[20] **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

[19] **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

[18] **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

[17] **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

[16] **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

[15] **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

[14] **viXra:1612.0278 [pdf]**
*replaced on 2017-04-10 05:59:50*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 77 Pages. In French, with minor corrections. Submitted to Journal of Number Theory. Comments welcome.

In 1997, Andrew Beal announced the following conjecture : \textit{Let $A, B,C, m,n$, and $l$ be positive integers with $m,n,l > 2$. If $A^m + B^n = C^l$ then $A, B,$ and $C$ have a common factor.} We begin to construct the polynomial $P(x)=(x-A^m)(x-B^n)(x+C^l)=x^3-px+q$ with $p,q$ integers depending of $A^m,B^n$ and $C^l$. We resolve $x^3-px+q=0$ and we obtain the three roots $x_1,x_2,x_3$ as functions of $p,q$ and a parameter $\theta$. Since $A^m,B^n,-C^l$ are the only roots of $x^3-px+q=0$, we discuss the conditions that $x_1,x_2,x_3$ are integers. Four numerical examples are given.

**Category:** Number Theory

[13] **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

[12] **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

[11] **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

[10] **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

[9] **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

[8] **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

[7] **viXra:1612.0176 [pdf]**
*replaced on 2017-03-08 18:56:06*

**Authors:** Stephen Crowley

**Comments:** 6 Pages.

It is conjectured that argζ(1/2+i t_n)=S_n(t_n)=π(3/2-frac((ϑ(t_n))/π)-⌊g~^(-1)(n)⌋-n) where 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 S(t)=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

[6] **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

[5] **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

[4] **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

[3] **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

[2] **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

[1] **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