimeNumbers are the atoms of mathematics and mathematics is needed to make sense of the real world. Finding the Prime number structure and eventually being able to crack their code is the ultimate goal in what is called Number Theory. From the evolution of species to cryptography, Nature finds help in Prime numbers.
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 the roots (zeros) of the zeta function, defined by:
(1)ζ(z) = ∑1/n^z
The Zeta function, ζ(z), is a function of a complex variable z that analytically continues the Dirichlet series.
Riemann also formulated a conjecture about the location of these zeros, which fall into two classes: the "trivial zeros" -2, -4, -6, etc., and those whose real part lies between 0 and 1. Riemann's conjecture Riemann hypothesis [RH] was formulated as this:
[RH]The real part of every non-trivial zero z* of the Riemann Zeta function is 1/2.
Thus, if the hypothesis is correct, all the non-trivial zeros lie on the critical line consisting of the complex numbers 1/2 + i ß, where ß is a real number and i is the imaginary unit.
In this paper, we will analyze the Riemann Zeta function and provide an analytical/geometrical proof of the Riemann Hypothesis. The proof will be based on the fact that if we decompose the ζ(z) in a difference of two functions, both functions need to be equal when ζ(z)=0, so their distance to the origin or modulus must be equal and we will prove that this can only happen when Re(z)=1/2 for certain values of Im(z).
We will also prove that all non-trivial zeros of ζ(z) in the form z=1/2+iß have all ß related by an algebraic expression. They are all connected and not independent.
Finally, we will show that as a consequence of this connection of all ß, the harmonic function Hn can be expressed as a function of each ß zero of ζ(z) with infinite representations.
We will use mathematical and computational methods available for engineers.

**Category:** Number Theory

[1419] **viXra:1703.0097 [pdf]**
*submitted on 2017-03-11 02:01:57*

**Authors:** Wu ShengPing

**Comments:** 4 Pages.

The main idea of this article is simply calculating integer
functions in module. The algebraic in the integer modules is studied in
completely new style. By a careful construction the result that
two finite numbers is with unequal logarithms in a corresponding module is proven, which result is applied to solving
a kind of diophantine equation: $c^q=a^p+b^p$.

**Category:** Number Theory

[1418] **viXra:1703.0086 [pdf]**
*submitted on 2017-03-09 09:40:06*

**Authors:** Stephen Marshall

**Comments:** 8 Pages.

This paper presents a complete and exhaustive proof of Landau's Fourth Problem. The approach to this proof uses same logic that Euclid used to prove there are an infinite number of prime numbers. Then we use a proof found in Reference 1, that if p > 1 and d > 0 are integers, that p and p+ d are both primes if and only if for integer m:
m = (p-1)!( 1/p + ((-1)^d (d!))/(p+d)) + 1/p + 1/(p+d)
We use this proof for d = 2n + 1 to prove the infinitude of Landau’s Fourth Problem prime numbers.
The author would like to give many thanks to the authors of 1001 Problems in Classical Number Theory, Jean-Marie De Koninck and Armel Mercier, 2004, Exercise Number 161 (see Reference 1). The proof provided in Exercise 6 is the key to making this paper on the Landau’s Fourth Problem possible.

**Category:** Number Theory

[1417] **viXra:1703.0078 [pdf]**
*submitted on 2017-03-08 10:45:40*

**Authors:** Wu ShengPing

**Comments:** 4 Pages.

The main idea of this article is simply calculating integer
functions in module. The algebraic in the integer modules is studied in
completely new style. By a careful construction a result is obtained on
two finite numbers with unequal logarithms, which result is applied to solving
a kind of diophantine equations.

**Category:** Number Theory

[1416] **viXra:1703.0048 [pdf]**
*submitted on 2017-03-05 22:06:49*

**Authors:** Stephen Crowley

**Comments:** 5 Pages.

It is proved that the limit of argζ(1/2+i ρ)=-1/2-frac((ϑ(ρ))/π) when ζ(ρ)=0. Therefore, the exact transcendental equation for the Riemann zeros has a solution for each positive integer n which proves that the Riemann hypothesis is true since the counting function for zeros on the critical line is equal to the counting function for zeros on the critical strip if the transcendental equation has a solution for each n, as Leclair has shown.

**Category:** Number Theory

[1415] **viXra:1703.0040 [pdf]**
*submitted on 2017-03-04 11:30:54*

**Authors:** Antoine Balan

**Comments:** 3 pages, written in French

We propose in the present paper to consider the Riemann Hypothesis asympotically (ARH) ; it means when the imaginary part of the zero in the critical band is great. We show that the problem, expressed in these terms, is equivalent to the fact that an equation called the * equation has only a finite number of solutions, but we have not proved it.

**Category:** Number Theory

[1414] **viXra:1703.0033 [pdf]**
*submitted on 2017-03-03 15:32:44*

**Authors:** Reuven Tint

**Comments:** 5 Pages. original papper in russian

Keywords: three-term equation, the method of infinite growth, elementary aspect.
Annotation. An infinitely lifting method for making certain types of three-term equations, which is completely refuted by the ABC conjecture.

**Category:** Number Theory

[1413] **viXra:1703.0022 [pdf]**
*submitted on 2017-03-03 10:27:01*

**Authors:** Peter Bissonnet

**Comments:** 5 Pages.

This paper elucidates the major points of the above referenced paper.
1. Emphasizes the derivation of the double helices and that they are not arbitrarily chosen.
2. Explains why multiples of 42 appear in prime number theory.
3. Why s in 6s-1 and 6s+1 is really a composite number.
4. Why 2 and 3 are not true prime numbers based upon characteristics.
5. Philosophical reason as to the double helices falling more into a discoverable category (as in experimental physics), as opposed to being postulate driven.

**Category:** Number Theory

[1412] **viXra:1703.0021 [pdf]**
*submitted on 2017-03-02 16:52:23*

**Authors:** Stephen Marshall

**Comments:** 15 Pages.

This paper presents a complete proof of the Pell Primes are infinite. We use a proof found in Reference 1, that if p > 1 and d > 0 are integers, that p and p + d are both primes if and only if for integer m:
m = (p-1)!( + ) + +
We use this proof for d = - to
prove the infinitude of Pell prime numbers. The author would like to give many thanks to the authors of 1001 Problems in Classical Number Theory, Jean-Marie De Koninck and Armel Mercier, 2004, Exercise Number 161 (see Reference 1). The proof provided in Exercise 6 is the key to making this paper on the Pell Prime Conjecture possible.

**Category:** Number Theory

[1411] **viXra:1703.0005 [pdf]**
*submitted on 2017-03-01 04:34:21*

**Authors:** Ricardo Gil

**Comments:** 3 Pages.

The purpose of this paper is to provide algorithm that is 5 lines of code and that finds P & Q when N is given. It will work for RSA-2048 if the computer can float large numbers in PyCharm or Python. Also, the P&Q from Part I of the algorithm becomes the range for a for loop in Part II that returns and solves P*Q=N (True).

**Category:** Number Theory

[1410] **viXra:1702.0335 [pdf]**
*submitted on 2017-02-27 15:30:38*

**Authors:** Stephen Marshall

**Comments:** 9 Pages.

This paper presents a complete proof of the Pierpont Primes are infinite, even though only 16 of them have been found as of 21 Feb 2017. We use a proof found in Reference 1, that if p > 1 and d > 0 are integers, that p and p+ d are both primes if and only if for integer m:
m = (p-1)!( + ) + +
We use this proof for d = 2u(n+1)3v(x+1) – 2u(n)3v(x) to prove the infinitude of Pierpont prime numbers. The author would like to give many thanks to the authors of 1001 Problems in Classical Number Theory, Jean-Marie De Koninck and Armel Mercier, 2004, Exercise Number 161 (see Reference 1). The proof provided in Exercise 6 is the key to making this paper on the Pierpont Prime Conjecture possible.

**Category:** Number Theory

[1409] **viXra:1702.0331 [pdf]**
*submitted on 2017-02-28 01:30:08*

**Authors:** A. A. Frempong

**Comments:** 4 Pages. Copyright © by A. A. Frempong

Using a direct construction approach, the author proves the original Beal conjecture that if
A^x + B^y = C^z , where A, B, C, x, y, z are positive integers and x, y, z > 2, then A, B and C have a common prime factor. Two equations are involved, namely, the equation, A^x + B^y = C^z, and a similar equation, G^m + H^n = I^p which will be called the tester equation. The tester equation has similar properties as A^x + B^y = C^z, and it is used to determine the properties of A^x + B^y = C^z. Each side of the two equations involved is reduced to unity by division. The non-unity sides are justifiably equated to each other to produce a new equation which will be called the master equation. The side of the master equation involving G^m, H^n and I^p will be called the tester side of the master equation. Two versions of the proof are presented. In Version 1 proof, the tester equation is a literal tester equation, but in Version 2 proof, the tester equation is a numerical tester equation. By inspection, using an approach in which the corresponding elements on the right and left sides of the master equation are equated to each other, it is determined that A, B and C have a common prime factor. The proof is very simple, and occupies a single page, and even high school students can learn it.

**Category:** Number Theory

[1408] **viXra:1702.0323 [pdf]**
*submitted on 2017-02-26 23:55:26*

**Authors:** Stephen Crowley

**Comments:** 3 Pages.

It is proved that the non-trivial roots of the Hardy Z function are simple having multiplicity 1 by showing that the fixed-points N_Z(α)=α of the Newton map N_Z(t)=t-(Z(t))/(Z˙(t)) must have a multiplier λ_(N_Z)(α)=|(N_Z)˙(α)|=|(Z(α)Z¨(α))/(Z˙(α))|=0 and therefore a multiplicity
m_Z(α)=1/(1-λ_(N_Z)(α))=1/(1-0)=1.

**Category:** Number Theory

[1407] **viXra:1702.0318 [pdf]**
*submitted on 2017-02-26 13:00:05*

**Authors:** Reza Farhadian

**Comments:** 5 Pages.

Let p_n be the nth prime number. We prove that p_(n+1)<〖p_n〗^((n+1)/n ((logp_(n+1))/(logp_n )) ) for every n≥1. This inequality is weaker than the Firoozbakht’s conjecture p_(n+1)<〖p_n〗^((n+1)/n).Afterward we prove that the new inequality is equivalent to Firoozbakht’s conjecture, as n→∞, and hence the Cramér’s conjecture p_(n+1)-p_n=O(log^2 p_n ) to be hold, because the Firoozbakht’s conjecture is stronger than the Cramér’s conjecture, see [5].

