**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(10) - 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(27) - 1404(10) - 1405(17) - 1406(20) - 1407(34) - 1408(51) - 1409(47) - 1410(17) - 1411(16) - 1412(18)

2015 - 1501(14) - 1502(14) - 1503(33) - 1504(23) - 1505(18) - 1506(12) - 1507(16) - 1508(14) - 1509(15) - 1510(12) - 1511(9) - 1512(26)

2016 - 1601(14) - 1602(18) - 1603(77) - 1604(55) - 1605(28) - 1606(18) - 1607(21) - 1608(18) - 1609(24) - 1610(24) - 1611(12) - 1612(20)

2017 - 1701(20) - 1702(12)

Any replacements are listed further down

[1409] **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

[1408] **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

[1407] **viXra:1702.0172 [pdf]**
*submitted on 2017-02-15 02:30:57*

**Authors:** Krzysztof Maslanka

**Comments:** 7 Pages.

Certain analytical expressions which "feel" the divisors of natural numbers are investigated. We show that these expressions encode in some way the well-known algorithm of the sieve of Eratosthenes.

**Category:** Number Theory

[1406] **viXra:1702.0166 [pdf]**
*submitted on 2017-02-14 10:18:35*

**Authors:** Chongjunhuang

**Comments:** 10 Pages.

Prime density formula

**Category:** Number Theory

[1405] **viXra:1702.0162 [pdf]**
*submitted on 2017-02-14 08:01:15*

[1404] **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

[1403] **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

[1402] **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

[1401] **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

[1400] **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

[1399] **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

[1398] **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

[1397] **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

[1396] **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

[1395] **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

[1394] **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

[1393] **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

[1392] **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

[1391] **viXra:1701.0600 [pdf]**
*submitted on 2017-01-24 02:35:20*

**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 + 3)*(4*j + 1)*(4*k + 3) is true that (k – h) and j must share a common factor (sometimes (k – h) is a multiple of j). 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

[1390] **viXra:1701.0588 [pdf]**
*submitted on 2017-01-25 02:44:01*

**Authors:** Andrei Lucian Dragoi

**Comments:** 15 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

[1389] **viXra:1701.0585 [pdf]**
*submitted on 2017-01-23 13:26:30*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that for any 2-Poulet number (Fermat pseudoprime to base 2 with two prime factors, see the sequence A214305 in OEIS) of the form (4*h + 1)*(4*k + 1) is true that h and k can not be relatively primes (in fact, for sixteen from the first twenty 2-Poulet numbers of this form is true that k is a multiple of h and this is also the case for four from a randomly chosen set of five consecutive, much larger, such numbers).

**Category:** Number Theory

[1388] **viXra:1701.0483 [pdf]**
*submitted on 2017-01-13 13:46:54*

**Authors:** Reuven Tint

**Comments:** 4 Pages. original papper in russian

Annotation. Are given in Section 1 the theorem and its proof, complementing the classical formulation of the ABC conjecture, and in Chapter 2 addressed the issue of communication with the elliptic curve Frey's "Great" Fermat's theorem.

**Category:** Number Theory

[1387] **viXra:1701.0482 [pdf]**
*submitted on 2017-01-13 09:00:42*

**Authors:** guilhem CICOLELLA

**Comments:** 4 Pages.

the only consecutives powers being 8 and 9 the probleme consisted in demonstrating that the quantities of primes numbers inferior to one billion depended on one single equation based on two different methods of calculation with congruent results,the ultimate purpose being to prove the existence of an algorithm capable of determining two intricate values more quickly than with computer(rapid mathematical system r.m.S)

**Category:** Number Theory

[1386] **viXra:1701.0478 [pdf]**
*submitted on 2017-01-12 13:25:43*

**Authors:** Tom Masterson

**Comments:** 1 Page. © 1965 by Tom Masterson

A number theory query related to Fermat's last theorem in higher dimensions.

**Category:** Number Theory

[1385] **viXra:1701.0475 [pdf]**
*submitted on 2017-01-12 10:27:06*

**Authors:** Nikolay Dementev

**Comments:** 5 Pages.

Based on the observation of randomly chosen primes it has been conjectured that the sum of digits that form any prime number should yield either even number or another prime number. The conjecture was successfully tested for the first 100 primes.

**Category:** Number Theory

[1384] **viXra:1701.0397 [pdf]**
*submitted on 2017-01-10 07:35:16*

**Authors:** Quang Nguyen Van

**Comments:** 1 Page.

We have found a solution of FLT for n = 3, so that FLT is wrong. In this paper, we give a counterexample ( the solution in integer for equation x^3 + y^3 = z^3 only. It is too large ( 18 digits).

**Category:** Number Theory

[1383] **viXra:1701.0329 [pdf]**
*submitted on 2017-01-08 11:02:17*

**Authors:** Marius Coman

**Comments:** 4 Pages.

In this paper I make the following conjecture: For any pair of consecutive primes [p1, p2], p2 > p1 > 43, p1 and p2 having the same number of digits, there exist a prime q, 5 < q < p1, such that the number n obtained concatenating (from the left to the right) q with p2, then with p1, then again with q is prime. Example: for [p1, p2] = [961748941, 961748947] there exist q = 19 such that n = 1996174894796174894119 is prime. Note that the least values of q that satisfy this conjecture for twenty consecutive pairs of consecutive primes with 9 digits are 19, 17, 107, 23, 131, 47, 83, 79, 61, 277, 163, 7, 41, 13, 181, 19, 7, 37, 29 and 23 (all twenty primes lower than 300!), the corresponding primes n obtained having 20 to 24 digits! This method appears to be a good way to obtain big primes with a high degree of ease and certainty.

**Category:** Number Theory

[1382] **viXra:1701.0320 [pdf]**
*submitted on 2017-01-07 12:05:30*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following conjecture: For any pair of twin primes [p, p + 2], p > 5, there exist a prime q, 5 < q < p, such that the number n obtained concatenating (from the left to the right) q with p + 2, then with p, then again with q is prime. Example: for [p, p + 2] = [18408287, 18408289] there exist q = 37 such that n = 37184082891840828737 is prime. Note that the least values of q that satisfy this conjecture for twenty consecutive pairs of twins with 8 digits are 19, 7, 19, 11, 23, 23, 47, 7, 47, 17, 13, 17, 17, 37, 83, 19, 13, 13, 59 and 97 (all twenty primes lower than 100!), the corresponding primes n obtained having 20 digits! This method appears to be a good way to obtain big primes with a high degree of ease and certainty.

