[77] **viXra:1603.0426 [pdf]**
*submitted on 2016-03-31 22:44:34*

**Authors:** Simon Plouffe

**Comments:** 14 Pages. news identities for primes, binomial sums and euler, bernoulli numbers

A survey is made based on finite sums of the polygamma function with rational arguments which are
D_(k,j)^n=∑_((m,n)=1)▒〖χ_j (n)ψ(k,m/n) 〗
Where, χ_j (n) is the j’th Dirichlet character and ψ(k,m/n) is the polygamma function of order k.
We use this representation to rewrite identities using a new notation for linear combinations of mathematical constants. Identities are given for prime numbers using irrational constants. For negative argument n we use the generalization of Espinosa and Moll[6], well implemented into Maple CAS.

**Category:** Number Theory

[76] **viXra:1603.0425 [pdf]**
*submitted on 2016-03-31 14:09:17*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following three conjectures: (I) there exist positive integers k such that the number (30*k)\\(30*k + p) is prime for an infinity of primes p. I used the operator “\\” with the meaning “concatenated to”; (II) there exist primes p such that the number (30*k)\\(30*k + p) is prime for an infinity of values of k; (III) there exist an infinity of primes of the form (30*k)\\(30*k + 1), where k positive integer.

**Category:** Number Theory

[75] **viXra:1603.0424 [pdf]**
*submitted on 2016-03-31 14:10:42*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I define a Smarandache reconcatenated sequence Sr(n) as “the sequence obtained from the terms of a Smarandache concatenated sequence S(n), terms for which was applied the operation of consecutive concatenation” and I present six such sequences. Example: for Smarandache consecutive numbers sequence (1, 12, 123, 1234, 12345...), the Smarandache reconcatenated consecutive numbers sequence has the terms: 1, 112, 112123, 1121231234, 112123123412345...). According to the same pattern, we can define back reconcatenated sequences (the terms of the Smarandache back reconcatenated consecutive numbers sequence, noted Sbr(n), are 1, 121, 123121, 1234123121...).

**Category:** Number Theory

[74] **viXra:1603.0414 [pdf]**
*submitted on 2016-03-30 18:19:23*

**Authors:** Brian Scannell

**Comments:** 28 Pages.

We look here at the geometry of zeta(3). By piling cubes a 3D shape is defined which has a volume of zeta(3). This shape is a double integral form for zeta(3). Considering the centroid of this shape leads to an experimental estimate for zeta(3). Cutting the shape parallel to the x axis reproduces the dilogarithmic relationship to zeta(3). Cutting the shape in the z axis reproduces the logarithmic version of Riemann’s formula for zeta(3). Geometrical considerations also reproduce formula for the polylog of a half Lin(1/2) for n=2 and 3.
These are illustrations of number geometry.

**Category:** Number Theory

[73] **viXra:1603.0403 [pdf]**
*submitted on 2016-03-30 07:26:23*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I present the following four Smarandache type sequences: (I) The sequence of numbers obtained concatenating the positive integers of the form 6*k – 1; (II) The sequence of numbers obtained concatenating the primes of the form 6*k – 1; (III) The sequence of numbers obtained concatenating the positive integers of the form 6*k + 1; (IV) The sequence of numbers obtained concatenating the primes of the form 6*k + 1.

**Category:** Number Theory

[72] **viXra:1603.0402 [pdf]**
*submitted on 2016-03-30 07:29:50*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: There exist an infinity of primes S(n+1) + S(n) - 1, where S(n) is a term in Smarandache concatenated odd sequence (which is defined as the sequence obtained through the concatenation of the first n odd primes).

**Category:** Number Theory

[71] **viXra:1603.0394 [pdf]**
*submitted on 2016-03-29 05:08:39*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I make the following two conjectures: (I) there exist an infinity of quintets of primes (p, p + 10, p + 30, p + 40, p + 60), where p is a prime of the form 6*k + 1; (II) there exist an infinity of primes of the form p\\(p + 10)\\(p + 30)\\(p + 40)\\(p + 60), where p is a number of the form 6*k + 1. I used the operator “\\” with the meaning “concatenated to”.

**Category:** Number Theory

[70] **viXra:1603.0393 [pdf]**
*submitted on 2016-03-29 06:17:41*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I make the following two conjectures: (I) there exist an infinity of quintets of primes (p, p + 20, p + 30, p + 50, p + 80), where p is a prime of the form 6*k - 1; (II) there exist an infinity of primes of the form p\\(p + 20)\\(p + 30)\\(p + 50)\\(p + 80), where p is a number of the form 6*k - 1. I used the operator “\\” with the meaning “concatenated to”.

**Category:** Number Theory

[69] **viXra:1603.0391 [pdf]**
*submitted on 2016-03-29 03:17:35*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I make the following conjecture: There exist an infinity of primes S(n) + S(n + 1) - 1, where S(n) is a term in Smarandache-Wellin sequence (which is defined as the sequence obtained through the concatenation of the first n primes).