**Category:** Number Theory

[1406] **viXra:1702.0313 [pdf]**
*submitted on 2017-02-26 02:45:31*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I conjecture that any number of the form 16^n – 4^n + 1, where n is positive integer, is either prime either divisible by a Poulet number (see the sequence A020520 in OEIS for the numbers of this form).

**Category:** Number Theory

[1405] **viXra:1702.0300 [pdf]**
*submitted on 2017-02-23 13:14:42*

**Authors:** Ralf Wüsthofen

**Comments:** 11 Pages. Older versions on http://vixra.org/abs/1403.0083

The present paper shows that a principle known as emergence lies beneath the strong Goldbach conjecture. Whereas the traditional approaches focus on the control over the distribution of the primes by means of circle method and sieve theory, we give a proof of the conjecture that is based on the constructive properties of the prime numbers, reflecting their multiplicative character within the natural numbers. With an equivalent but more convenient form of the conjecture in mind, we create a structure on the natural numbers. That structure leads to arithmetic identities which immediately imply the conjecture, more precisely, an even strengthened form of it. Moreover, we can achieve further results by generalizing the structuring. Thus, it turns out that the statement of the strong Goldbach conjecture is the special case of a general principle.

**Category:** Number Theory

[1404] **viXra:1702.0299 [pdf]**
*submitted on 2017-02-23 14:27:23*

**Authors:** Stephen Marshall

**Comments:** 7 Pages.

This paper presents a complete proof of the Factorial Primes are infinite, even though only 16 of them have been found as of 21 Feb 2017. We use a proof found in Reference 1, that if p > 1 and d > 0 are integers, that p and p+ d are both primes if and only if for integer m:
m = (p-1)!( 1/p + ((-1)^d(d!))/(p+d)) + 1/p + 1/(p+d)
We use this proof for d = n(n!) to prove the infinitude of Factorial prime numbers.
The author would like to give many thanks to the authors of 1001 Problems in Classical Number Theory, Jean-Marie De Koninck and Armel Mercier, 2004, Exercise Number 161 (see Reference 1). The proof provided in Exercise 6 is the key to making this paper on the Factorial Prime possible.

**Category:** Number Theory

[1403] **viXra:1702.0286 [pdf]**
*submitted on 2017-02-22 16:09:30*

**Authors:** Stephen Marshall

**Comments:** 3 Pages.

In mathematics, and in particular number theory, Grimm's Conjecture (named after Karl Albert Grimm) states that to each element of a set of consecutive composite numbers one can assign a distinct prime that divides it. It was first published in American Mathematical Monthly, 76(1969) 1126-1128.
The Formal statement defining Grimm’s Conjecture, still unproved, is as follows:
Suppose n + 1, n + 2, …, n + k are all composite numbers, then there are k distinct primes pi such that pi divides n + i for 1 ≤ i ≤ k.

**Category:** Number Theory

[1402] **viXra:1702.0285 [pdf]**
*submitted on 2017-02-22 16:11:33*

**Authors:** Stephen Marshall

**Comments:** 3 Pages.

In mathematics, Hall's conjecture is an open question, as of 2015, on the differences cube x3 that are not equal must lie a substantial distance apart. This question arose from consideration of the Mordell equation in the theory of integer points on elliptic curves. The original version of Hall's conjecture, formulated by Marshall Hall, Jr. in 1970, says that there is a positive constant C such that for any integers x and y for which y2 ≠ x3,

**Category:** Number Theory

[1401] **viXra:1702.0273 [pdf]**
*submitted on 2017-02-21 14:19:03*

**Authors:** Stephen Crowley

**Comments:** 5 Pages. The method described can be extended so that it converges to *all* the zeros. I am just posting this preliminary version in case I get hit with an asteroid before I finish writing it up.

A sequence of Cauchy sequences which converge to (almost all) the Riemann zeros is constructed.

**Category:** Number Theory

[1400] **viXra:1702.0271 [pdf]**
*submitted on 2017-02-21 16:20:17*

**Authors:** Stephen Marshall

**Comments:** 7 Pages.

This paper presents a complete proof of the Cullen Primes are infinite, even though only 16 of them have been found as of 21 Feb 2017. We use a proof found in Reference 1, that if p > 1 and d > 0 are integers, that p and p+ d are both primes if and only if for integer m:
See paper for this equation, as the text in this abstract does not support the mathematical format for this equation.
We use this proof for d = P2 + 1 to prove the infinitude of Cullen prime numbers.
The author would like to give many thanks to the authors of 1001 Problems in Classical Number Theory, Jean-Marie De Koninck and Armel Mercier, 2004, Exercise Number 161 (see Reference 1). The proof provided in Exercise 6 is the key to making this paper on the Cullen Prime Conjecture possible.

**Category:** Number Theory

[1399] **viXra:1702.0265 [pdf]**
*submitted on 2017-02-21 06:04:58*

**Authors:** Rédoane Daoudi

**Comments:** 7 Pages.

In our previous work (The distribution of prime numbers: overview of n.ln(n), (1) and (2)) we defined a new method derived from Rosser's theorem (2) and we used it in order to approximate the nth prime number. In this paper we improve our method to try to determine the next prime number if the previous is known. We use our method with five intervals and two values for n (see Methods and results). Our preliminary results show a reduced difference between the real next prime number and the number given by our algorithm. However long-term studies are required to better estimate the next prime number and to reduce the difference when n tends to infinity. Indeed an efficient algorithm is an algorithm that could be used in practical research to find new prime numbers for instance.

**Category:** Number Theory

[1398] **viXra:1702.0264 [pdf]**
*submitted on 2017-02-21 01:57:28*

**Authors:** Marius Coman

**Comments:** 2 Pages.

The Woodall numbers are defined by the formula W(n) = n*2^n – 1 (see the sequence A003261 in OEIS). In this paper I conjecture that any Woodall number of the form 2^k*2^(2^k) – 1, where k ≥ 3, is either prime either divisible by a Poulet number.

**Category:** Number Theory

[1397] **viXra:1702.0259 [pdf]**
*submitted on 2017-02-20 10:38:56*

**Authors:** Marius Coman

**Comments:** 3 Pages.

The Poulet numbers (or the Fermat pseudoprimes to base 2) are defined by the fact that are the only composites n for which 2^(n – 1) – 1 is divisible by n (so, of course, all Mersenne numbers 2^(n - 1) – 1 are divisible by Poulet numbers if n is a Poulet number; but these are not the numbers I consider in this paper). In a previous paper I conjectured that any composite Mersenne number of the form 2^m – 1 with odd exponent m is divisible by a 2-Poulet number but seems that the conjecture was infirmed for m = 49. In this paper I conjecture that any Mersenne number (with even exponent) 2^(p – 1) – 1 is divisible by at least a Poulet number for any p prime, p ≥ 11, p ≠ 13.

**Category:** Number Theory

[1396] **viXra:1702.0253 [pdf]**
*submitted on 2017-02-20 09:23:51*

**Authors:** Rédoane Daoudi

**Comments:** 12 Pages.

The empirical formula giving the nth prime number p(n) is p(n) = n.ln(n) (from ROSSER (2)). Other studies have been performed (from DUSART for example (1)) in order to better estimate the nth prime number. Unfortunately these formulas don't work since there is a significant difference between the real nth prime number and the number given by the formulas. Here we propose a new model in which the difference is effectively reduced compared to the empirical formula. We discuss about the results and hypothesize that p(n) can be approximated with a constant defined in this work. As prime numbers are important to cryptography and other fields, a better knowledge of the distribution of prime numbers would be very useful. Further investigations are needed to understand the behavior of this constant and therefore to determine the nth prime number with a basic formula that could be used in both theoretical and practical research.

**Category:** Number Theory

[1395] **viXra:1702.0226 [pdf]**
*submitted on 2017-02-17 04:27:18*

**Authors:** Predrag Terzic

**Comments:** 3 Pages.

Polynomial time probable prime test for specific class of N=k*b^n-1 is introduced .

**Category:** Number Theory

[1394] **viXra:1702.0191 [pdf]**
*submitted on 2017-02-16 10:26:00*

**Authors:** Zeraoulia Elhadj

**Comments:** 8 Pages.

This note is concerned with presenting sufficient conditions to proves that the number of elements of certain real sequences is infinite.

**Category:** Number Theory

[1393] **viXra:1702.0166 [pdf]**
*submitted on 2017-02-14 10:18:35*

**Authors:** Chongjunhuang

**Comments:** 10 Pages.

Prime density formula

**Category:** Number Theory

[1392] **viXra:1702.0162 [pdf]**
*submitted on 2017-02-14 08:01:15*

[1391] **viXra:1702.0160 [pdf]**
*submitted on 2017-02-13 16:00:14*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following conjecture: If F(2*p) is a Fibonacci number with an index equal to 2*p, where p is prime, p ≥ 5, then there exist a prime or a product of primes q1 and a prime or a product of primes q2 such that F(2*p) = q1*q2 having the property that q2 – 2*q1 is also a Fibonacci number with an index equal to 2^n*r, where r is prime or the unit and n natural. Also I observe that the ratio q2/q1 seems to be a constant k with values between 2.2 and 2.237; in fact, for p ≥ 17, the value of k seems to be 2.236067(...).

**Category:** Number Theory

[1390] **viXra:1702.0157 [pdf]**
*submitted on 2017-02-13 21:14:17*

**Authors:** Chongxi Yu

**Comments:** 8 Pages.

Prime numbers are the basic numbers and are crucial important. There are many conjectures concerning primes are challenging mathematicians for hundreds of years and many “advanced mathematics tools” are used to solve them, but they are still unsolved. Based on the fundamental theorem of arithmetic and Euclid’s proof of endless prime numbers, we have proved there are infinitely many twin primes.