**Category:** Number Theory

[1381] **viXra:1701.0281 [pdf]**
*submitted on 2017-01-04 06:46:28*

**Authors:** Ryujin Choe

**Comments:** 1 Page.

Every even integer greater than 2 can be expressed as the sum of two primes

**Category:** Number Theory

[1380] **viXra:1701.0014 [pdf]**
*submitted on 2017-01-03 01:34:45*

**Authors:** Barry Foster

**Comments:** 2 Pages.

This is a two page attempt using simple concepts

**Category:** Number Theory

[1379] **viXra:1701.0012 [pdf]**
*submitted on 2017-01-02 10:39:11*

**Authors:** Clive Jones

**Comments:** 2 Pages.

An exploration of prime-number summing grids

**Category:** Number Theory

[1378] **viXra:1701.0008 [pdf]**
*submitted on 2017-01-02 04:55:37*

**Authors:** Ryujin Choe

**Comments:** 2 Pages.

Twin primes are infinitely many

**Category:** Number Theory

[1377] **viXra:1612.0406 [pdf]**
*submitted on 2016-12-30 11:14:55*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that there exist an infinity of primes p = 30*h + j, where j can be 1, 7, 11, 13, 17, 19, 23 or 29, such that, concatenating to the left p with a number m, m < p, is obtained a number n having the property that the number of primes of the form 30*k + j up to n is equal to p. Example: such a number p is 67 = 30*2 + 7, because there are 67 primes of the form 30*k + 7 up to 3767 and 37 < 67. I also conjecture that there exist an infinity of primes q that don’t belong to the set above, i.e. doesn’t exist m, m < q, such that, concatenating to the left q with m, is obtained a number n having the property shown. Primes can be classified based on this criteria in two sets: primes p that have the shown property like 13, 17, 23, 31, 37, 41, 47, 59, 61, 67, 71, 73, 89, 103 (...) and primes q that don’t have it like 7, 11, 19, 29, 43, 53, 79, 83, 101 (...).

**Category:** Number Theory

[1376] **viXra:1612.0400 [pdf]**
*submitted on 2016-12-30 02:12:38*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that for any prime p, p > 5, there exist q prime, q > p, where p = 30*k + m1 and q = 30*h + m2, m1 and m2 distinct, having one from the values 1, 7, 11, 13, 17, 19, 23, 29, such that the number of primes congruent to m1 (mod 30) up to n, where n is the number obtained concatenating p with q, is equal to the number of primes congruent to m2 (mod 30) up to n. Example: for p = 17 there exist q = 23 such that there are 34 primes of the form 30*k + 17 up to 1723 and 34 primes of the form 30*k + 23 up to 1723.

**Category:** Number Theory

[1375] **viXra:1612.0395 [pdf]**
*submitted on 2016-12-29 16:06:30*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that there exist an infinity of numbers n obtained concatenating two primes p and q, where p = 30*k + m1 and q = 30*h + m2, p < q, m1 and m2 distinct, having one from the values 1, 7, 11, 13, 17, 19, 23, 29, such that the number of primes congruent to m1 (mod 30) up to n is equal to the number of primes congruent to m2 (mod 30) up to n. Example: for n = 1723 obtained concatenating the primes p = 17 and q = 23, there exist 34 primes of the form 30*k + 17 up to 1723 and 34 primes of the form 30*k + 23 up to 1723.

**Category:** Number Theory

[1374] **viXra:1612.0387 [pdf]**
*submitted on 2016-12-28 20:35:01*

**Authors:** Prashanth R. Rao

**Comments:** 1 Page.

In this paper we prove a simple theorem that is distantly related to the Even Goldbach conjecture and is weaker than Chen’s theorem regarding the expression of any even integer as the sum of a prime number and a semiprime number. We show that any even integer greater than six can be written as the sum of two odd integers coprime to one another and atleast one of them is a prime.

**Category:** Number Theory

[1373] **viXra:1612.0383 [pdf]**
*submitted on 2016-12-29 01:16:00*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I conjecture that there exist an infinity of palindromes n for which the number of primes up to n of the form 30k + 7 is equal to the number of primes up to n of the form 30k + 11 and I found the first 40 terms of the sequence of n (I also found few larger terms, as 99599, 816618 or 1001001 up to which the number of primes from the two sets, equally for each, is 1154, 8159, respectively 9817).

**Category:** Number Theory

[1372] **viXra:1612.0294 [pdf]**
*submitted on 2016-12-18 23:45:17*

**Authors:** Zhang Tianshu

**Comments:** 21 Pages.

The ABC conjecture is both likely of the wrong and likely of the right in the face of satisfactory many primes and satisfactory many odd numbers of 6K±1 from operational results of computer programs. So we find directly a specific equality 1+2N (2N-2)=(2N-1)2 with N≥2, then set about analyzing limits of values of ε to discuss the right and the wrong of the ABC conjecture in which case satisfying 2N-1>(Rad(1, 2N(2N-2), 2N-1))1+ε . Thereby supply readers to make with a judgment concerning a truth or a falsehood which the ABC conjecture is.

**Category:** Number Theory

[1371] **viXra:1612.0278 [pdf]**
*submitted on 2016-12-17 11:51:55*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 52 pages. In French. Submitted to the journal Functiones et Approximatio, Commentarii Mathematici. Comments welcome.