**Category:** Number Theory

[68] **viXra:1603.0380 [pdf]**
*submitted on 2016-03-27 16:43:22*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following four conjectures: (I) there exist always a prime of the form p^2 + 18*m between the squares p^2 and q^2 of a pair of twin primes [p, q = p + 2], beside the pair [3, 5]; examples: for [p^2, q^2] = [11^2, 13^2] = [121, 169] there exist the primes 139 = 121 + 1*18) and 157 = 121 + 2*18; for [p^2, q^2] = [17^2, 19^2] = [289, 361] there exist the prime 307 = 289 + 1*18); (II) there exist always a prime of the form q^2 – 18*n between the squares p^2 and q^2 of a pair of twin primes [p, q = p + 2], beside the pair [3, 5]; examples: for [p^2, q^2] = [11^2, 13^2] = [121, 169] there exist the prime 151 = 169 – 1*18); for [p^2, q^2] = [17^2, 19^2] = [289, 361] there exist the prime 307 = 361 – 3*18); (III) there exist an infinity of r primes of the form p^2 + 18*m or q^2 – 18*n between the squares p^2 and q^2 of a pair of twin primes [p, q = p + 2] such that the number obtained concatenating p^2 to the right with r is prime; example: 121139 is prime; (IV) there exist an infinity of r primes of the form p^2 + 18*m or q^2 - 18*n between the squares p^2 and q^2 of a pair of twin primes [p, q = p + 2] such that the number obtained concatenating q^2 to the left with r is prime; example: 139169 is prime. Of course, the conjectures (III) and (IV) imply that there exist an infinity of pairs of twin primes.

**Category:** Number Theory

[67] **viXra:1603.0379 [pdf]**
*submitted on 2016-03-27 16:45:25*

**Authors:** Marius Coman

**Comments:** 11 Pages.

In this paper I list a number of 33 sequences of primes obtained by the method of concatenation; some of these sequences are presented and analyzed in more detail in my previous papers, gathered together in five books of collected papers: “Two hundred conjectures and one hundred and fifty open problems on Fermat pseudoprimes”, “Two hundred and thirteen conjectures on primes”, “Conjectures on primes and Fermat pseudoprimes, many based on Smarandache function”, “Sequences of integers, conjectures and new arithmetical tools”, “Formulas and polynomials which generate primes and Fermat pseudoprimes”.

**Category:** Number Theory

[66] **viXra:1603.0370 [pdf]**
*submitted on 2016-03-26 23:53:47*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture on an infinity of subsequences of primes in Smarandache prime-partial-digital sequence, defined as the sequence of prime numbers which admit a deconcatenation into a set of primes: for any prime p which admits a deconcatenation in k primes larger than 3 is true that there exist a number of k sequences of primes P1, P2,...,Pk, each one having an infinity of prime terms which also admit a deconcatenation in prime numbers, obtained replacing a prime q in p with primes having the same digital root as q (example: for the prime 547 there exist an infinite sequence of primes obtained replacing 5 with primes having the digital root equal to 5 (2347, 13147, 14947, ...) and also an infinite sequence of primes obtained replacing 47 with primes having the digital root equal to 2 (5101, 5227, 5281,...).

**Category:** Number Theory

[65] **viXra:1603.0369 [pdf]**
*submitted on 2016-03-26 23:55:28*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: Let’s consider the primes p obtained from composite numbers in the following way: concatenating the prime factors of a composite number n (example: for 31941 = 3*3*3*7*13*13, the concatenation of its prime factors is 33371313) is obtained either a prime (in which case this prime is p), either a composite; if is obtained a composite, is reiterated the operation until is obtained a prime (in which case this prime is p). I conjecture that there exist such prime p for every composite number.

**Category:** Number Theory

[64] **viXra:1603.0367 [pdf]**
*submitted on 2016-03-27 04:47:03*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following two conjectures: (I) For any prime p, p > 5, there exist a pair of primes (q1, q2), both having the group of their last digits equal to p, and a positive integer n, such that p = (n + 1)*q1 – n*q2 (examples: for p = 11, there exist the primes q1 = 211 and q2 = 311 and also the number n = 2 such that 11 = 3*211 – 2*311; for p = 29, there exist the primes q1 = 829 and q2 = 929 and also the number n = 8 such that 29 = 9*829 – 8*929); (II) For any q1 prime, q1 > 5, and any n non-null positive integer, there exist an infinity of primes q2, having the group of their last digits equal to q1, such that p = (n + 1)*q2 – n*q1 is prime; (III) For any q1 prime, q1 > 5, and any q2 prime having the group of its last digits equal to q1, there exist an infinity of positive integers n such that p = (n + 1)*q2 – n*q1 is prime.