**Category:** Number Theory

[1389] **viXra:1702.0150 [pdf]**
*submitted on 2017-02-13 14:43:06*

**Authors:** Stephen Marshall

**Comments:** 4 Pages.

Christian Goldbach (March 18, 1690 – November 20, 1764) was a German mathematician. He is remembered today for Goldbach's conjecture.
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states: Every even integer greater than 2 can be expressed as the sum of two primes.
On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII) in which he proposed the following conjecture:
Every even integer which can be written as the sum of two primes (the strong conjecture)
He then proposed a second conjecture in the margin of his letter:
Every odd integer greater than 7 can be written as the sum of three primes (the weak conjecture).
A Goldbach number is a positive even integer that can be expressed as the sum of two odd primes. Since four is the only even number greater than two that requires the even prime 2 in order to be written as the sum of two primes, another form of the statement of Goldbach's conjecture is that all even integers greater than 4 are Goldbach numbers.
The “strong” conjecture has been shown to hold up through 4 × 1018, but remains unproven for almost 300 years despite considerable effort by many mathematicians throughout history.
In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that
Every odd number greater than 5 can be expressed as the sum of three primes. (A prime may be used more than once in the same sum). In 2013, Harald Helfgott proved Goldbach's weak conjecture.
The author would like to give many thanks to Helfgott’s proof of the weak conjecture, because this proof of the strong conjecture is completely dependent on Helfgott’s proof. Without Helfgott’s proof, this elementary proof would not be possible.

**Category:** Number Theory

[1388] **viXra:1702.0136 [pdf]**
*submitted on 2017-02-12 02:55:02*

**Authors:** Predrag Terzic

**Comments:** 1 Page.

Polynomial time primality test for safe primes is introduced .

**Category:** Number Theory

[1387] **viXra:1702.0090 [pdf]**
*submitted on 2017-02-07 08:27:42*

**Authors:** Chongxi Yu

**Comments:** 29 Pages.

Prime numbers are the basic numbers and are crucial important. There are many conjectures concerning primes are challenging mathematicians for hundreds of years. Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics. We give a clear proof for Goldbach's conjecture based on the fundamental theorem of arithmetic and Euclid's proof that the set of prime numbers is endless.

**Category:** Number Theory

[1386] **viXra:1702.0030 [pdf]**
*submitted on 2017-02-02 11:56:36*

**Authors:** Stephen Marshall

**Comments:** 8 Pages. This is an update to my proff subitted in 2014, I have simpified the submission by removing uneccessary material from the proof.

This paper presents a complete and exhaustive proof of the Polignac Prime Conjecture. The approach to this proof uses same logic that Euclid used to prove there are an infinite number of prime numbers. Then we use a proof found in Reference 1, that if p > 1 and d > 0 are integers, that p and p+ d are both primes if and only if for integer n:
n = (p-10!(1/p + ((-1)^d(d!))/(p+d)) + 1/p + 1/(p+d)
We use this proof for d = 2k to prove the infinitude of Polignac prime numbers.
The author would like to give many thanks to the authors of 1001 Problems in Classical Number Theory, Jean-Marie De Koninck and Armel Mercier, 2004, Exercise Number 161 (see Reference 1). The proof provided in Exercise 6 is the key to making this paper on the Polignac Prime Conjecture possible.
Additionally, our proof of the Polignac Prime Conjecture leads to proofs of several other significant number theory conjectures such as the Goldbach Conjecture, Twin Prime Conjecture, Cousin Prime Conjecture, and Sexy Prime Conjecture. Our proof of Polignac’s Prime Conjecture provides significant accomplishments to Number Theory, yielding proofs to several conjectures in number theory that has gone unproven for hundreds of years.

**Category:** Number Theory

[1385] **viXra:1702.0027 [pdf]**
*submitted on 2017-02-02 09:19:28*

**Authors:** Dragan Turanyanin

**Comments:** 3 Pages.

Three real numbers are introduced via related infinite series. With e, together they complete a quadruplet.

**Category:** Number Theory

[1384] **viXra:1701.0682 [pdf]**
*submitted on 2017-01-30 17:11:35*

**Authors:** Federico Gabriel

**Comments:** 2 Pages.

In this article, a prime number distribution formula is given. The formula is based on the periodic property of the sine function and an important trigonometric limit.

**Category:** Number Theory

[1383] **viXra:1701.0664 [pdf]**
*submitted on 2017-01-29 15:23:52*

**Authors:** Andrei Lucian Dragoi

**Comments:** 7 Pages.

This article proposes the generalization of the both binary (strong) and ternary (weak) Goldbach’s Conjectures (BGC and TGC), briefly called “the Vertical Goldbach’s Conjectures” (VBGC and VTGC), discovered in 2007[1] and perfected until 2016[2] by using the arrays (S_p and S_o,p) of Matrix of Goldbach index-partitions (GIPs) (simple M_p,n and recursive M_o,p,n, with order o ≥ 0), which are a useful tool in studying BGC by focusing on prime indexes (as the function P_n that numbers the primes is a bijection). Simple M (M_p,n) and recursive M (M_o,p,n) are related to the concept of generalized “primeths” (a term first used by Fernandez N. in his “The Exploring Primeness Project”), which is the generalization with order o≥0 of the known “higher-order prime numbers” (alias “superprime numbers”, “super-prime numbers”, ”super-primes”, ” super-primes” or “prime-indexed primes[PIPs]”) as a subset of (simple or recursive) primes with (also) prime indexes (oPx is the x-th o-primeth, with order o ≥ 0 as explained later on).
The author of this article also brings in a S-M-synthesis of some Goldbach-like conjectures (GLC) (including those which are “stronger” than BGC) and a new class of GLCs “stronger” than BGC, from which VBGC (which is essentially a variant of BGC applied on a serial array of subsets of primeths with a general order o ≥ 0) distinguishes as a very important conjecture of primes (with great importance in the optimization of the BGC experimental verification and other potential useful theoretical and practical applications in mathematics [including cryptography and fractals] and physics [including crystallography and M-Theory]), and a very special self-similar propriety of the primes subset of (noted/abbreviated as or as explained later on in this article).
Keywords: Prime (number), primes with prime indexes, the o-primeths (with order o≥0), the Binary Goldbach Conjecture (BGC), the Ternary Goldbach Conjecture (TGC), Goldbach index-partition (GIP), fractal patterns of the number and distribution of Goldbach index-partitions, Goldbach-like conjectures (GLC), the Vertical Binary Goldbach Conjecture (VBGC) and Vertical Ternary Goldbach Conjecture (VTGC) the as applied on o-primeths

**Category:** Number Theory

[1382] **viXra:1701.0647 [pdf]**
*submitted on 2017-01-28 03:12:53*

**Authors:** M. MADANI Bouabdallah

**Comments:** 7 Pages. Seul M. Andrzej Schinzel (IMPAN) a accepté d'examiner mon texte début janvier,il en a résulté 3 observations.Les 2 premières ont été solutionnées (lemmes 1 et 2) et la 3ème a fait l'objet d'un désaccord.J'ai demandé l'arbitrage à MM. Pierre Deligne,E. Bom

J.P. Gram (1903)writes p.298 of his paper
'Note sur les zéros de la fonction zéta de Riemann' :
'Mais le résultat le plus intéressant qu'ait donné ce calcul consiste en ce qu'il révèle l'irrégularité qui se trouve dans la série des α. Il est très probable que ces racines sont liées intimement aux nombres premiers.
La recherche de cette dépendance, c'est-à-dire la manière dont une α donnée est exprimée au moyen des nombres premiers sera l'objet d'études ultérieures.'
Also the proof of the Riemann hypothesis is based on the definition of an application between the set P of the prime numbers and the set S of the zeros of ζ.

**Category:** Number Theory

[1381] **viXra:1701.0630 [pdf]**
*submitted on 2017-01-26 22:23:47*

**Authors:** Kelvin Kian Loong Wong

**Comments:** 17 Pages. French translation for abstract and keywords

This paper provides a potential pathway to a formal simple proof of Fermat's Last Theorem. The geometrical formulations of n-dimensional hypergeometrical models in relation to Fermat's Last Theorem are presented. By imposing geometrical constraints pertaining to the spatial allowance of these hypersphere configurations, it can be shown that a violation of the constraints confirms the theorem for n equal to infinity to be true.

**Category:** Number Theory

[1380] **viXra:1701.0618 [pdf]**
*submitted on 2017-01-25 20:40:28*

**Authors:** Juan G. Orozco

**Comments:** 8 Pages.

Abstract. This paper introduces proofs to several open problems in number theory, particularly the Goldbach Conjecture and the Twin Prime Conjecture. These two conjectures are proven by using a greedy elimination algorithm, and incorporating Mertens' third theorem and the twin prime constant. The argument is extended to Germain primes, Cousin Primes, and other prime related conjectures. A generalization is provided for all algorithms that result in a Euler product\prod{1-\frac{a}{p}}.

**Category:** Number Theory

[1379] **viXra:1701.0602 [pdf]**
*submitted on 2017-01-24 00:00:25*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that for any 3-Carmichael number (absolute Fermat pseudoprime with three prime factors, see the sequence A087788 in OEIS) of the form (4*h + 1)*(4*j + 1)*(4*k + 1) is true that h, j and k must share a common factor (in fact, for seven from a randomly chosen set of ten consecutive, reasonably large, such numbers it is true that both j and k are multiples of h). The conjecture is probably true even for the larger set of 3-Poulet numbers (Fermat pseudoprimes to base 2 with three prime factors, see the sequence 215672 in OEIS).

**Category:** Number Theory

[658] **viXra:1706.0421 [pdf]**
*replaced on 2017-06-23 12:02:29*

**Authors:** Andrea OSSICINI

**Comments:** 11 Pages.

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

**Category:** Number Theory

[657] **viXra:1706.0022 [pdf]**
*replaced on 2017-06-04 01:31:19*

**Authors:** T.Nakashima

**Comments:** 2 Pages.

This paper is the answer of collatz's Problem.

**Category:** Number Theory

[656] **viXra:1705.0471 [pdf]**
*replaced on 2017-06-05 13:24:31*

**Authors:** V.I.Saenko

**Comments:** 3 Pages. This is a corrected Russian variant. In the previous version, the reviewer found a significant error. Systems (10) and (11) can be excluded. The left-hand side of the first equation of system (9) is certainly not zero for even M and odd G.

A perfect cuboid, i.e., a rectangular parallelepiped having integer edges, integer face diagonals, and integer space diagonal, is proved to be not possible

**Category:** Number Theory

[655] **viXra:1705.0392 [pdf]**
*replaced on 2017-05-31 13:11:14*

**Authors:** M. A. Thomas

**Comments:** 71 Pages. A PDF copy of a slide presentation containing 71 slides Final pion and electron forms match particle ratios

A PDF copy of a Slide Presentation. Relationship of number theory to physics forms are established using a 'wholly trinity' consisting of the first odd Riemann Zeta function, the Euler-Mascheroni constant and the imaginary part of the first non-trivial Zeta zero. Quark flavor changes occur under extremal gravities with only slight asymmetric changes to dimensionless ratios. Symmetric invariance is maintained throughout changes.