En 1997, Andrew Beal \cite{B1} avait annonc\'e la conjecture suivante : \textit{Soient $A, B,C, m,n$, et $l$ des entiers positifs avec $m,n,l > 2$. Si $A^m + B^n = C^l$ alors $A, B,$ et $C$ ont un facteur en commun}. Nous consid\'erons le polyn\^ome $P(x)=(x-A^m)(x-B^n)(x+C^l)=x^3-px+q$ avec $p,q$ des entiers qui d\'ependent de $A^m,B^n$ et $C^l$. La r\'esolution de $x^3-px+q=0$ nous donne les trois racines $x_1,x_2,x_3$ comme fonctions de $p,q$ et d'un param\`etre $\theta$. Comme $A^m,B^n,-C^l$ sont les seules racines de $x^3-px+q=0$, nous discutons les conditions pour que $x_1,x_2,x_3$ soient des entiers. Quatre exemples num\'eriques sont pr\'esent\'es.
\\

**Category:** Number Theory

[1370] **viXra:1612.0262 [pdf]**
*submitted on 2016-12-16 09:29:19*

**Authors:** Marius Coman

**Comments:** 1 Page.

In my previous paper “Conjecture involving repunits, repdigits, repnumbers and also the primes of the form 30k + 11 and 30k + 13” I conjectured that there exist an infinity of repnumbers n (repunits, repdigits and numbers obtained concatenating not the unit or a digit but a number) for which the number of primes up to n of the form 30k + 11 is equal to the number of primes up to n of the form 30k + 13 and I found the first 18 terms of the sequence of n (I also found few larger terms, as 11111, 888888 and 11111111 up to which the number of primes from the two sets, equally for each, is 167, 8816, respectively 91687). In this paper I extend the search to first 40 terms of the sequence.

**Category:** Number Theory

[1369] **viXra:1612.0260 [pdf]**
*submitted on 2016-12-15 16:20:52*

**Authors:** Marius Coman

**Comments:** 1 Page.

In my previous paper “Conjecture on semiprimes n = p*q related to the number of primes up to n” I was wondering if there exist a class of numbers n for which the number of primes up to n of the form 30k + 1, 30k + 7, 30k + 11, 30k + 13, 30k + 17, 30k + 19, 30k + 23 and 30k + 29 is equal in each of these eight sets. I didn’t yet find such a class, but I observed that around the repdigits, repunits and repnumbers (numbers obtained concatenating not the unit or a digit but a number) the distribution of primes in these eight sets tends to draw closer and I made a conjecture about it.

**Category:** Number Theory

[1368] **viXra:1612.0257 [pdf]**
*submitted on 2016-12-15 10:18:33*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that there exist an infinity of semiprimes n = p*q, where p = 30*k + m1 and q = 30*h + m2, m1 and m2 distinct, having one from the values 1, 7, 11, 13, 17, 19, 23, 29, such that the number of primes congruent to m1 (mod 30) up to n is equal to the number of primes congruent to m2 (mod 30) up to n. Example: for n = 91 = 7*13, there exist 3 primes of the form 30*k + 7 up to 91 (7, 37 and 67) and 3 primes of the form 30*k + 13 up to 91 (13, 43 and 73).

**Category:** Number Theory

[1367] **viXra:1612.0253 [pdf]**
*submitted on 2016-12-15 06:24:20*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I conjecture that: (I) for any prime p of the form 6*k + 1 there are obtained at least n primes concatenating p to the left with the (p – 1) integers lesser than p, where n ≥ (p - 10)/3; (II) for any prime p of the form 6*k – 1, p ≥ 11, there are obtained at least n primes concatenating p to the left with the (p – 1) integers lesser than p, where n ≥ (p - 8)/3.

**Category:** Number Theory

[1366] **viXra:1612.0223 [pdf]**
*submitted on 2016-12-11 17:29:09*

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

The even Goldbach conjecture suggests that every even integer greater than four may be written as the sum of two odd primes. This conjecture remains unproven. We explore whether two probable primes satisfying the Fermat’s little theorem can potentially exist for every even integer greater than four. Our results suggest that there are no obvious constraints on this possibility.

**Category:** Number Theory

[1365] **viXra:1612.0200 [pdf]**
*submitted on 2016-12-11 02:20:30*

**Authors:** Simon Plouffe

**Comments:** 28 Pages.

A presentation is made on the numerical world of mathematics. Round table on the numerical data.
Une présentation du numérique à Nantes, table ronde organisée par ADN ouest au Lycée Clémenceau

**Category:** Number Theory

[1364] **viXra:1612.0142 [pdf]**
*submitted on 2016-12-09 02:54:12*

**Authors:** Brian Ekanyu

**Comments:** 6 Pages.

This paper proves an identity for generating a special kind of Pythagorean quadruples by conjecturing that the shortest is defined by a=1,2,3,4...... and b=a+1, c=ab and d=c+1. It also shows that a+d=b+c and that the surface area to volume ratio of these Pythagorean boxes is given by 4/a where a is the length of the shortest edge(side).

**Category:** Number Theory

[1363] **viXra:1612.0140 [pdf]**
*submitted on 2016-12-09 03:46:53*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In a previous paper I conjectured that for any largest prime factor of a Poulet number p1 with two prime factors exists a series with infinite many Poulet numbers p2 formed this way: p2 mod (p1 - d) = d, where d is the largest prime factor of p1 (see the sequence A214305 in OEIS). In this paper I conjecture that for any least prime factor of an odd Harshad number h1 with two prime factors, not divisible by 3, exists a series with infinite many Harshad numbers h2 formed this way: h2 mod (h1 - d) = d, where d is the least prime factor of p1.

**Category:** Number Theory

[1362] **viXra:1612.0138 [pdf]**
*submitted on 2016-12-08 15:52:25*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following two conjectures: (I) For any prime p, p > 5, there exist n positive integer such that the sum of the digits of the number p*2^n is divisible by p; (II) For any prime p, p > 5, there exist an infinity of positive integers m such that the sum of the digits of the number p*2^m is prime.

**Category:** Number Theory

[1361] **viXra:1612.0101 [pdf]**
*submitted on 2016-12-07 11:18:19*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I conjecture that for any pair of sexy primes (p, p + 6) there exist a prime q = p + 6*n, where n > 1, such that the number p*(p + 6)*(p + 6*n) is a Harshad number.

**Category:** Number Theory

[1360] **viXra:1612.0072 [pdf]**
*submitted on 2016-12-07 05:45:46*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that for any prime p of the form 6*k + 1 there exist an infinity of Harshad numbers of the form p*q1*q2, where q1 and q2 are distinct primes, q1 = p + 6*m and q2 = p + 6*n.