**Category:** Number Theory

[63] **viXra:1603.0362 [pdf]**
*submitted on 2016-03-25 21:27:44*

**Authors:** Octavian Cira, Florentin Smarandache

**Comments:** 400 Pages.

Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. These notions, definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc. on integer sequences, numbers, quotients, residues, exponents, sieves, pseudo-primes squares cubes factorials, almost primes, mobile
periodicals, functions, tables, prime square factorial bases, generalized factorials, generalized palindromes, so on, have been extracted from the Archives of American Mathematics (University of Texas at Austin) and Arizona State University (Tempe): "The Florentin Smarandache papers" special collections, and Arhivele Statului (Filiala Vâlcea & Filiala Dolj, Romania).
This book was born from the collaboration of the two authors, which started in 2013. The first common work was the volume "Solving Diophantine Equations", published in 2014. The contribution of the authors can be summarized as follows: Florentin Smarandache came with his extraordinary ability to propose new areas of study in number theory, and Octavian Cira - with his algorithmic thinking and knowledge of Mathcad.

**Category:** Number Theory

[62] **viXra:1603.0356 [pdf]**
*submitted on 2016-03-25 12:28:28*

**Authors:** Prashanth R. Rao

**Comments:** 1 Page.

In this paper, we show that for all even integers “2n”, there exists infinite
positive integers “d” greater than one, such that their product “2nd” is a sum of two
primes. Any two odd primes add to give even integers. However this general method
does not allow us to understand the property or relationship among even numbers
numbers derived in this manner. On the other hand, our results suggests existence
of even integers of the specific form “2nd” that can be written as a sum of two
primes.

**Category:** Number Theory

[61] **viXra:1603.0355 [pdf]**
*submitted on 2016-03-25 09:59:25*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following three conjectures: (I) for any k having one of the values 1, 2, 4, 5, 7 or 8, there exist an infinity of primes obtained concatenating two primes that both have the digital root equal to k; (II) for any n positive integer, not divisible by 3, n ≥ 4, there exist primes obtained concatenating two primes that both have the digital sum equal to n; (III) there exist an infinity of values of n, positive integer, for which exist an infinity of primes obtained concatenating two primes that both have the digital sum equal to n.

**Category:** Number Theory

[60] **viXra:1603.0353 [pdf]**
*submitted on 2016-03-25 02:50:23*

**Authors:** Marius Coman

**Comments:** 5 Pages.

In this paper I make the following eight conjectures: (Ia) for any p prime, p > 3, there exist an infinity of primes q such that the number n obtained concatenating p – 1 to the right with q^2 is prime; (Ib) there exist an infinity of terms in any of the sequences above (for any p) such that r = (p – 1)*q^2 + 1 is prime; (IIa) for any q prime, q > 3, there exist an infinity of primes p such that the number n obtained concatenating q^2 to the left with p – 1 is prime; (IIb) there exist an infinity of terms in any of the sequences above (for any q) such that r = (p – 1)*q^2 + 1 is prime; (IIIa) for any Poulet number P, not divisible by 3, there exist an infinity of primes q such that the number n obtained concatenating P – 1 to the right with q^2 is prime; (IIIb) there exist an infinity of terms in any of the sequences above (for any P) such that r = (P – 1)*q^2 + 1 is prime; (IVa) for any Poulet number Q, not divisible by 3 or 5, there exist an infinity of primes p such that the number n obtained concatenating Q^2 to the left with p - 1 is prime; (IVb) there exist an infinity of terms in any of the sequences above (for any Q) such that r = (p – 1)*Q^2 + 1 is prime.

**Category:** Number Theory

[59] **viXra:1603.0347 [pdf]**
*submitted on 2016-03-24 07:52:47*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following four conjectures: (I) there exist an infinity of primes p of the form (6*k - 1)]c[(6*k + 1)]c[(6*k - 1), where “]c[“ means “concatenated to" (example: for k = 4, the number p = 232523 is prime); (II) there exist an infinity of primes q of the form (6*k + 1)]c[(6*k - 1)]c[(6*k + 1) (example: for k = 2, the number p = 131113 is prime); (III) there exist an infinity of pairs of primes (p, q) = ((6*k - 1)]c[(6*k + 1)]c[(6*k - 1), ((6*k + 1)]c[(6*k – 1)]c[(6*k + 1)); example: for k = 5, (p, q) = (293129, 32931); note that, for such a pair (p, q), q – p = 19802; 1998002; 199980002 and so on; (IV) there exist, for any h positive integer, an infinity of primes q = p + m, where p is prime and m is the number obtained concatenating 1 with a number of h digits of 9 then with 8 then with the same number of h digits of 0 then with 2.