**Category:** Number Theory

[654] **viXra:1705.0360 [pdf]**
*replaced on 2017-05-25 07:46:44*

**Authors:** Maik Becker-Sievert

**Comments:** 1 Page.

This Identity proofs direct Fermats Last Theorem

**Category:** Number Theory

[653] **viXra:1705.0142 [pdf]**
*replaced on 2017-05-13 03:50:38*

**Authors:** Carlos Castro

**Comments:** 13 Pages. Submitted to Mod. Phys. Letts A

An approach to solving the Riemann Hypothesis is revisited within the framework of the special properties of $\Theta$ (theta) functions, and the notion of $ {\cal C } { \cal T} $ invariance. The conjugation operation $ {\cal C }$ amounts to complex scaling transformations, and the $ {\cal T } $ operation
$ t \rightarrow ( 1/ t ) $ amounts to the reversal $ log (t) \rightarrow - log ( t ) $. A judicious scaling-like operator is constructed whose spectrum $E_s = s ( 1 - s ) $ is real-valued, leading to $ s = {1\over 2} + i \rho$,
and/or $ s $ = real. These values are the location of the non-trivial and trivial zeta zeros, respectively.
A thorough analysis of the one-to-one correspondence among the zeta zeros, and the orthogonality conditions among pairs of eigenfunctions, reveals that $no$ zeros exist off the critical line. The role of the $ {\cal C }, {\cal T } $ transformations, and the properties of the Mellin transform of $ \Theta$ functions were essential in our construction.

**Category:** Number Theory

[652] **viXra:1705.0142 [pdf]**
*replaced on 2017-05-09 20:04:55*

**Authors:** Carlos Castro

**Comments:** 11 Pages. Submitted to Mod. Phys. Letts A

An approach to solving the Riemann Hypothesis is revisited within the framework of
the special properties of $\Theta$ (theta) functions, and the notion of $ {\cal C } { \cal T} $ invariance.
The conjugation operation $ {\cal C }$ amounts to complex scaling transformations, and the $ {\cal T } $ operation
$ t \rightarrow ( 1/ t ) $ amounts to the reversal $ log (t) \rightarrow - log ( t ) $.
A judicious scaling-like operator is constructed whose spectrum $E_s = s ( 1 - s ) $ is real-valued, leading to $ s = {1\over 2} + i \rho$,
and/or $ s $ = real. These values are the location of the non-trivial and trivial zeta zeros, respectively.
A thorough analysis of the one-to-one correspondence among the zeta zeros, and the orthogonality conditions among pairs of eigenfunctions,
reveals that $no$ zeros exist off the critical line. The role of the $ {\cal C }, {\cal T } $ transformations, and the properties of the Mellin transform of $ \Theta$ functions were essential in our construction.

**Category:** Number Theory

[651] **viXra:1705.0117 [pdf]**
*replaced on 2017-05-31 18:39:21*

**Authors:** Jason Cole

**Comments:** 4 Pages.

There is exciting research trying to connect the nontrivial zeros of the Riemann Zeta function to Quantum mechanics as a breakthrough towards proving the 160-year-old Riemann Hypothesis. This research offers a radically new approach.
Most research up to this point have focused only on mapping the nontrivial zeros directly to eigenvalues. Those attempts have failed or didn’t yield any new breakthrough. This research takes a radically different approach by focusing on the quantum mechanical properties of the wave graph of Zeta as ζ(0.5+it) and not the nontrivial zeros directly. The conjecture is that the wave forms in the graph of the Riemann Zeta function ζ(0.5+it) is a wave function ψ. It is made of a Complex version of the Parity Operator wave function. The Riemann Zeta function consists of linked Even and Odd Parity Operator wave functions on the critical line. From this new approach, it shows the Complex version of the Parity Operator wave function is Hermitian and it eigenvalues matches the zeros of the Zeta function.

**Category:** Number Theory

[650] **viXra:1705.0117 [pdf]**
*replaced on 2017-05-22 09:41:44*

**Authors:** Jason Cole

**Comments:** 4 Pages.

There is exciting research trying to connect the nontrivial zeros of the Riemann Zeta function to Quantum mechanics as a breakthrough towards proving the 160-year-old Riemann Hypothesis. This research offers a radically new approach.
Most research up to this point have focused only on mapping the nontrivial zeros directly to eigenvalues. Those attempts have failed or didn’t yield any new breakthrough. This research takes a radically different approach by focusing on the quantum mechanical properties of the wave graph of Zeta as ζ(0.5+it) and not the nontrivial zeros directly. The conjecture is that the wave forms in the graph of the Riemann Zeta function ζ(0.5+it) is a wave function ψ. It is made of a Complex version of the Parity Operator wave function. The Riemann Zeta function consists of linked Even and Odd Parity Operator wave functions on the critical line. From this new approach, it shows the Complex version of the Parity Operator wave function is Hermitian and it eigenvalues matches the zeros of the Zeta function.

**Category:** Number Theory

[649] **viXra:1705.0117 [pdf]**
*replaced on 2017-05-18 08:17:38*

**Authors:** Jason Cole

**Comments:** 4 Pages.

**Category:** Number Theory

[648] **viXra:1705.0117 [pdf]**
*replaced on 2017-05-17 08:56:58*

**Authors:** Jason Cole

**Comments:** 4 Pages.

**Category:** Number Theory

[647] **viXra:1705.0117 [pdf]**
*replaced on 2017-05-10 19:37:27*

**Authors:** Jason Cole

**Comments:** 4 Pages.

**Category:** Number Theory

[646] **viXra:1704.0274 [pdf]**
*replaced on 2017-04-22 06:37:27*

**Authors:** François Mendzina Essomba

**Comments:** 1 Page. extreme fomulas

I present in this small article two algorithms of calculation of pi, they are characterized by two extremities, one is the most convergent and the other the slowest of the imaginable formulas.

**Category:** Number Theory

[645] **viXra:1704.0146 [pdf]**
*replaced on 2017-04-25 10:45:44*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 6 Pages.

In this paper we define a new Mellin discrete convolution, which is related to Perron's formula. Also we introduce new explicit formulae for arithmetic function which generalize the explicit formulae of Weil.

**Category:** Number Theory

[644] **viXra:1704.0129 [pdf]**
*replaced on 2017-04-19 11:21:34*

**Authors:** Barry Foster

**Comments:** 2 Pages.

This attempt uses Bertrand’s postulate.

**Category:** Number Theory

[643] **viXra:1704.0129 [pdf]**
*replaced on 2017-04-18 01:14:32*

**Authors:** Barry Foster

**Comments:** 2 Pages.

This attempt uses Bertrand’s postulate.

**Category:** Number Theory

[642] **viXra:1704.0101 [pdf]**
*replaced on 2017-05-11 09:44:47*

**Authors:** Chongxi Yu

**Comments:** 22 Pages.

Prime numbers are the basic numbers and are crucially important. There are many conjectures concerning primes that have been challenging mathematicians for hundreds of years. Goldbach's conjecture is one of the oldest and most well-known unsolved problems in number theory and in all of mathematics. A kaleidoscope can produce an endless variety of colorful patterns and it looks like magic, but when you open one and examine it, it contains only very simple, loose, colored objects such as beads or pebbles and bits of glass. Humans are very easily cheated by 2 words, infinite and anything, because we never see infinite and anything, and so we always make a simple thing complex. Goldbach’s conjecture is about all very simple numbers, with the pattern of prime numbers similar to a “kaleidoscope” of numbers. If we divided all even numbers into 5 groups and primes into 4 groups, Goldbach’s conjecture becomes much simpler. Here we give a clear proof for Goldbach's conjecture based on the fundamental theorem of arithmetic, the prime number theorem, and Euclid's proof that the set of prime numbers is endless.

**Category:** Number Theory

[641] **viXra:1704.0101 [pdf]**
*replaced on 2017-04-29 05:43:31*

**Authors:** Chongxi Yu

**Comments:** 22 Pages.

Prime numbers are the basic numbers and are crucial important. There are many conjectures concerning primes are challenging mathematicians for hundreds of years. Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics. A kaleidoscope can produce an endless variety of colorful patterns and it looks like a magic, but when you open it, it contains only very simple, loose, colored objects such as beads or pebbles and bits of glass. Human is very easily cheated by 2 words, infinite and anything, because we never see infinite and anything, so we always make simple thing complex. Goldbach’s conjecture is about all very simple numbers, the pattern of prime numbers likes a “kaleidoscope” of numbers, we divided any even numbers into 5 groups and primes into 4 groups, Goldbach’s conjecture becomes much simpler. Here we give a clear proof for Goldbach's conjecture based on the fundamental theorem of arithmetic, the prime number theorem, and Euclid's proof that the set of prime numbers is endless.
Key words: Goldbach's conjecture, fundamental theorem of arithmetic, Euclid's proof of infinite primes, the prime number theorem

**Category:** Number Theory

[640] **viXra:1704.0101 [pdf]**
*replaced on 2017-04-14 23:03:01*

**Authors:** Chongxi Yu

**Comments:** 19 Pages.

Prime numbers are the basic numbers and are crucial important. There are many conjectures concerning primes are challenging mathematicians for hundreds of years. Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics. A kaleidoscope can produce an endless variety of colorful patterns and it looks like a magic, but when you open it, it contains only very simple, loose, colored objects such as beads or pebbles and bits of glass. Human is very easily cheated by 2 words, infinite and anything, because we never see infinite and anything, so we always make simple thing complex. Goldbach’s conjecture is about all very simple numbers, the pattern of prime numbers likes a “kaleidoscope” of numbers, we divided any even numbers into 5 groups and primes into 4 groups, Goldbach’s conjecture becomes much simpler. Here we give a clear proof for Goldbach's conjecture based on the fundamental theorem of arithmetic, the prime number theorem, and Euclid's proof that the set of prime numbers is endless.
Key words: Goldbach's conjecture, fundamental theorem of arithmetic, Euclid's proof of infinite primes, the prime number theorem