**Category:** Number Theory

[1359] **viXra:1612.0042 [pdf]**
*submitted on 2016-12-03 10:57:30*

**Authors:** Safaa Abdallah Moallim

**Comments:** 5 Pages.

In this paper we prove that there exist infinitely many twin prime numbers by studying n when 6n±1 are primes. By studying n we show that for every n that generates a twin prime number, there has to be m>n that generates a twin prime number too.

**Category:** Number Theory

[1358] **viXra:1611.0410 [pdf]**
*submitted on 2016-11-30 07:48:39*

**Authors:** Zhang Tianshu

**Comments:** 18 Pages.

The ABC conjecture seemingly is difficult to carry conviction in the face of satisfactory many primes and satisfactory many odd numbers of 6K±1 from operational results of computer programs. So we select and adopt a specific equality 1+2N(2N-2)=(2N-1)2 with N≥2 satisfying 2N-1>(Rad(2N-2))1+ ε. Then, proceed from the analysis of the limits of values of ε to find its certain particular values, thereby finally negate the ABC conjecture once and for all.

**Category:** Number Theory

[1357] **viXra:1611.0390 [pdf]**
*submitted on 2016-11-29 03:29:40*

**Authors:** Robert Deloin

**Comments:** 13 Pages.

Bunyakovsky's conjecture states that under special conditions, polynomial integer functions of degree greater than one generate infinitely many primes.
The main contribution of this paper is to introduce a new approach that enables to prove Bunyakovsky's conjecture. The key idea of this new
approach is that there exists a general method to solve this problem by using only arithmetic progressions and congruences.
As consequences of Bunyakovsky's proven conjecture, three Landau's problems are resolved: the n^2+1 problem, the twin primes conjecture and
the binary Goldbach conjecture.
The method is also used to prove that there are infinitely many primorial and factorial primes.

**Category:** Number Theory

[1356] **viXra:1611.0373 [pdf]**
*submitted on 2016-11-27 08:39:53*

**Authors:** Victor Christianto

**Comments:** 4 Pages. This paper will be submitted to Annals of Mathematics

In this paper we will give an outline of proof of Fermat’s Last Theorem using a graphical method. Although an exact proof can be given using differential calculus, we choose to use a more intuitive graphical method.

**Category:** Number Theory

[1355] **viXra:1611.0224 [pdf]**
*submitted on 2016-11-14 18:05:57*

**Authors:** Jonas Kaiser

**Comments:** 11 Pages.

The sieve of Collatz is a new algorithm to trace back the non-linear Collatz problem to a linear cross out algorithm. Until now it is unproved.

**Category:** Number Theory

[1354] **viXra:1611.0178 [pdf]**
*submitted on 2016-11-12 09:51:56*

**Authors:** Aaron Chau

**Comments:** 3 Pages.

十分幸运，本文应用的是永不改变的定律（多与少），而不再是重复那类受局限的定理。
感谢数学的美妙，因为多与少的个数区别永远会造成二个质数的距离= 2。简述，= 2。

**Category:** Number Theory

[1353] **viXra:1611.0176 [pdf]**
*submitted on 2016-11-12 04:58:51*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In a previous paper I defined the notion of Harshad-Coman numbers as the numbers n with the property that (n – 1)/(s(n) – 1), where s(n) is the sum of the digits of n, is integer. In this paper I conjecture that there exist an infinity of even numbers n for which n^2 is a Harshad-Coman number and I also make a classification in four classes of all the even numbers.

**Category:** Number Theory

[1352] **viXra:1611.0175 [pdf]**
*submitted on 2016-11-12 05:01:08*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In a previous paper I defined the notion of Harshad-Coman numbers as the numbers n with the property that (n – 1)/(s(n) – 1), where s(n) is the sum of the digits of n, is integer. In this paper I conjecture that there exist an infinity of odd numbers n for which n^2 is a Harshad-Coman number and I also make a classification in three classes of all the odd numbers greater than 1.

**Category:** Number Theory

[1351] **viXra:1611.0172 [pdf]**
*submitted on 2016-11-11 15:58:33*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following three conjectures: (I) If P is both a Poulet number and a Harshad number, than the number P – 1 is also a Harshad number; (II) If P is a Poulet number divisible by 5 under the condition that the sum of the digits of P – 1 is not divisible by 5 than P – 1 is a Harshad number; (III) There exist an infinity of Harshad numbers of the form P – 1, where P is a Poulet number.

**Category:** Number Theory

[1350] **viXra:1611.0171 [pdf]**
*submitted on 2016-11-11 16:00:16*

**Authors:** Marius Coman

**Comments:** 2 Pages.

OEIS defines the notion of Harshad numbers as the numbers n with the property that n/s(n), where s(n) is the sum of the digits of n, is integer (see the sequence A005349). In this paper I define the notion of Harshad-Coman numbers as the numbers n with the property that (n – 1)/(s(n) – 1), where s(n) is the sum of the digits of n, is integer and I make the conjecture that there exist an infinity of Poulet numbers which are also Harshad-Coman numbers.

**Category:** Number Theory

[1349] **viXra:1611.0120 [pdf]**
*submitted on 2016-11-09 07:22:21*

**Authors:** Jian Ye

**Comments:** 3 Pages.

Goldbach’s conjecture: symmetrical primes exists in natural numbers. the generalized Goldbach’s conjecture: symmetry of prime number in the former and tolerance coprime to arithmetic progression still exists.

**Category:** Number Theory

[1348] **viXra:1611.0089 [pdf]**
*submitted on 2016-11-07 11:29:42*

**Authors:** W.B. Vasantha Kandasamy, K. Ilanthenral, Florentin Smarandache

**Comments:** 10 Pages.

The Collatz conjecture is an open conjecture in mathematics named so after Lothar Collatz who proposed it in 1937. It is also known as 3n + 1 conjecture, the Ulam conjecture (after Stanislaw Ulam), Kakutanis problem (after Shizuo
Kakutani) and so on. Several various generalization of the Collatz conjecture
has been carried. In this paper a new generalization of the Collatz conjecture
called as the 3n ± p conjecture; where p is a prime is proposed. It functions on
3n + p and 3n - p, and for any starting number n, its sequence eventually enters
a finite cycle and there are finitely many such cycles. The 3n ± 1 conjecture, is
a special case of the 3n ± p conjecture when p is 1.

