[20] **viXra:1401.0241 [pdf]**
*submitted on 2014-01-31 15:47:02*

**Authors:** Marius Coman

**Comments:** 3 Pages.

Studying the relation between the two prime factors of a 2-Poulet number I found an interesting recurrent formula involving these numbers that seems to lead often to a value which is semiprime; based on this observation I made three conjectures about semiprimes.

**Category:** Number Theory

[19] **viXra:1401.0235 [pdf]**
*submitted on 2014-01-30 16:23:31*

**Authors:** Marius Coman

**Comments:** 2 Pages.

Combining two of my favorite topics of study, the recurrence relations and the Smarandache function, I discovered a very interesting pattern: seems like the recurrent formula f(n) = S(f(n – 2)) + S(f(n – 1)), where S is the Smarandache function and f(1), f(2) are any given different non-null positive integers, leads every time to a set of seven values (i.e. 11, 17, 28, 24, 11, 15, 16) which is then repeating infinitely.

**Category:** Number Theory

[18] **viXra:1401.0222 [pdf]**
*submitted on 2014-01-30 02:45:36*

**Authors:** Marius Coman

**Comments:** 4 Pages.

In this paper are made five conjectures about a type of pairs of primes respectively Fermat pseudoprimes which have the property to generate an infinity of primes respectively Fermat pseudoprimes via a recurrence formula that will be defined in this paper; we name the pairs with this property Coman pairs of primes respectively Coman pairs of pseudoprimes. Because it is easy to show that two given primes respectively pseudoprimes do not form such a pair and it is very difficult to prove that they form such a pair, the correct expression about two odd primes (or pseudoprimes) p, q, where p = 30*k + d and q = 30*h + d, where k, h are non-null positive integers and d has the values 1, 7, 11, 13, 17, 19, 23, 29, is that the pair (p,q) is not a Coman pair respectively that the pair (p,q) is a possible Coman pair of primes (or pseudoprimes).

**Category:** Number Theory

[17] **viXra:1401.0204 [pdf]**
*submitted on 2014-01-28 07:59:49*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In my previous paper «Twenty-four conjectures about “the eight essential subsets of primes”» are made three conjectures about each one from the following eight subsets: the primes of the form 30*k + 1, 30*k + 7, 30*k + 11, 30*k + 13, 30*k + 17, 30*k + 19, 30*k + 23 respectively 30*k + 29. The conjectures from that paper state that each from these eight sets of primes has an infinity of terms and also that each one of them can be entirely defined with a recurrent formula starting from just three given terms. In this paper are generalized the three conjectures for an infinity of subsets, each having possibly an infinity of terms which are primes or squares of primes, subsets of integers of the form 2*p(1)*p(2)*...*p(m)*k + d, where p(1), p(2), ..., p(m) are the first m odd primes, k is a non-null positive integer and d an odd positive integer satisfying certain conditions.

**Category:** Number Theory

[16] **viXra:1401.0203 [pdf]**
*submitted on 2014-01-28 05:00:08*

[15] **viXra:1401.0199 [pdf]**
*submitted on 2014-01-27 14:42:23*

**Authors:** Barar Stelian Liviu

**Comments:** 10 Pages.

sieve

**Category:** Number Theory

[14] **viXra:1401.0164 [pdf]**
*submitted on 2014-01-24 09:49:32*

**Authors:** Marius Coman

**Comments:** 6 Pages.

In this paper are made twenty-four conjectures about eight subsets of prime numbers, i.e. the primes of the form 30*k + 1, 30*k + 7, 30*k + 11, 30*k + 13, 30*k + 17, 30*k + 19, 30*k + 23 respectively 30*k + 29. Because we strongly believe that this classification of primes can have many applications, we refered in the title of this paper to these subsets of primes as to “the eight essential subsets of primes”. The conjectures state that each from these eight sets of primes has an infinity of terms and also that each one of them can be entirely defined with a recurrent formula starting from just three given terms.

**Category:** Number Theory

[13] **viXra:1401.0150 [pdf]**
*submitted on 2014-01-22 04:28:22*

**Authors:** Jiang Shan

**Comments:** 2 Pages.

I was the only solution to form indefinite equation and according to the Catalan theorem to the proof of the bill conjecture.

**Category:** Number Theory

[12] **viXra:1401.0137 [pdf]**
*replaced on 2015-07-28 08:48:19*

**Authors:** Bertrand Wong

**Comments:** 5 Pages.

The Riemann hypothesis is an important outstanding problem in number theory as its validity will affirm the manner of the distribution of the prime numbers. It posits that all the non-trivial zeros of the zeta function ζ lie on the critical strip between Re(s) = 0 and Re(s) = 1 at the critical line Re(s) = 1/2. The important question is whether there would be zeros appearing at other locations on this critical strip, e.g., at Re(s) = 1/4, 1/3, 3/4, or, 4/5, etc., which would disprove the Riemann hypothesis. This paper provides an indirect proof or proof by contradiction (reductio ad absurdum) of the Riemann hypothesis.

**Category:** Number Theory

[11] **viXra:1401.0132 [pdf]**
*submitted on 2014-01-17 19:31:35*

**Authors:** Zhang Tianshu

**Comments:** 13 Pages.

We know that every positive even number 2n(n≥3) can express in a sum which 3 plus an odd number 2k+1(k≥1) makes. And then, for any odd point 2k+1 (k≥1)at the number axis, if 2k+1 is an odd prime point, of course even number 3+(2k+1) is equal to the sum which odd prime number 2k+1 plus odd prime number 3 makes; If 2k+1 is an odd composite point, then let 3<B<2k+1, where B is an odd prime point, and enable line segment B(2k+1) to equal line segment 3C. If C is an odd prime point, then even number 3+(2k+1) is equal to the sum which odd prime number B plus odd prime number C makes. So the proof for Goldbach’s Conjecture is converted to prove there be certainly such an odd prime point B at the number axis’s a line segment which take odd point 3 and odd point 2k+1 as ends, so as to prove the conjecture by such a method indirectly.