**Category:** Number Theory

[639] **viXra:1704.0101 [pdf]**
*replaced on 2017-04-10 22:29:48*

**Authors:** Chongxi Yu

**Comments:** 18 Pages.

**Category:** Number Theory

[638] **viXra:1704.0029 [pdf]**
*replaced on 2017-06-02 04:16:44*

**Authors:** A. Zaganidis

**Comments:** 16 Pages. I have chosen the category Mathematics-Number Theory since most of the consequences of the present article are inside the number theory

In this work, we introduce the $n$-formal sequents and the formal numbers defined with the help of the second order logic. We give many concrete examples of formal numbers and $n$-formal sequents with the Peano's axioms and the axioms of the real numbers. Shortly, a sequent is $n$-formal iff the sequent is composed by some closed hypotheses and a $n$-formal formula (a close formula with one internal variable such that the formula is only true when we set that variable to the unique natural number $n$), and it does not exist some strict sub-sequent which are composed by some closed sub-hypotheses and some sub-$m$-formal formula with $m>1$. The definition is motivated by the intuition that the ``Nature's hypotheses'' do not carry natural numbers or "hidden natural numbers" except for the numbers $0$ and $1$, i.e., they can be used in a $n$-formal sequent. Moreover, we postulate at second order of logic that the ``Nature's hypotheses'' are not chosen randomly: the ``Nature's hypotheses'' are the only hypotheses which give the largest formal number $N_Z\cong 2^{1.0\times 10^4}-2^{2.4\times 10^6}$. The Goldbach's conjecture, the Polignac's conjecture, the Firoozbakht's conjecture, the Oppermann's conjecture, the Agoh-Giuga conjecture, the generalized Fermat's conjecture and the Schinzel's hypothesis H are reviewed with this new (second order logic) formal axiom. Finally, three open questions remain: Can we prove that a natural number is not formal? If a formal number $n$ is found with a function symbol $f$ where its outputs values are only $0$ and $1$, can we always replace the function symbol $f$ by a another function symbol $\tilde{f}$ such that $\tilde{f}=1-f$ and the new sequent is still $n$-formal? Does a sequent exist to make a difference between the definition of the $n$-formal sequents and the following weaker variant of that definition: we look at the explicit sub-formulas of $\phi$ which induce the $m$-formal formulas instead of looking at the explicit sub-formulas of $\phi_{n-formal}$ which are $m$-formal formulas?

**Category:** Number Theory

[637] **viXra:1703.0304 [pdf]**
*replaced on 2017-06-22 07:22:56*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 7 Pages. In French. Submitted to Journal Annales Scientifiques de l'Ecole Normale Supérieure. Comments welcome.

In 1898, Georg Friedrich Bernhard Riemann had announced the following conjecture, called Riemann Hypothesis : The nontrivial roots (zeros) s=\sigma+it of the zeta function, defined by:
\zeta(s) = \sum_{n=1}^{+\infty}\frac{1}{n^s},\,for \Re(s)>1
have real part \sigma= \frac{1}{2}.
We give a proof that \sigma= \frac{1}{2} using an equivalent statement of Riemann Hypothesis.

**Category:** Number Theory

[636] **viXra:1703.0243 [pdf]**
*replaced on 2017-03-29 16:58:21*

**Authors:** Wes Hansen

**Comments:** 9 Pages.

In what follows we develop foundations for a set of non-standard natural numbers we call q-naturals, where q stands for quanta, by the recursive generation of reflexive sets. From the practical perspective, these q-naturals correspond to ordered pairs of natural numbers with the lexicographic ordering, hence, they are isomorphic to ω^2. In addition, we demonstrate a novel definition of the arithmetical operation, multiplication, which turns out to be recursive. This, in turn, enables our demonstration of a counter-example to Tennenbaum’s Theorem.

**Category:** Number Theory

[635] **viXra:1703.0243 [pdf]**
*replaced on 2017-03-27 15:53:09*

**Authors:** Wes Hansen

**Comments:** 9 Pages.

In what follows we develop foundations for a set of non-standard natural numbers we call q-naturals, where q stands for quanta, by the recursive generation of reflexive sets. From the practical perspective, these q-naturals correspond to ordered pairs of natural numbers with the lexicographic ordering, hence, they are isomorphic to ω^2. In addition, we demonstrate a novel definition of the arithmetical operation, multiplication, which turns out to be recursive. This, in turn, enables our demonstration of a counter-example to Tennenbaum’s Theorem.

**Category:** Number Theory

[634] **viXra:1703.0241 [pdf]**
*replaced on 2017-04-01 04:31:10*

**Authors:** Barry Foster

**Comments:** 2 Pages.

This attempt is essentially an inductive approach.

**Category:** Number Theory

[633] **viXra:1703.0220 [pdf]**
*replaced on 2017-03-29 20:38:48*

**Authors:** Pedro Caceres

**Comments:** 23 Pages.

Abstract: A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.
The crucial importance of prime numbers to number theory and mathematics in general stems from the fundamental theorem of arithmetic, which states that every integer larger than 1 can be written as a product of one or more primes in a way that is unique except for the order of the prime factors. Primes can thus be considered the “basic building blocks”, the atoms, of the natural numbers.
There are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no known simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, the statistical behavior of primes in the large, can be modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or to the logarithm of n.
The way to build the sequence of prime numbers uses sieves, an algorithm yielding all primes up to a given limit, using only trial division method which consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime.
This paper introduces a new way to approach prime numbers, called the DNA-prime structure because of its intertwined nature, and a new process to create the sequence of primes without direct division or multiplication, which will allow us to introduce a new primality test, and a new factorization algorithm.
As a consequence of the DNA-prime structure, we will be able to provide a potential proof of Golbach’s conjecture.

**Category:** Number Theory

[632] **viXra:1703.0147 [pdf]**
*replaced on 2017-06-01 20:27:28*

**Authors:** Philip A. Bloom

**Comments:** 2 Pages.

This simple proof of Fermat's last theorem rewrites x^n + y^n = z^n as a corresponding equation, explicitly showing that no integral triple (x, y, z) exists for n > 2.

**Category:** Number Theory

[631] **viXra:1703.0147 [pdf]**
*replaced on 2017-05-20 20:42:12*

**Authors:** Philip A. Bloom

**Comments:** 1 Page.

This simple, direct proof of Fermat's last theorem uses an algebraic equation that has integral triples corresponding to integral (x,y,z) for which x^n + y^n = z^n holds.

**Category:** Number Theory

[630] **viXra:1703.0147 [pdf]**
*replaced on 2017-05-05 22:00:11*

**Authors:** Philip A. Bloom

**Comments:** 2 Pages.

This simple, direct proof of Fermat's last theorem uses an algebraic equation that has integral triples equal to integral (z,y,x) for which z^n- y^n = x^n holds.

**Category:** Number Theory

[629] **viXra:1703.0147 [pdf]**
*replaced on 2017-04-28 19:06:48*

**Authors:** Philip A. Bloom

**Comments:** 2 Pages.

This two-page proof, by contraposition, of Fermat's last theorem, uses two analogous forms of a previously overlooked equation, each of which is equivalent to rational z^n-y^n = x^n. A relationship held in common by these two equations reveals the crucial n = 1, 2 limitation.

**Category:** Number Theory

[628] **viXra:1703.0115 [pdf]**
*replaced on 2017-04-17 05:58:07*

**Authors:** John Yuk Ching Ting

**Comments:** 21 Pages. This research paper outline the rigorous proofs for Polignac's and Twin prime conjectures. It is cross-related to Solving Riemann Hypothesis Using Sigma-Power Laws (http://viXra.org/abs/1703.0114).

Prime numbers and composite numbers are intimately related simply because the complementary set of composite numbers constitutes the set of natural numbers with the exact set of prime numbers excluded in its entirety. In this research paper, we use our 'Virtual container' (which predominantly incorporates the novel mathematical tool coined Information-Complexity conservation with its core foundation based on this [complete] prime-composite number relationship) to solve the intractable open problem of whether prime gaps are infinite (arbitrarily large) in magnitude with each individual prime gap generating prime numbers which are again infinite in magnitude. This equates to solving Polignac's conjecture which involves analysis of all possible prime gaps = 2, 4, 6,... and [the subset] Twin prime conjecture which involves analysis of prime gap = 2 (for twin primes). In conjunction with our cross-referenced 2017-dated research paper entitled "Solving Riemann Hypothesis Using Sigma-Power Laws" (http://viXra.org/abs/1703.0114), we advocate for our ambition that the Virtual container research method be considered a new method of mathematical proof especially for solving the 'Special-Class-of-Mathematical-Problems with Solitary-Proof-Solution'.

**Category:** Number Theory

[627] **viXra:1703.0115 [pdf]**
*replaced on 2017-04-02 00:28:44*

**Authors:** John Yuk Ching Ting

**Comments:** 19 Pages. This research paper outline the rigorous proofs for Polignac's and Twin prime conjectures.

Prime numbers and composite numbers are intimately related simply because the complementary set of composite numbers constitutes the set of natural numbers with the exact set of prime numbers excluded in its entirety. In this research paper, we use our 'Virtual container', which predominantly incorporates the novel mathematical tool coined Information-Complexity conservation with its core foundation based on this [complete] prime-composite number relationship, to solve the intractable open problem of whether prime gaps are infinite (arbitrarily large) in magnitude with each individual prime gap generating prime numbers which are again infinite in magnitude. This equates to solving Polignac's conjecture which involves analysis of all possible prime gaps = 2, 4, 6,... and [the subset] Twin prime conjecture which involves analysis of prime gap = 2 (for twin primes). In conjunction with our cross-referenced 2017-dated research paper entitled "Rigorous Proof for Riemann Hypothesis Using Sigma-Power Laws" http://viXra.org/abs/1703.0114, we advocate for our ambition that the Virtual container technique be considered as a new method of mathematical proof especially for the ’Special-Class-of-Mathematical-Problems with Solitary-Proof-Solution’.