**Category:** Number Theory

[1347] **viXra:1611.0085 [pdf]**
*submitted on 2016-11-07 06:46:24*

**Authors:** Predrag Terzic

**Comments:** 32 Pages.

Some theorems and conjectures concerning prime numbers .

**Category:** Number Theory

[1346] **viXra:1610.0356 [pdf]**
*submitted on 2016-10-29 14:52:21*

**Authors:** Caitherine Gormaund

**Comments:** 2 Pages.

In which the Collatz Conjecture is proven using fairly simple mathematics.

**Category:** Number Theory

[1345] **viXra:1610.0349 [pdf]**
*submitted on 2016-10-28 13:23:48*

**Authors:** Reza Farhadian

**Comments:** 4 Pages.

In this paper we offer the some details and particulars about some famous conjectures in relative to consecutive primes.

**Category:** Number Theory

[1344] **viXra:1610.0313 [pdf]**
*submitted on 2016-10-26 05:42:56*

**Authors:** Jared Beal

**Comments:** 14 Pages.

This paper describes an algorithm for finding all the prime numbers. It also describes how this pattern among primes can be used to show the ratio of primes to not primes in an infinite set of X integers. It can also be used to show that the ratio of twin primes to not twin primes in an infinite set of X integers is always going to be greater than zero.

**Category:** Number Theory

[1343] **viXra:1610.0284 [pdf]**
*submitted on 2016-10-24 03:05:49*

**Authors:** Reuven Tint

**Comments:** Updates: 4.3.2 - 4.3.5.. page 7

Аннотация. Предложен вариант решения гипотезы Била с помощью прямого доказательства» Великой» теоремы Ферма элементарными методами. Новыми являются «инвариантное тождество « (ключевое слово) и полученные нами приведенные в тексте работы тождества, позволившие напрямую решить ВТФ и гипотезу Била,и ряд других. Предложены также новая формулировка теорем ( п.2.1.4.), ,доказательства для n= 1,2,3,..n>2 и x,y,z>2.

**Category:** Number Theory

[1342] **viXra:1610.0276 [pdf]**
*submitted on 2016-10-24 00:02:00*

**Authors:** John Smith

**Comments:** 19 Pages.

Riemann's prime-counting function R(x) looks good for every value of x we can compute, but in the light of Littlewood's result its superiority over li(x) is illusory: Ingram (1938) pointed out that 'for special values of x (as large as we please), the one approximation will deviate as widely as the other from the true value'. This note introduces a type of prime-counting function that is always better than li(x)...

**Category:** Number Theory

[1341] **viXra:1610.0275 [pdf]**
*submitted on 2016-10-23 13:15:42*

**Authors:** Reuven Tint

**Comments:** 2 Pages.

Аннотация. Интерес к названной в заглавии проблеме вызван следующими соображениями:
1) Возьмем, к примеру, «пифагорово» уравнение, все взаимно простые решения которого опре-
деляются формулами A= a^2- b^2 и B=2ab. Но если мы выберем A≠a^2- b^2 и B≠2ab как гипо-
тетически «верные» решения этого уравнения, то, наверное, можно будет доказать, что, в этом
случае, «пифагорово» уравнение не существует. Но оно действительно не существует для гипотетически выбранных «верных» решений.
2) Уравнение A^N+B^N = C^N и уравнение эллиптической кривой Фрея (как будет показано ниже для предложенного варианта их решения) не совместны.
3) Поэтому, как представляется, выглядит не совсем убедительной связь между уравнением
эллиптической кривой Фрея и соответствующим уравнением Ферма.
4) Приведено приложение.

**Category:** Number Theory

[1340] **viXra:1610.0274 [pdf]**
*submitted on 2016-10-23 13:19:39*

**Authors:** Reuven Tint

**Comments:** 2 Pages.

Annotation. Interest in the title problem is caused by the following considerations:
1) Take, for example, "Pythagoras' equation, all of which are relatively prime solutions determined
Delyan formulas A= a^2- b^2 and B=2ab. But if we choose A≠a^2- b^2 and B≠2ab both hypo-
Tethyan "correct" solutions of this equation, then perhaps it will be possible to prove that, in this
case, "Pythagoras" equation exists. But it really does not exist for the selected hypothetically "true" solutions.
2) The equation A^N+B^N = C^N and the equation of the elliptic curve Frey (as will be shown below for the proposed options to solve them) are not compatible.
3) Therefore, it seems, it does not look quite convincing relationship between the equation
elliptic curve Frey Farm and the corresponding equation.
4) Supplement.

**Category:** Number Theory

[1339] **viXra:1610.0272 [pdf]**
*submitted on 2016-10-23 13:58:45*

**Authors:** Luca Nascimbene

**Comments:** 13 Pages.

In this paper the author continue the works [6] [11] [12] and present a proposal for a demonstration on the Riemann Hypothesis and the conjecture on the multiplicity of non-trivial zeros of the Riemann s zeta.

**Category:** Number Theory

[1338] **viXra:1610.0253 [pdf]**
*submitted on 2016-10-21 18:17:51*

**Authors:** Filippos Nikolaidis

**Comments:** 10 Pages. fil_nikolaidis@yahoo.com

The present study is an effort for giving some evidence that the goldbach conjecture is not true, by showing that not all even natural numbers greater than two can be expressed as a sum of two primes. This conclusion can be drawn by showing that prime numbers are not enough –in population- so that, when added in couples, to give all the even numbers.

**Category:** Number Theory

[1337] **viXra:1610.0183 [pdf]**
*submitted on 2016-10-17 05:37:47*

**Authors:** Edward Szaraniec

**Comments:** 5 Pages.

Equation constituting the Beal conjecture is rearranged and squared, then rearranged
again and raised to power 4. The result, standing as an equivalent having the same
property, is emerging as a singular primitive Pythagorean equation with no solution.
So, the conjecture is proved. General line of proving the Pythagorean equation is
observed as a moving spirit.

**Category:** Number Theory

[1336] **viXra:1610.0172 [pdf]**
*submitted on 2016-10-16 05:13:25*

**Authors:** Mugur B. Răuţ

**Comments:** 5 Pages.

In this paper we propose another proof for Fermat’s Last Theorem (FLT). We found a simpler approach through Pythagorean Theorem, so our demonstration would be close to the times FLT was formulated. On the other hand it seems the Pythagoras’ Theorem was the inspiration for FLT. It resulted one of the most difficult mathematical problem of all times, as it was considered. Pythagorean triples existence seems to support the claims of the previous phrase.