**Category:** Number Theory

[58] **viXra:1603.0342 [pdf]**
*submitted on 2016-03-24 05:01:47*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following three conjectures: (I) there exist an infinity of primes p obtained concatenating even numbers n with 0 then with n + 2, then again with 0, then with n + 5 (example: for n = 44, the number p = 44046049 is prime). It is notable that are found chains with 4 primes p obtained for 4 consecutive even numbers n (example: 17201740177, 17401760177, 17601780181, 17801800183, obtained for 172, 174, 176, 178); (II) there exist an infinity of pairs of primes (p, q) obtained applying on two consecutive even numbers (m, n) the method of concatenation showed in the conjecture above (note that q – p = 20202; 2002002; 200020002 and so on); (III) there exist, for any k positive integer, an infinity of primes q = p + n, where p is prime and n is the number obtained concatenating 2 with a number of k digits of 0 then with 2 then again with the same number of k digits of 0 then again with 2.

**Category:** Number Theory

[57] **viXra:1603.0341 [pdf]**
*submitted on 2016-03-23 15:58:15*

**Authors:** Prashanth R. Rao

**Comments:** 1 Page.

The Kurepa conjecture states that the gcd (!n,n!) is equal to 2 for all n=2,3,….Although this conjecture has not been proven, in this paper we study the implication of a true Kurepa conjecture. We use (!n)/2 and (n!)/2 as components of two distinct arithmetic progressions and propose that both should have infinitely many primes as known from Dirichlet’s theorem of Arithmetic progressions and these are named as Dirichlet-Kurepa primes Type 1 and Type 2 in their honor.

**Category:** Number Theory

[56] **viXra:1603.0336 [pdf]**
*submitted on 2016-03-23 09:12:36*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: there exist an infinity of primes q = 2*n – 1, where n is the sum of a reversible prime p of the form 6*k + 1 concatenated to the left with 1 and its reversal, also concatenated to the left with 1 (example: for p = 13, n = 113 + 131 = 244 and q = 244*2 – 1 = 487, prime).

**Category:** Number Theory

[55] **viXra:1603.0328 [pdf]**
*submitted on 2016-03-22 20:13:29*

**Authors:** Constantin Dumitrescu, Vasile Seleacu

**Comments:** 137 Pages.

The Smarandache Function is defined as the smallest integer S(n) such that S(n)! is divisible by n.
The authors study properties of this function.

**Category:** Number Theory

[54] **viXra:1603.0327 [pdf]**
*submitted on 2016-03-23 01:41:55*

**Authors:** Marius Coman

**Comments:** 13 Pages.

In this paper I list a number of 20 Smarandache concatenated sequences (for other lists and analyses on these sequences see “Smarandache Sequences” on Wolfram MathWorld and “The math encyclopedia of Smarandache type notions”, Educational Publishing, 2013) and I highlight the sets of primes distinguished among the terms of these sequences, but also I list 25 “sets of primes which can be obtained from the terms of Smarandache sequences using any arithmetical operation” (I named such primes Smarandache-Coman sequences of primes, see my previous papers “Fourteen Smarandache-Coman sequences of primes” and “Seven Smarandache-Coman sequences of primes”).

**Category:** Number Theory

[53] **viXra:1603.0322 [pdf]**
*submitted on 2016-03-22 07:20:59*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following conjecture: for any p prime, p > 5, there exist an infinity of k positive integers such that the number q obtained concatenating to the right p with p + 30*k is prime (examples: for p = 13, the least k for which q is prime is 2 because 1373 is prime; for p = 104729, the least k for which q is prime is 3 because 104729104819 is prime). It is notable the small values of k for which primes q are obtained, even in the case of primes p having 20 digits, so this formula could be a way to easily find, starting from a prime p, a prime q having twice as many digits!

**Category:** Number Theory

[52] **viXra:1603.0321 [pdf]**
*submitted on 2016-03-21 17:06:19*

**Authors:** Prashanth R. Rao

**Comments:** 1 Page.

In this paper we show that any two successive left factorials can only represent the first and second terms and not any other pair of successive terms in any arithmetic progression of positive integers.

**Category:** Number Theory

[51] **viXra:1603.0314 [pdf]**
*submitted on 2016-03-21 15:06:30*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following conjecture: for any prime p of the form 6*k + 1 there exist an infinity of primes n obtained concatenating p to the left with 3 and to the right with a square of prime q^2 (examples: for p = 13, the numbers n = 313289, 313961, 3131369 – obtained for q = 17, 31, 37 – are primes).

**Category:** Number Theory

[50] **viXra:1603.0307 [pdf]**
*submitted on 2016-03-21 11:58:26*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following three conjectures on primes: (I) there exist an infinity of primes q obtained concatenating to the left a prime p with the number (p – 1)/2 (example: for p = 23, q is the number obtained concatenating 23 to the left with (p – 1)/2 = 11, i.e. q = 1123, prime); (II) there exist an infinity of primes q obtained concatenating to the left a prime p with the number (p + 1)/2 (example: for p = 41, q is the number obtained concatenating 41 to the left with (p + 1)/2 = 21, i.e. q = 2141, prime); (III) there exist an infinity of pairs of primes (q1, q2) where q1 is obtained concatenating to the left a prime p with the number (p – 1)/2 and q2 is obtained concatenating to the left the same prime p with the number (p + 1)/2.