**Category:** Number Theory

[626] **viXra:1703.0115 [pdf]**
*replaced on 2017-03-17 20:27:33*

**Authors:** John Yuk Ching Ting

**Comments:** 18 Pages. This research paper contains the rigorous proofs for Polignac's and Twin prime conjectures.

Prime numbers and composite numbers are intimately related simply because the complementary set of composite numbers constitutes the set of natural numbers with the exact set of prime numbers excluded in its entirety. In this research paper, we use our 'Virtual container' method, which predominantly incorporates the novel mathematical tool coined Information-Complexity conservation with its core foundation based on this [complete] prime-composite number relationship, to solve the intractable open problem of whether prime gaps are infinite (arbitrarily large) in magnitude with each individual prime gap generating prime numbers which are again infinite in magnitude. This equates to solving Polignac's conjecture which involves analysis of all possible prime gaps = 2, 4, 6,... and [the subset] Twin prime conjecture which involves analysis of prime gap = 2 (for twin primes). In conjunction with our cross-referenced 2017-dated research paper entitled "Rigorous proof for Riemann hypothesis using Sigma-Power Laws" http://viXra.org/abs/1703.0114, we advocate for our ambition that the Virtual container method be considered as a new method of mathematical proof especially for ’Special-Class-of-Mathematical-Problems with Solitary-Proof-Solution’.

**Category:** Number Theory

[625] **viXra:1703.0114 [pdf]**
*replaced on 2017-04-17 06:11:01*

**Authors:** John Yuk Ching Ting

**Comments:** 23 Pages. This research paper contains the rigorous proof for Riemann hypothesis and explanation for Gram points. It is cross-referenced to "Solving Polignac's and Twin Prime Conjectures using Information-Complexity Conservation" (http://viXra.org/abs/1703.0115).

The triple countable infinite sets of (i) x-axis intercepts, (ii) y-axis intercepts, and (iii) both x- and y-axes [formally known as the 'Origin'] intercepts in Riemann zeta function are intimately related to each other simply because they all constitute complementary points of intersection arising from the single [exact same] countable infinite set of curves generated by this function. Recognizing this [complete] relationship amongst all three sets of intercepts enable the simultaneous study on important intrinsic properties and behaviors arising from our derived key formulae coined Sigma-Power Laws in a mathematically consistent manner. This then permit the rigorous proof for Riemann hypothesis to mature as well as allows explanations for x-axis intercepts (which is the usual traditionally-dubbed 'Gram points') and y-axis intercepts. Riemann hypothesis involves analysis of all nontrivial zeros of Riemann zeta function and refers to the celebrated proposal by famous German mathematician Bernhard Riemann in 1859 whereby all nontrivial zeros are conjectured to be located on the critical line [or equivalently stated as all nontrivial zeros are conjectured to exactly match the Origin intercepts]. Concepts from the Hybrid method of Integer Sequence classification, together with our 'Virtual container' incorporating the novel Sigma-Power Laws, are some of the important mathematical tools employed in this research paper to successfully achieve our proof. Not least in [again] using the same Virtual container research method in this paper, there are other additional deeply inseparable mathematical connections between the contents of this paper and our cross-referenced 2017-dated publication on the dual source of prime number infiniteness entitled "Solving Polignac's and Twin Prime Conjectures using Information-Complexity Conservation" (http://viXra.org/abs/1703.0115).

**Category:** Number Theory

[624] **viXra:1703.0114 [pdf]**
*replaced on 2017-04-02 00:16:28*

**Authors:** John Yuk Ching Ting

**Comments:** 21 Pages. This research paper contains the rigorous proof for Riemann hypothesis and explanation for Gram points.

The triple countable infinite sets of (i) x-axis intercepts, (ii) y-axis intercepts, and (iii) both x- and y-axes [formally known as the 'Origin'] intercepts in Riemann zeta function are intimately related to each other simply because they all constitute complementary points of intersection arising from the single [exact same] countable infinite set of curves generated by this function. This [complete] relationship amongst all three sets of intercepts will enable us to simultaneously study important intrinsic properties derived from all those intercepts in a mathematically consistent manner which then provides the rigorous proof for Riemann hypothesis as well as fully explain x-axis intercepts (which is the usual traditionally-dubbed 'Gram points') and y-axis intercepts. Riemann hypothesis involves analysis of all nontrivial zeros of Riemann zeta function and refers to the celebrated proposal by famous German mathematician Bernhard Riemann in 1859 whereby all nontrivial zeros are conjectured to be located on the critical line [or equivalently stated as all nontrivial zeros are conjectured to exactly match the Origin intercepts]. Concepts from the Hybrid method of Integer Sequence classification, together with our key formulae coined Sigma-Power Laws, are some of the important mathematical tools employed in this paper to successfully achieve our proof. Not least in [again] using the same 'Virtual container' method in this current research paper, there are other additional deeply inseparable mathematical connections between the content of this paper and our 2017-dated publication on the dual source of prime number infiniteness entitled "Rigorous Proofs for Polignac's and Twin Prime Conjectures using Information-Complexity Conservation" http://viXra.org/abs/1703.0115.

**Category:** Number Theory

[623] **viXra:1703.0114 [pdf]**
*replaced on 2017-03-18 01:01:54*

**Authors:** John Yuk Ching Ting

**Comments:** 20 Pages. This research paper contains the rigorous proof for Riemann hypothesis.

The triple countable infinite sets of (i) x-axis intercepts, (ii) y-axis intercepts, and (iii) both x- and y-axes [formally known as the 'Origin'] intercepts in Riemann zeta function are intimately related to each other simply because they all constitute complementary points of intersection arising from the single [exact same] countable infinite set of curves generated by this function. This [complete] relationship amongst all three sets of intercepts will enable us to simultaneously study important intrinsic properties derived from all those intercepts in a mathematically consistent manner which then provides the rigorous proof for Riemann hypothesis as well as fully explain x-axis intercepts (which is the usual traditionally-dubbed 'Gram points') and y-axis intercepts. Riemann hypothesis involves analysis of all nontrivial zeros of Riemann zeta function and refers to the celebrated proposal by famous German mathematician Bernhard Riemann in 1859 whereby all nontrivial zeros are conjectured to be located on the critical line [or equivalently stated as all nontrivial zeros are conjectured to exactly match the Origin intercepts]. Concepts from the Hybrid method of Integer Sequence classification, together with our key formulae coined Sigma-Power Laws, are some of the important mathematical tools employed in this paper to successfully achieve our proof. Not least in [again] using the same 'Virtual container' method in this current research paper, there are other additional deeply inseparable mathematical connections between the content of this paper and our 2017-dated publication on the dual source of prime number infiniteness entitled "Rigorous proofs for Polignac's and Twin prime conjectures using Information-Complexity conservation" http://viXra.org/abs/1703.0115.

**Category:** Number Theory

[622] **viXra:1703.0048 [pdf]**
*replaced on 2017-03-14 18:11:01*

**Authors:** Stephen Crowley

**Comments:** 6 Pages.

Abstract. It is conjectured that when t=t_n is the imaginary part of the n-th zero of ζ on the critical line, the normalised argument S(t_)_=π^(-1)argζ(1/2+i t__) is equal to S(t)=S_n(t_n)=_n-3/2-(ϑ(t_n_))/π where ϑ(t) is the Riemann-Siegel ϑ function. If S(t_n)=S_n(t_n)∀n∈ℤ^+ then the exact transcendental equation for the Riemann zeros has a solution for each positive integer n which proves that Riemann's hypothesis is true since the counting function for zeros on the critical line is equal to the counting function for zeros on the critical strip in that case.

**Category:** Number Theory

[621] **viXra:1703.0048 [pdf]**
*replaced on 2017-03-09 16:06:14*

**Authors:** Stephen Crowley

**Comments:** 6 Pages.

It is conjectured that argζ(1/2+i t_n)=S_n(t_n) where S_n(t_n)=π(3/2-frac((ϑ(t_n))/π)-⌊g~^(-1)(n)⌋-n) and g~^(-1)(t)=(t ln(t/(2 π e)))/(2 π)+7/8 is the inverse of g~(n)=((8n-7)π)/(4 W((8n-7)/(8 e))) which accurately approximates the Gram points g(n) and that all of the non-trivial zeros of ζ, enumerated by n, are on the critical line. Therefore, if argζ(1/2+i t_n)=S_n(t_n) then the exact transcendental equation for the Riemann zeros has a solution for each positive integer n which proves that Riemann's hypothesis is true since the counting function for zeros on the critical line is equal to the counting function for zeros on the critical strip if the transcendental equation has a solution for each n.

**Category:** Number Theory

[620] **viXra:1703.0048 [pdf]**
*replaced on 2017-03-07 22:44:56*

**Authors:** Stephen Crowley

**Comments:** 5 Pages.

It is conjectured that argζ(1/2+i t_n)=π(1/2-frac((ϑ(t_n))/π)-(floor(g~^(-1)(n))-n+1))∀n⩾2 where g~^(-1)(n)=(t ln(t/(2 π e)))/(2 π)+7/8 is the inverse of g~(n)=((8n-7)π)/(4 W((8n-7)/(8 e))) which accurately approximates the Gram points g(n) and that all of the non-trivial zeros of ζ, enumerated by n, are on the critical line. Therefore, the exact transcendental equation for the Riemann zeros has a solution for each positive integer n which proves that Riemann's hypothesis is true since the counting function for zeros on the critical line is equal to the counting function for zeros on the critical strip if the transcendental equation has a solution for each n.

**Category:** Number Theory

[619] **viXra:1702.0331 [pdf]**
*replaced on 2017-03-07 17:04:09*

**Authors:** A. A. Frempong

**Comments:** 5 Pages. Copyright © by A. A. Frempong

Using a direct construction approach, the author proved the original Beal conjecture that if A^x + B^y = C^z , where A, B, C, x, y, z are positive integers and x, y, z > 2, then A, B and C have a common prime factor. Two main types of equations were involved, namely, the equation A^x + B^y = C^z and an equation which was called a tester equation. A tester equation has similar properties as A^x + B^y = C^z and was used to determine the properties of A^x + B^y = C^z . Also, two types of tester equations, namely, a literal tester equation and a numerical tester equation were applied. Each side of A^x + B^y = C^z and a tester equation was reduced to unity by division. The non-unity sides were justifiably equated to each other to produce a new equation which was called the master equation. The side of the master equation involving the terms of the tester equation was called the tester side of the master equation. Three versions of the proof were presented. In Version 1 proof, the tester equation was the literal equation G^m + H^n = I^p, but in Versions 2 and 3 proofs, the tester equations were the numerical tester equations, 2^9 + 8^3 = 4^5 and 3^3 + 6^3 = 3^5, respectively. By a comparative analysis, in which the corresponding "terms" on the right and left sides of the master equation were equated to each other, it was determined that if A^x + B^y = C^z , then A, B and C have a common prime factor. The proof is very simple, and occupies a single page, and even, high school students can learn it.

