[20] **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

[19] **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

[18] **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

[17] **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

[16] **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

[15] **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

[14] **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

[13] **viXra:1701.0588 [pdf]**
*replaced on 2017-04-14 07:26:33*

**Authors:** Andrei-Lucian Dragoi

**Comments:** 30 Pages.

This paper proposes the generalization of the both binary (strong) and ternary (weak) Goldbach’s Conjectures (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
(VBGC 1.5e - the conjecture only - 23.02.2017 - 21 pages) The "Vertical" (generalization of) the Binary Goldbach's Conjecture (VBGC) as applied on “iterative” primes with (recursive) prime indexes (i-primeths). Available from: https://www.researchgate.net/publication/313038562_VBGC_15e_-_the_conjecture_only_-_23022017_-_21_pages_The_Vertical_generalization_of_the_Binary_Goldbach%27s_Conjecture_VBGC_as_applied_on_iterative_primes_with_recursive_prime_indexes_i-primeths [accessed Apr 14, 2017].

**Category:** Number Theory

[12] **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

[11] **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

[10] **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

[9] **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

[8] **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

[7] **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

[6] **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

[5] **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

[4] **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

[3] **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

[2] **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

[1] **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