**Category:** Number Theory

[49] **viXra:1603.0303 [pdf]**
*submitted on 2016-03-21 10:33:02*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following three conjectures on squares of primes: (I) there exist an infinity of primes q obtained concatenating to the left a square of a prime p^2 with the number (p^2 + 1)/2 (example: for p = 17, p^2 = 289 and q is the number obtained concatenating 289 to the left with (p^2 + 1)/2 = 145, i.e. q = 145289, prime); (II) there exist an infinity of primes q obtained concatenating to the left a square of a prime p^2 with the number p + 12 (example: for p = 7, p^2 = 49 and q = 1949, prime); (III) there exist an infinity of primes q obtained concatenating to the left a square of a prime p^2 with the number p^ + 12 (example: for p = 11, p^2 = 121 and q = 133121, prime).

**Category:** Number Theory

[48] **viXra:1603.0302 [pdf]**
*submitted on 2016-03-21 07:08:43*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following four conjectures: (I) there exist an infinity of primes p such that 3*p – 10 is also prime; (II) there exist an infinity of triplets of primes (p, 2*p – 1, 3*p – 10); (III) there exist an infinity of primes q obtained concatenating a prime p to the right with 2*p – 1 and to the left with 3 (example: for p = 11, q = 31121, prime; (IV) there exist, for any n positive integer, n > 1, an infinity of primes q obtained concatenating a prime p to the right with n*p – n + 1 and to the left with 3 (examples: for n = 5 and p = 19, q = 31991, prime; for n = 8 and p = 13, q = 31397, prime).

**Category:** Number Theory

[47] **viXra:1603.0299 [pdf]**
*submitted on 2016-03-20 20:48:07*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

In this paper, we transform the zeros in the critical line.

**Category:** Number Theory

[46] **viXra:1603.0290 [pdf]**
*submitted on 2016-03-21 03:18:13*

**Authors:** Marius Coman

**Comments:** 4 Pages.

Though the well known Fermat’s conjecture on the diophantine equation x^n + y^n = z^n is named “Fermat’s big theorem”, in fact probably much more important for number theory is what is called “Fermat’s little theorem” which was the most important step up to that time in order to discover a primality criterion. This exceptional criterion of primality still has its exceptions: Fermat pseudoprimes, numbers which “behave” like primes though they are no primes; but they are still a class of numbers at least as interesting as the class of primes. Among Fermat pseudoprimes two classes of numbers are particularly distinguished: Poulet numbers (relative Fermat pseudoprimes) and Carmichael numbers (absolute Fermat pseudoprimes). The initial aim of this paper was only to see which Poulet numbers can be obtained concatenating primes (or, in other words, whichever admit a deconcatenation in prime numbers) but, inspired by a characteristic of a subset of Poulet numbers, I also made the following conjecture: there exist an infinity of primes p obtained concatenating to the right a prime q having the sum of the digits s(q) equal to a multiple of 5 with 3.

**Category:** Number Theory

[45] **viXra:1603.0288 [pdf]**
*submitted on 2016-03-20 14:56:25*

**Authors:** Charles Ashbacher

**Comments:** 62 Pages.

The Smarandache Function is the smallest integer S(n) such that S(n)! is divisibil by n.

**Category:** Number Theory

[44] **viXra:1603.0286 [pdf]**
*submitted on 2016-03-20 14:59:03*

**Authors:** Marius Coman

**Comments:** 113 Pages.

This book brings together fifty-two papers regarding primes and Fermat pseudoprimes, submitted by the author to the scientific database Research Gate. Part One of this book, “Sequences of primes and conjectures on them”, contains papers on sequences of primes, squares of primes, semiprimes, pairs, triplets and quadruplets of primes and conjectures on them. This part also contains papers on possible methods to obtain large primes, some of them based on concatenation, some of them on other arithmetical operations. It is also introduced a new notion, “Smarandache-Coman sequences of primes”, defined as “all sequences of primes obtained from the Smarandache sequences using any arithmetical operation”. Part Two of this book, “Sequences of Fermat pseudoprimes and conjecture on them”, contains sequences of Poulet and Carmichael numbers. Among these papers there is a list of thirty-six polynomials and formulas that generate sequences of Fermat pseudoprimes. Part Three of this book, “Prime producing quadratic polynomials”, contains three papers which list few already known such polynomials, that generate more than 20, 30 or even 40 primes in a row, and few such polynomials discoverd by the author himself (in a review of records in the field of prime generating polynomials, written by Dress and Landreau, two mathematicians well known for their contributions in this field, the author is mentioned with 18 prime producing quadratic polynomials). One of these three papers proposes 17 generic formulas that may generate prime producing quadratic polynomials.