**Category:** Number Theory

[1335] **viXra:1610.0106 [pdf]**
*submitted on 2016-10-10 03:35:21*

**Authors:** W.B. Vasantha Kandasamy, K. Ilanthenral, Florentin Smarandache

**Comments:** 9 Pages.

The Collatz conjecture is an open conjecture in mathematics named so after Lothar Collatz who proposed it in 1937. It is also known as 3n + 1 conjecture, the Ulam conjecture (after Stanislaw Ulam), Kakutani's problem (after Shizuo Kakutani) and so on.
In this paper a new conjecture called as the 3n-1 conjecture which is akin to the Collatz conjecture is proposed. It functions on 3n -1, for any starting number n, its sequence eventually reaches either 1, 5 or 17. The 3n-1 conjecture is compared with the Collatz conjecture.

**Category:** Number Theory

[1334] **viXra:1610.0099 [pdf]**
*submitted on 2016-10-08 17:28:15*

**Authors:** Idriss Olivier Bado

**Comments:** Dans ce présent document nous donnons la preuve de la conjecture de Sophie Germain en utilisant le theoreme de densité de Chebotarev ,le principe d' inclusion d'exclusion de Moivre ,la formule de Mertens . en 13 pages nous donnons une preuve convaincante

In this paper We give Sophie Germain 's conjecture proof by using Chebotarev density theorem, principle inclusion -exclusion of Moivre, Mertens formula

**Category:** Number Theory

[1333] **viXra:1610.0083 [pdf]**
*submitted on 2016-10-07 06:34:33*

**Authors:** Ricardo Gil

**Comments:** 2 Pages.

ζ(s)=1/(((1/(2))/log(2)))+ 1/(((1/(3))/log(3)))+ 1/(((1/(4))/log(4)))+1/(((1/(5))/log(5))) is a form of Riemann Zeta Function and it shows an approximate relationship between the Riemann Zeta Function and Prime Numbers.

**Category:** Number Theory

[1332] **viXra:1610.0082 [pdf]**
*submitted on 2016-10-07 06:37:51*

**Authors:** Ricardo Gil

**Comments:** 1 Page.

The classical Distribution of Primes Equation can be modified to make an Nth Prime Equation which generates the Nth Prime.

**Category:** Number Theory

[1331] **viXra:1610.0065 [pdf]**
*submitted on 2016-10-05 09:48:06*

**Authors:** Bing He

**Comments:** 14 Pages.

In this paper we give a finite field analogue of the Lauricella hypergeometric series and
obtain some transformation and reduction formulae and several generating functions for the Lauricella hypergeometric series over finite fields. These generalize some known results of Li \emph{et al} as well as several other well-known results.

**Category:** Number Theory

[1330] **viXra:1610.0042 [pdf]**
*submitted on 2016-10-04 12:01:31*

**Authors:** Idriss Olivier Bado

**Comments:** Dans ce présent document nous donnons la preuve du théorème de Mertens en utilisant le théorème de densité de Chebotarev ,principle d'inclusion - exclusion de Moivre,formule de Mertens en 15 pages nous donnons une élégante preuve

In this paper we give the proof of Sophie Germain's conjecture by using Chebotarev density theorem, the principle inclusion-exclusion of Moivre, Mertens formula

**Category:** Number Theory

[1329] **viXra:1610.0034 [pdf]**
*submitted on 2016-10-03 19:56:15*

**Authors:** Chunxuan Jiang

**Comments:** 6 Pages.

using complex hyperbolic function we prove Fermat last theorem

**Category:** Number Theory

[1328] **viXra:1610.0033 [pdf]**
*submitted on 2016-10-03 20:01:14*

**Authors:** Chunxuan Jiang

**Comments:** 5 Pages.

using trogonometric function we prove Fermat last theorem

**Category:** Number Theory

[1327] **viXra:1610.0024 [pdf]**
*submitted on 2016-10-03 09:06:13*

**Authors:** Ricardo Gil

**Comments:** 2 Pages.

(1/2 Part)>1.002 (1.002, 2.16, 4.008 & 6.012) Generate Riemann Non Trivial Zero’s Off Of Critical Line. A Riemann Non Trivial Zero off the Critical Line occurs between 1 /2 or .50 and Gamma 0.577215664901532860606512090 08240243104 215 93 359399.When (1/2 Part) = (1.002 , 2.16, 4.008 & 6.012) Riemann Non Trivial Zero’s Are Off .001 To The Rt. Of The Critical Line & When (1/2 Part)= (1 / 2) A Riemann Non Trivial Zero’s Will Be On Critical Line.

**Category:** Number Theory

[1326] **viXra:1610.0016 [pdf]**
*submitted on 2016-10-02 14:25:25*

**Authors:** Philip E Gibbs

**Comments:** 14 Pages.

A rational Diophantine m-tuple is a set of m distinct positive rational numbers such that the product of any two is one less than a rational number squared. A computational search is used to find over 300 examples of rational Diophantine sextuples of low height which are then analysed in terms of algebraic relationships between entries. Three examples of near-septuples are found where a rational Diophantine quintuple can be extended to sextuples in two different ways so that the combination fails to be a rational Diophantine septuple only in one pair.