**Category:** Number Theory

[618] **viXra:1702.0331 [pdf]**
*replaced on 2017-03-04 01:18:52*

**Authors:** A. A. Frempong

**Comments:** 5 Pages. Copyright © by A. A. Frempong

Using a direct construction approach, the author proves the original Beal conjecture that if A^x + B^y = C^z , where A, B, C, x, y, z are positive integers and x, y, z > 2, then A, B and C have a common prime factor. Two main types of equations are involved, namely, the equation A^x + B^y = C^z and an equation which will be called a tester equation. A tester equation has similar properties as A^x + B^y = C^z and will be used to determine the properties of A^x + B^y = C^z . Also, two types of tester equations, namely, a literal tester equation and a numerical tester equation will be applied. Each side of A^x + B^y = C^z and a tester equation is reduced to unity by division. The non-unity sides are justifiably equated to each other to produce a new equation which will be called the master equation. The side of the master equation involving the terms of the tester equation will be called the tester side of the master equation. Three versions of the proof are presented. In Version 1 proof, the tester equation was the literal equation G^m + H^n = I^p, but in Versions 2 and 3 proofs, the tester equations were the numerical tester equations, 2^9 + 8^3 = 4^5 and 3^3 + 6^3 = 3^5, respectively. By inspection, using an approach in which the corresponding elements on the right and left sides of the master equation are equated to each other, it is determined that if A^x + B^y = C^z , then A, B and C have a common prime factor. The proof is very simple, and occupies a single page, and even, high school students can learn it.

**Category:** Number Theory

[617] **viXra:1702.0323 [pdf]**
*replaced on 2017-02-27 13:42:01*

**Authors:** Stephen Crowley

**Comments:** 3 Pages.

It is proved that the non-trivial roots of the Hardy Z function are simple having multiplicity 1 by showing that the fixed-points N_Z(α)=α of the Newton map N_Z(t)=t-(Z(t))/(Z˙(t)) must have a multiplier λ_(N_Z)(α)=|(N_Z)˙(α)|=|(Z(α)Z¨(α))/(Z˙(α))|=0 and therefore a multiplicity m_Z(α)=1/(1-λ_(N_Z)(α))=1/(1-0)=1.

**Category:** Number Theory

[616] **viXra:1702.0300 [pdf]**
*replaced on 2017-05-13 01:29:59*

**Authors:** Ralf Wüsthofen

**Comments:** 11 Pages. Older versions on http://vixra.org/abs/1403.0083

The present paper shows that a principle known as emergence lies beneath the strong Goldbach conjecture. Whereas the traditional approaches focus on the control over the distribution of the primes by means of circle method and sieve theory, we give a proof of the conjecture that involves the constructive properties of the prime numbers, reflecting their multiplicative character within the natural numbers. With an equivalent but more convenient form of the conjecture in mind, we create a structure on the natural numbers which is based on the prime factorization. Then, we realize that the characteristics of this structure immediately imply the conjecture and, in addition, an even strengthened form of it. Moreover, we can achieve further results by generalizing the structuring. Thus, it turns out that the statement of the strong Goldbach conjecture is the special case of a general principle.

**Category:** Number Theory

[615] **viXra:1702.0300 [pdf]**
*replaced on 2017-04-02 17:21:31*

**Authors:** Ralf Wüsthofen

**Comments:** 12 Pages. Older versions on http://vixra.org/abs/1403.0083

The present paper shows that a principle known as emergence lies beneath the strong Goldbach conjecture. Whereas the traditional approaches focus on the control over the distribution of the primes by means of circle method and sieve theory, we give a proof of the conjecture that involves the constructive properties of the prime numbers, reflecting their multiplicative character within the natural numbers. With an equivalent but more convenient form of the conjecture in mind, we create a structure on the natural numbers which is based on the prime factorization. Then, we realize that the characteristics of this structure immediately imply the conjecture and, in addition, an even strengthened form of it. Moreover, we can achieve further results by generalizing the structuring. Thus, it turns out that the statement of the strong Goldbach conjecture is the special case of a general principle.

**Category:** Number Theory

[614] **viXra:1702.0300 [pdf]**
*replaced on 2017-03-14 15:04:19*

**Authors:** Ralf Wüsthofen

**Comments:** 11 Pages. Older versions on http://vixra.org/abs/1403.0083

The present paper shows that a principle known as emergence lies beneath the strong Goldbach conjecture. Whereas the traditional approaches focus on the control over the distribution of the primes by means of circle method and sieve theory, we give a proof of the conjecture that involves the constructive properties of the prime numbers, reflecting their multiplicative character within the natural numbers. With an equivalent but more convenient form of the conjecture in mind, we create a structure on the natural numbers which is based on the prime factorization. Then, we realize that the characteristics of this structure immediately imply the conjecture and, in addition, an even strengthened form of it. Moreover, we can achieve further results by generalizing the structuring. Thus, it turns out that the statement of the strong Goldbach conjecture is the special case of a general principle.

**Category:** Number Theory

[613] **viXra:1702.0273 [pdf]**
*replaced on 2017-02-28 13:59:01*

**Authors:** Stephen Crowley

**Comments:** 10 Pages.

A sequence of Cauchy sequences which conjecturally converge to the Riemann zeros is constructed and related to the LeClair-França criteria for the Riemann hypothesis.

**Category:** Number Theory

[612] **viXra:1702.0273 [pdf]**
*replaced on 2017-02-24 19:33:35*

**Authors:** Stephen Crowley

**Comments:** 9 Pages. no b.s. this time, it should be impossible to argue with this one ;)

An iteration function which has fixed-points at the zeros of the Hardy Z function is constructed and it is shown that it is impossible for this function converge to a non-real number when started with a real number. If there were any zeros of ζ(t) with Re(t)≠1/2 they would correspond to zeros of Z(t) with Im(t)≠0 and thus the constructed interation function must be able to converge for at least one real-valued starting point to a number with non-zero imaginary part, but this is impossible because the iteration function is real-valued when its argument is real. Thus, the Riemann hypothesis is shown to be true.

**Category:** Number Theory

[611] **viXra:1702.0273 [pdf]**
*replaced on 2017-02-23 13:25:44*

**Authors:** Stephen Crowley

**Comments:** 7 Pages.

A sequence of Cauchy sequences which converge to the Riemann zeros is constructed and related to the LeClair-França criteria for the Riemann hypothesis.

**Category:** Number Theory

[610] **viXra:1702.0157 [pdf]**
*replaced on 2017-05-11 09:42:42*

**Authors:** Chongxi Yu

**Comments:** 15 Pages.

Prime numbers are the basic numbers and are crucial important. There are many conjectures concerning primes are challenging mathematicians for hundreds of years and many “advanced mathematics tools” are used to solve them, but they are still unsolved. A kaleidoscope can produce an endless variety of colorful patterns and it looks like magic, but when you open one and examine it, it contains only very simple, loose, colored objects such as beads or pebbles and bits of glass. Humans are very easily cheated by 2 words, infinite and anything, because we never see infinite and anything, and so we always make a simple thing complex. The pattern of prime numbers similar to a “kaleidoscope” of numbers, if we divide primes into 4 groups, twin primes conjecture becomes much simpler. Based on the fundamental theorem of arithmetic and Euclid’s proof of endless prime numbers, we have proved there are infinitely many twin primes.

**Category:** Number Theory

[609] **viXra:1702.0157 [pdf]**
*replaced on 2017-04-29 06:52:43*

**Authors:** Chongxi Yu

**Comments:** 14 Pages.

Prime numbers are the basic numbers and are crucial important. There are many conjectures concerning primes are challenging mathematicians for hundreds of years and many “advanced mathematics tools” are used to solve them, but they are still unsolved. Based on the fundamental theorem of arithmetic and Euclid’s proof of endless prime numbers, we have proved there are infinitely many twin primes.

**Category:** Number Theory

[608] **viXra:1702.0157 [pdf]**
*replaced on 2017-03-05 03:16:45*

**Authors:** Chongxi Yu

**Comments:** 8 Pages.

Prime numbers are the basic numbers and are crucial important. There are many conjectures concerning primes are challenging mathematicians for hundreds of years and many “advanced mathematics tools” are used to solve them, but they are still unsolved. Based on the fundamental theorem of arithmetic and Euclid’s proof of endless prime numbers, we have proved there are infinitely many twin primes.

**Category:** Number Theory

[607] **viXra:1702.0157 [pdf]**
*replaced on 2017-02-17 19:44:31*

**Authors:** Chongxi Yu

**Comments:** 8 Pages.

**Category:** Number Theory

[606] **viXra:1702.0136 [pdf]**
*replaced on 2017-02-15 03:23:14*

**Authors:** Predrag Terzic

**Comments:** 2 Pages.

Polynomial time primality test for safe primes is introduced .

**Category:** Number Theory

[605] **viXra:1702.0136 [pdf]**
*replaced on 2017-02-14 00:07:47*

**Authors:** Predrag Terzic

**Comments:** 2 Pages.

Polynomial time primality test for safe primes is introduced .

**Category:** Number Theory

[604] **viXra:1702.0090 [pdf]**
*replaced on 2017-02-22 08:26:54*

**Authors:** Chongxi Yu

**Comments:** 33 Pages.

Prime numbers are the basic numbers and are crucial important. There are many conjectures concerning primes are challenging mathematicians for hundreds of years. Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics. A kaleidoscope can produce an endless variety of colorful patterns and it looks like a magic, but when you open it, it contains only very simple, loose, colored objects such as beads or pebbles and bits of glass. Goldbach’s conjecture is about all numbers, the pattern of prime numbers likes a “kaleidoscope” of numbers, we divided any even numbers into 10 groups and primes into 4 groups, Goldbach’s conjecture becomes much simpler. Here we give a clear proof for Goldbach's conjecture based on the fundamental theorem of arithmetic and Euclid's proof that the set of prime numbers is endless.