**Category:** Number Theory

[43] **viXra:1603.0268 [pdf]**
*submitted on 2016-03-20 03:45:41*

**Authors:** Marius Coman

**Comments:** 4 Pages.

In this paper I make the following four conjectures on the Smarandache prime-partial-digital sequence defined as the sequence of prime numbers which admit a deconcatenation into a set of primes: (I) there exist an infinity of primes p obtained concatenating two primes m and n, both of the form 6*k + 1, such that n = m*h – h + 1 , where h positive integer; (II) there exist an infinity of primes p obtained concatenating two primes m and n, both of the form 6*k - 1, such that n = m*h + h - 1 , where h positive integer; (III) there exist an infinity of primes p obtained concatenating two primes m and n, both of the form 6*k + 1, such that n + m - 1 is prime or power of prime; (IV) there exist an infinity of primes p obtained concatenating two primes m and n, both of the form 6*k - 1, such that n - m + 1 is prime or power of prime. Note that almost all from the first 65 primes obtained concatenating two primes of the form 6k + 1 (exceptions: 3779, 4373, 6173, 6719, 6779), and all the first 65 primes obtained concatenating two primes of the form 6k - 1, belong to one of the four sequences considered by the conjectures above.

**Category:** Number Theory

[42] **viXra:1603.0266 [pdf]**
*submitted on 2016-03-18 23:32:28*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following two conjectures: (I) there exist an infinity of primes q obtained concatenating a prime p with 9 then with p itself (example: p = 104593 is prime and q = 1045939104593 is also prime); (II) there exist an infinity of primes q obtained concatenating a prime p of the form 6*k – 1 with 9 then with p itself and subtracting 2 (example: p = 104471 is prime and q = 1044719104471 – 2 = 1044719104469 is also prime).

**Category:** Number Theory

[41] **viXra:1603.0264 [pdf]**
*submitted on 2016-03-19 02:53:33*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: there exist, for any m prime of the form 6*k + 1, an infinity of primes n obtained concatenating a prime p with a prime q where q – p + 1 = m (example: for m = 457, prime, we have q - p + 1 = 457 for [p, q] = [11, 467], both primes, and the number n = 11467 is prime).

**Category:** Number Theory

[40] **viXra:1603.0263 [pdf]**
*submitted on 2016-03-18 14:33:20*

**Authors:** Marius Coman

**Comments:** 4 Pages.

In this paper I make the following two conjectures: (I) there exist an infinity of primes obtained concatenating, once or repeatedly, an odd multiple n of 3 with 111, then raising the number obtained to the power 2, adding to it n and subtracting 1 (Examples: 3111^2 + 3 – 1 = 9678323, prime; 27111111^2 + 27 – 1 = 735012339654347, prime); (I) there exist an infinity of semiprimes obtained concatenating, once or repeatedly, an odd multiple n of 3 with 111, then raising the number obtained to the power 2, adding to it n and subtracting 1.

**Category:** Number Theory

[39] **viXra:1603.0262 [pdf]**
*submitted on 2016-03-18 16:23:50*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I make the following conjecture: there exist an infinity of primes obtained concatenating the square of a prime p with p then with 1 and then subtracting 2 from the resulting number (example: 127^2 = 16129 and the number 161291271 – 2 = 161291269 is prime).

**Category:** Number Theory

[38] **viXra:1603.0256 [pdf]**
*submitted on 2016-03-18 00:45:36*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following four conjectures: (I) there exist an infinity of primes of the form 2*(p*q*r) + 1, where p, q = p + 6, r = q + 6 are odd numbers of the form 6*k – 1; (II) there exist an infinity of semiprimes m*n of the form 2*(p*q*r) + 1, where p, q = p + 6, r = q + 6 are odd numbers of the form 6*k – 1, semiprimes having the property that n – m + 1 is prime; (III) there exist an infinity of primes of the form 2*(p*q*r) - 1, where p, q = p + 6, r = q + 6 are odd numbers of the form 6*k + 1; (IV) there exist an infinity of semiprimes m*n of the form 2*(p*q*r) + 1, where p, q = p + 6, r = q + 6 are odd numbers of the form 6*k + 1, semiprimes having the property that n – m + 1 is prime.

**Category:** Number Theory

[37] **viXra:1603.0254 [pdf]**
*submitted on 2016-03-18 03:26:29*

**Authors:** Kunle Adegoke

**Comments:** 22 Pages.

Using a clear and straightforward approach, we prove new ternary (base 3) digit extraction BBP-type formulas for polylogarithm constants. Some known results are also rediscovered in a more direct and elegant manner. A previously unproved degree~4 ternary formula is also proved. Finally, a couple of ternary zero relations are established, which prove two known but hitherto unproved formulas.