**Category:** Number Theory

[1325] **viXra:1610.0009 [pdf]**
*submitted on 2016-10-01 19:37:45*

**Authors:** Liujingru

**Comments:** 4 Pages.

This work reveals the intrinsic relationship of numbers with the conception of “prime multiple” to prove the “hypothesis of twin primes”. Based on this proof, “Goldbach conjecture” is proved with the “Odd-Gaussian Corresponding”. The nature of “prime number” can be thus obtained.Paper is using the axiom Ⅶ twice. For the first time: high high more than nonsingular group, according to the axiom Ⅶ get there will be a (high + high group). Second: high + high group) will be (prime number + prime)

**Category:** Number Theory

[1324] **viXra:1610.0008 [pdf]**
*submitted on 2016-10-01 20:19:40*

**Authors:** 刘静儒

**Comments:** 4 Pages.

通过“素数的倍数”这一概念，揭示了数的内在关系，论证了“孪生素数猜想”，并在此基础上给出了“奇高组”的定义，并结合“高斯对应”，论文只是两次运用公理Ⅶ。第一次：奇高组多于非奇高组，根据公理Ⅶ得到必有这样的结果：（奇高组+奇高组）。第二次：（奇高组+奇高组）必有这样的结果：（素数+素数），这就证明了“哥德巴赫猜想”。

**Category:** Number Theory

[1323] **viXra:1610.0001 [pdf]**
*submitted on 2016-10-01 01:46:45*

**Authors:** Zhang Tianshu

**Comments:** 13 Pages.

Let us consider positive integers which have a common prime factor as a kind, then the positive half line of the number axis consists of infinite many recurring line segments of same permutations of c kinds of integers’ points, where c≥1. In this article we proved Grimm’s conjecture by stepwise change symbols of each kind of composite numbers’ points at the number axis, so as to form consecutive composite numbers’ points under the qualification of proven Legendre-Zhang conjecture as the true.

**Category:** Number Theory

[1322] **viXra:1609.0425 [pdf]**
*submitted on 2016-09-29 11:39:24*

**Authors:** Philip E Gibbs

**Comments:** 13 Pages.

A polynomial equation in six variables is given that generalises the definition of regular rational Diophantine triples, quadruples and quintuples to regular rational Diophantine sextuples. The definition can be used to extend a rational Diophantine quintuple to a weak rational Diophantine sextuple. In some cases a regular sextuple is a full rational Diophantine sextuple. Ten examples of this are provided

**Category:** Number Theory

[1321] **viXra:1609.0398 [pdf]**
*submitted on 2016-09-27 14:41:12*

**Authors:** BERKOUK Mohamed

**Comments:** 12 Pages.

Ceci est une démonstration de la conjecture de C.Goldbach émise en 1742 , aussi bien la faible que la forte , elle repose essentiellement sur le théorème fondamentales des nombres premiers , et quatre autres théorèmes plus quatre lemmes ...la démarche consiste à démontrer pour chaque pair ou impair l’existence d’au moins un couplet ou un triplet dont les éléments sont premiers qui répondent aux deux énoncés de la conjecture , et que plus ce nombre pair ou impair est grand , plus le nombre de couplets ou triplets premiers est grand ...

**Category:** Number Theory

[1320] **viXra:1609.0384 [pdf]**
*submitted on 2016-09-26 21:46:39*

**Authors:** Bing He, Long Li

**Comments:** 16 Pages.

In this paper we give a finite field analogue of one of the Appell series and obtain some transformation and reduction formulae and the generating functions for the Appell series over finite fields.

**Category:** Number Theory

[1319] **viXra:1609.0383 [pdf]**
*submitted on 2016-09-26 23:16:52*

**Authors:** A. A. Frempong

**Comments:** 6 Pages. Copyright © by A. A. Frempong

Beal conjecture has been proved on a single page; and the proof has been specialized to prove Fermat's last theorem, on half of a page. The approach used in the proof is exemplified by the following system. If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one would first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^z = a^x + b^y when x, y, z > 2, one should first determine why there are solutions when x, y, z = 2, and note the necessary condition in the solutions for x, y, z = 2. The necessary condition in the solutions for x, y, z = 2, will guide one to determine if there are solutions when x, y, z > 2. The proof in this paper is based on the identity (a^2 + b^2 )/c^2 = 1 for a primitive Pythagorean triple (a, b, c). It is shown by contradiction that the uniqueness of the x, y, z = 2, identity excludes all other x, y, z-values, x, y, z > 2 from satisfying the equation c^z = a^x + b^y. One will first show that if x, y, z = 2, c^z = a^x + b^y holds, noting the necessary condition in the solution; followed by showing that if x, y, z > 2 ( x, y, z integers), c^z = a^x + b^y has no solutions. Two proof versions are covered. The first version begins with only the terms in the given equation, but the second version begins with the introduction of ratio terms which are subsequently and "miraculously" eliminated to allow the introduction of a much needed term for the necessary condition for c^z = a^x + b^y to have solutions or to be true. Each proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system.

**Category:** Number Theory

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

**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

[615] **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

[614] **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

[613] **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

[612] **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

[611] **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

[610] **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

[609] **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

[608] **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

[607] **viXra:1701.0588 [pdf]**
*replaced on 2017-02-02 03:50:49*

**Authors:** Andrei Lucian Dragoi

**Comments:** 21 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

[606] **viXra:1701.0588 [pdf]**
*replaced on 2017-01-31 05:09:49*

**Authors:** Andrei Lucian Dragoi

**Comments:** 19 Pages.

**Category:** Number Theory

[605] **viXra:1701.0014 [pdf]**
*replaced on 2017-02-06 00:15:30*

**Authors:** Barry Foster

**Comments:** 2 Pages.

This is a two page attempt using simple concepts

**Category:** Number Theory

[604] **viXra:1701.0014 [pdf]**
*replaced on 2017-02-02 06:02:17*

**Authors:** Barry Foster

**Comments:** 2 Pages.

This is a two page attempt using simple concepts

**Category:** Number Theory

[603] **viXra:1701.0014 [pdf]**
*replaced on 2017-01-12 06:19:40*

**Authors:** Barry Foster

**Comments:** 2 Pages.

This is a two page attempt using simple concepts

**Category:** Number Theory

[602] **viXra:1612.0296 [pdf]**
*replaced on 2016-12-24 13:07:51*

**Authors:** Armando M. Evangelista Jr.

**Comments:** 4 Pages. typographical error on the abstract

ABSTRACT
Riemann Hypothesis states that all the non-trivial zeros of the zeta function ζ(s) have real part equal to 1⁄2. It is the purpose of this present work to prove that the Riemann Hypothesis is true.

**Category:** Number Theory

[601] **viXra:1612.0296 [pdf]**
*replaced on 2016-12-20 03:29:10*

**Authors:** Armando M. Evangelista Jr.

**Comments:** 4 Pages.

In Riemann’s 1859 paper he conjecture that all the zeros of the zeta funtion ζ(s) are real in the critical strip, 0 ≤ σ ≤ 1; or equivalently, if ζ(s) is a complex quantity in the said strip, then it has no zero. It is the purpose of this present work to prove that the Riemann Hypothesis is true.