**Category:** Number Theory

[603] **viXra:1702.0090 [pdf]**
*replaced on 2017-02-22 02:59:46*

**Authors:** Chongxi Yu

**Comments:** 33 Pages.

Prime numbers are the basic numbers and are crucial important. There are many conjectures concerning primes are challenging mathematicians for hundreds of years. Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics. A kaleidoscope can produce an endless variety of colorful patterns and it looks like a magic, but when you open it, it contains only very simple, loose, colored objects such as beads or pebbles and bits of glass. Goldbach’s conjecture is about all numbers, the pattern of prime numbers likes a “kaleidoscope” of numbers, here we divided any even numbers into 10 groups and primes into 4 groups, Goldbach’s conjecture will be much simpler. Here we give a clear proof for Goldbach's conjecture based on the fundamental theorem of arithmetic and Euclid's proof that the set of prime numbers is endless.

**Category:** Number Theory

[602] **viXra:1702.0090 [pdf]**
*replaced on 2017-02-13 23:18:35*

**Authors:** Chongxi Yu

**Comments:** 24 Pages.

Prime numbers are the basic numbers and are crucial important. There are many conjectures concerning primes are challenging mathematicians for hundreds of years. Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics. We give a clear proof for Goldbach's conjecture based on the fundamental theorem of arithmetic and Euclid's proof that the set of prime numbers is endless.
Key words: Goldbach's conjecture , fundamental theorem of arithmetic, Euclid's proof of infinite primes

**Category:** Number Theory

[601] **viXra:1702.0090 [pdf]**
*replaced on 2017-02-12 00:48:46*

**Authors:** Chongxi Yu

**Comments:** 21 Pages.

Prime numbers are the basic numbers and are crucial important. There are many conjectures concerning primes are challenging mathematicians for hundreds of years. Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics. We give a clear proof for Goldbach's conjecture based on the fundamental theorem of arithmetic and Euclid's proof that the set of prime numbers is endless.
Key words: Goldbach's conjecture , fundamental theorem of arithmetic, Euclid's proof of infinite primes

**Category:** Number Theory

[600] **viXra:1702.0027 [pdf]**
*replaced on 2017-02-09 15:34:07*

**Authors:** Dragan Turanyanin

**Comments:** 3 Pages.

Three real numbers are introduced via related infinite series. With e, together they complete a quadruplet.

**Category:** Number Theory

[599] **viXra:1701.0664 [pdf]**
*replaced on 2017-04-16 16:16:48*

**Authors:** Andrei Lucian Dragoi

**Comments:** 32 Pages.

(BGC and TGC) [1,2,3,4] [5,6,7], briefly called “the Vertical Goldbach’s Conjectures” (VBGC and VTGC), which are essentially meta-conjectures (as VBGC states an infinite number of conjectures stronger than BGC). VBGC was discovered in 2007[1] and perfected until 2016[2] by using the arrays (S_p and S_i,p) of Matrix of Goldbach index-partitions (GIPs) (simple M_p,n and recursive M_i,p,n, with iteration order i ≥ 0), which are a useful tool in studying BGC by focusing on prime indexes (as the function P_n that numbers the primes is a bijection). Simple M (M_p,n) and recursive M (M_i,p,n) are related to the concept of generalized “primeths” (a term first used by Fernandez N. in his “The Exploring Primeness Project”), which is the generalization with iteration order i≥0 of the known “higher-order prime numbers” (alias “superprime numbers”, “super-prime numbers”, ”super-primes”, ” super-primes” or “prime-indexed primes[PIPs]”) as a subset of (simple or recursive) primes with (also) prime indexes (iPx is the x-th o-primeth, with iteration order i ≥ 0 as explained later on). The author of this article also brings in a S-M-synthesis of some Goldbach-like conjectures (GLC) (including those which are “stronger” than BGC) and a new class of GLCs “stronger” than BGC, from which VBGC (which is essentially a variant of BGC applied on a serial array of subsets of primeths with a general iteration order i ≥ 0) distinguishes as a very important conjecture of primes (with great importance in the optimization of the BGC experimental verification and other potential useful theoretical and practical applications in mathematics [including cryptography and fractals] and physics [including crystallography and M-Theory]), and a very special self-similar propriety of the primes subset of (noted/abbreviated as or as explained later on in this article). Keywords: Prime (number), primes with prime indexes, the i-primeths (with iteration order i≥0), the Binary Goldbach Conjecture (BGC), the Ternary Goldbach Conjecture (TGC), Goldbach index-partition (GIP), fractal patterns of the number and distribution of Goldbach index-partitions, Goldbach-like conjectures (GLC), the Vertical Binary Goldbach Conjecture (VBGC) and Vertical Ternary Goldbach Conjecture (VTGC) the as applied on i-primeths

**Category:** Number Theory

[598] **viXra:1701.0664 [pdf]**
*replaced on 2017-02-02 03:48:33*

**Authors:** Andrei Lucian Dragoi

**Comments:** 10 Pages.

This article proposes the generalization of the both binary (strong) and ternary (weak) Goldbach’s Conjectures (BGC and TGC), briefly called “the Vertical Goldbach’s Conjectures” (VBGC and VTGC), discovered in 2007[1] and perfected until 2016[2] by using the arrays (S_p and S_i,p) of Matrix of Goldbach index-partitions (GIPs) (simple M_p,n and recursive M_i,p,n, with iteration order i ≥ 0), which are a useful tool in studying BGC by focusing on prime indexes (as the function P_n that numbers the primes is a bijection). Simple M (M_p,n) and recursive M (M_i,p,n) are related to the concept of generalized “primeths” (a term first used by Fernandez N. in his “The Exploring Primeness Project”), which is the generalization with iteration order i≥0 of the known “higher-order prime numbers” (alias “superprime numbers”, “super-prime numbers”, ”super-primes”, ” super-primes” or “prime-indexed primes[PIPs]”) as a subset of (simple or recursive) primes with (also) prime indexes (iPx is the x-th o-primeth, with iteration order i ≥ 0 as explained later on).
The author of this article also brings in a S-M-synthesis of some Goldbach-like conjectures (GLC) (including those which are “stronger” than BGC) and a new class of GLCs “stronger” than BGC, from which VBGC (which is essentially a variant of BGC applied on a serial array of subsets of primeths with a general iteration order i ≥ 0) distinguishes as a very important conjecture of primes (with great importance in the optimization of the BGC experimental verification and other potential useful theoretical and practical applications in mathematics [including cryptography and fractals] and physics [including crystallography and M-Theory]), and a very special self-similar propriety of the primes subset of (noted/abbreviated as or as explained later on in this article).
Keywords: Prime (number), primes with prime indexes, the i-primeths (with iteration order i≥0), the Binary Goldbach Conjecture (BGC), the Ternary Goldbach Conjecture (TGC), Goldbach index-partition (GIP), fractal patterns of the number and distribution of Goldbach index-partitions, Goldbach-like conjectures (GLC), the Vertical Binary Goldbach Conjecture (VBGC) and Vertical Ternary Goldbach Conjecture (VTGC) the as applied on i-primeths

**Category:** Number Theory

[597] **viXra:1701.0664 [pdf]**
*replaced on 2017-01-31 05:14:05*

**Authors:** Andrei Lucian Dragoi

**Comments:** 10 Pages.

This article proposes the generalization of the both binary (strong) and ternary (weak) Goldbach’s Conjectures (BGC and TGC), briefly called “the Vertical Goldbach’s Conjectures” (VBGC and VTGC), discovered in 2007[1] and perfected until 2016[2] by using the arrays (S_p and S_o,p) of Matrix of Goldbach index-partitions (GIPs) (simple M_p,n and recursive M_o,p,n, with order o ≥ 0), which are a useful tool in studying BGC by focusing on prime indexes (as the function P_n that numbers the primes is a bijection). Simple M (M_p,n) and recursive M (M_o,p,n) are related to the concept of generalized “primeths” (a term first used by Fernandez N. in his “The Exploring Primeness Project”), which is the generalization with order o≥0 of the known “higher-order prime numbers” (alias “superprime numbers”, “super-prime numbers”, ”super-primes”, ” super-primes” or “prime-indexed primes[PIPs]”) as a subset of (simple or recursive) primes with (also) prime indexes (oPx is the x-th o-primeth, with order o ≥ 0 as explained later on). The author of this article also brings in a S-M-synthesis of some Goldbach-like conjectures (GLC) (including those which are “stronger” than BGC) and a new class of GLCs “stronger” than BGC, from which VBGC (which is essentially a variant of BGC applied on a serial array of subsets of primeths with a general order o ≥ 0) distinguishes as a very important conjecture of primes (with great importance in the optimization of the BGC experimental verification and other potential useful theoretical and practical applications in mathematics [including cryptography and fractals] and physics [including crystallography and M-Theory]), and a very special self-similar propriety of the primes subset of (noted/abbreviated as or as explained later on in this article). Keywords: Prime (number), primes with prime indexes, the o-primeths (with order o≥0), the Binary Goldbach Conjecture (BGC), the Ternary Goldbach Conjecture (TGC), Goldbach index-partition (GIP), fractal patterns of the number and distribution of Goldbach index-partitions, Goldbach-like conjectures (GLC), the Vertical Binary Goldbach Conjecture (VBGC) and Vertical Ternary Goldbach Conjecture (VTGC) the as applied on o-primeths

**Category:** Number Theory

[596] **viXra:1701.0618 [pdf]**
*replaced on 2017-03-11 10:01:21*

**Authors:** Juan G. Orozco

**Comments:** 9 Pages.

This paper introduces proofs to several open problems in number theory, particularly the Goldbach Conjecture and the Twin Prime Conjecture. These two conjectures are proven by using a greedy elimination algorithm, and incorporating Mertens' third theorem and the twin prime constant. The argument is extended to Germain primes, Cousin Primes, and other prime related conjectures. A generalization is provided for all algorithms that result in an Euler product like\prod{\left(1-\frac{a}{p}\right)}.

**Category:** Number Theory

[595] **viXra:1701.0618 [pdf]**
*replaced on 2017-01-26 21:10:42*

**Authors:** Juan G. Orozco

**Comments:** 9 Pages. Image of algorithm implementation example added.

This paper introduces proofs to several open problems in number theory, particularly the Goldbach Conjecture and the Twin Prime Conjecture. These two conjectures are proven by using a greedy elimination algorithm, and incorporating Mertens' third theorem and the twin prime constant. The argument is extended to Germain primes, Cousin Primes, and other prime related conjectures. A generalization is provided for all algorithms that result in a Euler product\prod{1-\frac{a}{p}}.

**Category:** Number Theory