**Category:** Number Theory

[36] **viXra:1603.0248 [pdf]**
*replaced on 2016-04-14 03:21:54*

**Authors:** Prem kumar

**Comments:** 3 Pages.

In this paper I discuss an algorithm which will solve a very famous puzzle involving a monkey, few men and some coconuts. The puzzle involves a group of n men who have an unknown number of coconuts among them. At night, while the others are asleep, one of the men divides the coconuts in n parts and hides his share. While dividing, he discovers that there is one extra coconut, which he gives away to a monkey. Exactly the same thing happens with the rest of the men, one by one. They all hide their share, are left with one extra coconut that cannot be divided, which they give to the monkey. The next morning they again divide the coconuts together equally among themselves, with no extra coconut remaining this time. The puzzle is to find out the initial number of coconuts.

**Category:** Number Theory

[35] **viXra:1603.0238 [pdf]**
*submitted on 2016-03-16 10:05:23*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following two conjectures on the Smarandache’s proper divisor products sequence where a term P(n) of the sequence is defined as the product of the proper divisors of n: (1) there exist an infinity of numbers n divisible by 3 such that the number obtained concatenating the value of P(n) to the right with 1 is prime; (2) there exist an infinity of numbers n divisible by 3 such that the number obtained concatenating the value of P(n) to the right with 1 is semiprime p*q with the property that q – p + 1 is prime.

**Category:** Number Theory

[34] **viXra:1603.0237 [pdf]**
*submitted on 2016-03-16 10:07:03*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following three conjectures: (I) there exist an infinity of primes obtained concatenating the number P – 1 to the right with 1, where P is a Poulet number; (II) there exist an infinity of primes obtained concatenating the number P – 1 to the right with 11, where P is a Poulet number; (III) there exist an infinity of primes obtained concatenating the number P + 1 to the right with 11, where P is a Poulet number.

**Category:** Number Theory

[33] **viXra:1603.0236 [pdf]**
*submitted on 2016-03-16 10:09:42*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following four conjectures: (I) there exist, for any prime p having the value of the last digit d equal to 1, respectively to 3, 7 or 9, an infinity of primes obtained concatenating p – 1 with the value of d; (II) there exist, for any prime p having the value of the last digit d equal to 1, respectively to 3, 7 or 9, an infinity of primes obtained concatenating twice p – 1 with the value of d; (III) there exist, for any prime p having the value of the last digit d equal to 1, respectively to 3, 7 or 9, an infinity of primes obtained concatenating p + 1 with the value of d; (II) there exist, for any prime p having the value of the last digit d equal to 1, respectively to 3, 7 or 9, an infinity of primes obtained concatenating twice p + 1 with the value of d.

**Category:** Number Theory

[32] **viXra:1603.0233 [pdf]**
*submitted on 2016-03-15 17:51:42*

**Authors:** Allen D Allen

**Comments:** 4 Pages.

The only way to prove Fermat’s Last Theorem with logical rigor is to first prove Fermat’s Extended Last Theorem (FELT): If n is an integer greater than 2, then there cannot exist positive rational fractions r, s, and t, neither integral nor non-integral, such that r^n + s^n = t^n.

**Category:** Number Theory

[31] **viXra:1603.0227 [pdf]**
*submitted on 2016-03-16 01:25:03*

**Authors:** Kunle Adegoke

**Comments:** 6 Pages.

Hitherto only a base 5 BBP-type formula is known for $\sqrt 5\log\phi$, where \mbox{$\phi=(\sqrt 5+1)/2$}, the golden ratio, ( i.e. Formula 83 of the April 2013 edition of Bailey's Compendium of \mbox{BBP-type} formulas). In this paper we derive a new binary BBP-type formula for this constant. The formula is obtained as a particular case of a BBP-type formula for a family of logarithms.

**Category:** Number Theory

[30] **viXra:1603.0194 [pdf]**
*submitted on 2016-03-12 17:01:37*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state two conjectures: (I) There exist an infinity of primes p of the form n – 1, where n is the number obtained concatenating the digits of a prime q, each one of them multiplied by 6 (example: for q = 239, n = 121854 and p = n – 1 = 121853, prime); (II) There exist an infinity of primes p of the form n + 1, where n is the number obtained concatenating the digits of a prime q, each one of them multiplied by 6 (example: for q = 283, n = 124818 and p = n + 1 = 124819, prime).

**Category:** Number Theory

[29] **viXra:1603.0191 [pdf]**
*submitted on 2016-03-13 04:41:01*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: there exist an infinity of 2-Poulet numbers P = m*n having the property that the number q = 6*(m + n) + 1 is prime.