**Category:** Number Theory

[10] **viXra:1401.0129 [pdf]**
*submitted on 2014-01-17 19:53:36*

**Authors:** Zhang Tianshu

**Comments:** 19 Pages.

Let us consider positive odd numbers which share a prime factor1 as a kind, then the positive directional half line of the number axis consists of infinite many equivalent line segments on same permutation of c kinds’ odd points plus odd points amongst the c kinds’ odd points, where c≥1. We will prove together that there are infinitely many sets of n-odd prime numbers and pairs of consecutive odd prime numbers by the mathematical induction with aid of such equivalent line segments and odd points thereof, in this article.

**Category:** Number Theory

[9] **viXra:1401.0128 [pdf]**
*submitted on 2014-01-17 19:59:53*

**Authors:** Zhang Tianshu

**Comments:** 13 Pages.

Let us consider odd numbers which share a prime factor1 as a kind, then the number axis’s positive half line which begins with odd point 3 consists of infinite many equivalent line segments on same permutation of c kinds’ odd points plus odd points amongst the c kinds’ odd points, where c≥1. In this article, we shall prove the unproved half of the Polignac’s conjecture by mathematical induction with the aid of such equivalent line segments and kinds of odd points thereon.

**Category:** Number Theory

[8] **viXra:1401.0127 [pdf]**
*submitted on 2014-01-17 20:04:31*

**Authors:** Zhang Tianshu

**Comments:** 17 Pages.

If reduce limits which contain odd primes by a half for Legendre’s conjecture, then there is at least an odd prime within the either half likewise, this is exactly the Legendre-Zhang’s conjecture. We shall first prove the Legendre-Zhang’s conjecture by mathematical induction with the aid of two number axes’ positive half lines whose directions reverse from each other. Successively, prove the Gilbreath’s conjecture by mathematical induction with the aid of the got result.

**Category:** Number Theory

[7] **viXra:1401.0126 [pdf]**
*submitted on 2014-01-17 20:08:12*

**Authors:** Zhang Tianshu

**Comments:** 10 Pages.

In this article, we shall prove the Beal’s conjecture by certain usual mathematical fundamentals with the aid of proven Fermat’s last theorem, and finally reach a conclusion that the Beal’s conjecture is tenable.

**Category:** Number Theory

[6] **viXra:1401.0125 [pdf]**
*submitted on 2014-01-17 20:11:18*

**Authors:** Zhang Tianshu

**Comments:** 14 Pages.

If every positive integer is able to be operated into 1 by set operational rule of the Collatz conjecture, then begin with 1, we can get all positive integers after pass infinite many operations by another operational rule on the contrary to the set operational rule. Thereby, substitute some unknown proof of the former proposition by a poof of the latter proposition, in addition apply mathematical induction, then this is exactly an effective way for proving Collatz conjecture, thus we use it in this article justly.

**Category:** Number Theory

[5] **viXra:1401.0124 [pdf]**
*submitted on 2014-01-18 05:00:59*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In a previous paper (“Five conjectures on Sophie Germain primes and Smarandache function and the notion of Smarandache-Germain primes”) I defined two notions: the Smarandache-Germain pairs of primes and the Coman-Germain primes of the first and second degree. The few conjectures that I made on these particular types of primes inspired me to make two other conjectures regarding two sets of primes that are generalizations of the set of Sophie Germain primes. And, based on the observation of the first few primes from these two possible infinite sets of primes, I also made a conjecture regarding the primes q of the form q = p*2^n + 31 = r*2^m + 3, where p, r are primes an m, n are non-null positive integers.

**Category:** Number Theory

[4] **viXra:1401.0083 [pdf]**
*submitted on 2014-01-11 03:37:29*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I define a new type of pairs of primes, id est the Smarandache-Germain pairs of primes, notion related to Sophie Germain primes and also to Smarandache function, and I conjecture that for all pairs of Sophie Germain primes but a definable set of them there exist corespondent pairs of Smarandache-Germain primes. I also make a conjecture that attributes to the set of Sophie Germain primes but a definable subset of them a corespondent set of smaller primes, id est Coman-Germain primes.

**Category:** Number Theory

[3] **viXra:1401.0080 [pdf]**
*submitted on 2014-01-10 16:38:19*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In a previous article I defined the Smarandache-Coman congruence on primes. In this paper I present few sequences of primes that are congruent sco n.

**Category:** Number Theory

[2] **viXra:1401.0077 [pdf]**
*submitted on 2014-01-10 00:11:34*

**Authors:** Ocean Yu

**Comments:** 6 Pages.

By following Riemann'a method, this paper extended
discussion on J(x) prime formula. We showed relationship between this
formula with Jq(x). Further more, we presented a new expression for Euler-
Mascheroni constant
, and showed a way for approaching proving Riemann
Hypothesis.

**Category:** Number Theory

[1] **viXra:1401.0064 [pdf]**
*submitted on 2014-01-08 12:36:24*

**Authors:** Marius Coman

**Comments:** 4 Pages.

In two previous articles I defined the Smarandache-Coman divisors of order k of a composite integer n with m prime factors and I made few conjectures about few possible infinite sequences of Poulet numbers, characterized by a certain set of Smarandache-Coman divisors. In this paper I define a very related notion, the Smarandache-Coman congruence on primes, and I also make four conjectures regarding Poulet numbers based on this new notion.

**Category:** Number Theory