**Category:** Number Theory

[600] **viXra:1612.0296 [pdf]**
*replaced on 2016-12-19 05:18:45*

**Authors:** Armando M. Evangelista Jr.

**Comments:** 4 Pages.

In Riemann’s 1859 paper he conjecture that all the zeros of ξ(s) are real in the critical strip 0 ≤ σ ≤ 1, or equivalently, if ξ(s) is a complex quantity in the said strip, then it has no zero. It is the purpose of this present work to prove that the Riemann Hypothesis is true.

**Category:** Number Theory

[599] **viXra:1612.0223 [pdf]**
*replaced on 2016-12-15 14:31:17*

**Authors:** Prashanth R. Rao

**Comments:** 1 Page.

The even Goldbach conjecture suggests that every even integer greater than four may be written as the sum of two odd primes. This conjecture remains unproven. We explore whether two probable primes satisfying the Fermat’s little theorem can potentially exist for every even integer greater than four. Our results suggest that there are no obvious constraints on this possibility.

**Category:** Number Theory

[598] **viXra:1612.0042 [pdf]**
*replaced on 2016-12-19 03:14:49*

**Authors:** Safa Abdallah Moallim

**Comments:** 8 Pages.

In this paper we prove that there exist infinitely many twin
prime numbers by studying n when 6n ± 1 are primes. By studying n we
show that for every n that generates a twin prime number, there has to be
m > n that generates a twin prime number too.

**Category:** Number Theory

[597] **viXra:1611.0390 [pdf]**
*replaced on 2016-12-08 03:13:44*

**Authors:** Robert Deloin

**Comments:** 10 Pages. This is version 2 with important changes.

Bunyakovsky's conjecture states that under special conditions, polynomial integer functions of degree greater than one generate innitely
many primes.
The main contribution of this paper is to introduce a new approach that enables to prove Bunyakovsky's conjecture. The key idea of this new
approach is that there exists a general method to solve this problem by using only arithmetic progressions and congruences.
As consequences of Bunyakovsky's proven conjecture, three Landau's problems are resolved: the n^2+1 problem, the twin primes conjecture and
the binary Goldbach conjecture.
The method is also used to prove that there are infinitely many primorial and factorial primes.

**Category:** Number Theory

[596] **viXra:1610.0065 [pdf]**
*replaced on 2016-10-10 23:28:04*

**Authors:** Bing He

**Comments:** 22 Pages.

In this paper we give a finite field analogue of the Lauricella hypergeometric series and
obtain some transformation and reduction formulae and several generating functions for the Lauricella hypergeometric series over finite fields. Some of these generalize some known results of Li \emph{et al} as well as several other well-known results.

**Category:** Number Theory

[595] **viXra:1610.0016 [pdf]**
*replaced on 2016-10-26 05:46:31*

**Authors:** Philip Gibbs

**Comments:** Pages. DOI: 10.13140/RG.2.2.29253.65761

A rational Diophantine m-tuple is a set of m distinct positive rational numbers such that the product of any two is one less than a rational number squared. A computational search has been used to find over 1000 examples of rational Diophantine sextuples of low height which are then analysed in terms of algebraic relationships between entries. Three examples of near-septuples are found where a rational Diophantine quintuple can be extended to sextuples in two different ways so that the combination fails to be a rational Diophantine septuple only in one pair.

**Category:** Number Theory

[594] **viXra:1609.0425 [pdf]**
*replaced on 2016-10-27 10:44:46*

**Authors:** Philip Gibbs

**Comments:** 13 Pages.

A polynomial equation in six variables is given that generalises the definition of regular rational Diophantine triples, quadruples and quintuples to regular rational Diophantine sextuples. The definition can be used to extend a rational Diophantine quintuple to a weak rational Diophantine sextuple. In some cases a regular sextuple is a full rational Diophantine sextuple. Ten examples of this are provided.

**Category:** Number Theory

[593] **viXra:1609.0398 [pdf]**
*replaced on 2016-10-26 15:46:30*

**Authors:** BERKOUK Mohamed

**Comments:** 12 Pages.

Ceci est une démonstration de la conjecture de C.Goldbach émise en 1742 , aussi bien la faible que la forte , elle repose essentiellement sur le théorème fondamentale des nombres premiers , ...la démarche consiste à démontrer pour chaque pair ou impair l’existence d’au moins un couplet ou un triplet dont les éléments sont premiers qui répondent aux deux énoncés de la conjecture à savoir la Sommation et la primalité des ses éléments, ...et que plus ce nombre pair ou impair est grand , plus le nombre de couplets ou triplets premiers est grand .

**Category:** Number Theory

[592] **viXra:1609.0383 [pdf]**
*replaced on 2016-10-01 23:01:09*

**Authors:** A. A. Frempong

**Comments:** 6 Pages. Copyright © by A. A. Frempong

Beal conjecture has been proved on a single page; and the proof has been specialized to prove Fermat's last theorem, on half of a page. The approach used in the proof is exemplified by the following system. If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one would first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^z = a^x + b^y when x, y, z > 2, one should first determine why there are solutions when x, y, z = 2, and note the necessary condition in the solutions for x, y, z = 2. The necessary condition in the solutions for x, y, z = 2, will guide one to determine if there are solutions when x, y, z > 2. The proof in this paper is based on the identity (a^2 + b^2 )/c^2 = 1 for a primitive Pythagorean triple (a, b, c). It is shown by contradiction that the uniqueness of the x, y, z = 2, identity excludes all other x, y, z-values, x, y, z > 2 from satisfying the equation c^z = a^x + b^y. One will first show that if x, y, z = 2, c^z = a^x + b^y holds, noting the necessary condition in the solution; followed by showing that if x, y, z > 2 ( x, y, z integers), c^z = a^x + b^y has no solutions. Two proof versions are covered. The first version begins with only the terms in the given equation, but the second version begins with the introduction of ratio terms which are subsequently and "miraculously" eliminated to allow the introduction of a much needed term for the necessary condition for c^z = a^x + b^y to have solutions or to be true. Each proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system.

**Category:** Number Theory