**Category:** Number Theory

[28] **viXra:1603.0186 [pdf]**
*submitted on 2016-03-12 05:31:28*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I state the following two conjectures: (I) For any prime q greater than 5 there exist an infinity of primes p obtained subtracting the square of q or the square of r from the number obtained concatenating the square of q with the square of r, where r prime, r = q + 18*n, and adding 1 (for example, p = 121841 – 121 + 1 = 121721, prime, also p = 121841 – 841 + 1 = 121001, prime, where q^2 = 11^2 = 121, r^2 = 29^2 = 841 and 29 = 11 + 18*1); (II) For any positive integer n there exist an infinity of triplets of primes [p, q, r] such that r = q + 18*n and p is obtained subtracting the square of q or the square of r from the number obtained concatenating the square of q with the square of r and adding 1.

**Category:** Number Theory

[27] **viXra:1603.0173 [pdf]**
*submitted on 2016-03-11 16:10:53*

**Authors:** Marius Coman

**Comments:** 5 Pages.

In this paper I list a number of eight formulas that generate two certain types of semiprimes, i.e. semiprimes p*q with the property that n = q – p + 1 is prime respectively with the property that P + q – 1 is prime.

**Category:** Number Theory

[26] **viXra:1603.0167 [pdf]**
*replaced on 2016-03-14 22:23:28*

**Authors:** Marius Coman

**Comments:** 112 Pages.

To make an introduction to a book about arithmetic it is always difficult, because even most apparently simple assertions in this area of study may hide unsuspected inaccuracies, so one must always approach arithmetic with attention and care; and seriousness, because, in spite of the many games based on numbers, arithmetic is not a game. For this reason, I will avoid to do a naive and enthusiastic apology of arithmetic and also to get into a scholarly dissertation on the nature or the purpose of arithmetic. Instead of this, I will summarize this book, which brings together several articles regarding primes and Fermat pseudoprimes, submitted by the author to the preprint scientific database Research Gate. Part One of this book, “Sequences of primes and conjectures on them”, brings together thirty-two papers regarding sequences of primes, sequences of squares of primes, sequences of certain types of semiprimes, also few types of pairs, triplets and quadruplets of primes and conjectures on all of these sequences. There are also few papers regarding possible methods to obtain large primes or very large numbers with very few prime factors, some of them based on concatenation, some of them on other arithmetic operations. It is also introduced a new notion: “Smarandache-Coman sequences of primes”, defined as “all sequences of primes obtained from the terms of Smarandache sequences using any arithmetical operation” (for instance, the sequence of primes obtained concatenating to the right with the digit one the terms of Smarandache consecutive numbers sequence). Part Two of this book, “Sequences of Fermat pseudoprimes and conjectures on them”, brings together seventeen papers on sequences of Poulet numbers and Carmichael numbers, i.e. the Fermat pseudoprimes to base 2 and the absolute Fermat pseudoprimes, two classes of numbers that fascinated the author for long time. Among these papers there is a list of thirty-six polynomials and formulas that generate sequences of Fermat pseudoprimes. Part Three of this book, “Prime producing quadratic polynomials”, contains three papers which list some already known such polynomials, that generate more than 20, 30 or even 40 primes in a row, and few such polynomials discovered by the author himself (in a review of records in the field of prime generating polynomials, written by Dress and Landreau, two French mathematicians well known for records in this field, review that can be found on the web address

**Category:** Number Theory

[25] **viXra:1603.0162 [pdf]**
*submitted on 2016-03-10 07:19:11*

**Authors:** George Gregory

**Comments:** 1 Page.

Developing interesting Generalized Smarandache Palindromes formed from smarandacheian sequences.

**Category:** Number Theory

[24] **viXra:1603.0159 [pdf]**
*submitted on 2016-03-10 07:22:55*

**Authors:** M. Perez

**Comments:** 2 Pages.

Problems in Number Theory.

**Category:** Number Theory

[23] **viXra:1603.0146 [pdf]**
*submitted on 2016-03-09 12:02:50*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following two conjectures on the Smarandache’s divisor products sequence where a term P(n) of the sequence is defined as the product of the positive divisors of n: (1) there exist an infinity of n composites such that the number m = P(n) + n – 1 is prime; (2) there exist an infinity of n composites such that the number m = P(n) – n + 1 is prime.

**Category:** Number Theory

[22] **viXra:1603.0142 [pdf]**
*submitted on 2016-03-09 07:18:33*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make two conjectures on the numbers m created concatenating to the right an odd number n, not divisible by 3, with 3*n – 4 and then, if n is of the form 6*k + 1, with 11, respectively, if n is of the form 6*k – 1, with 1: (I) there exist an infinity of m primes; (II) there exist an infinity of m = p*q composites such that p + q – 1 is prime (where p and q may be, or may be not, primes). Note that for 25 from the first 30 odd numbers n not divisible by 3 the number m obtained belongs to one of the two sequences considered by the conjectures above.

**Category:** Number Theory

[21] **viXra:1603.0141 [pdf]**
*submitted on 2016-03-09 07:24:23*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make four conjectures on the numbers n created concatenating to the right the product p*q with number 11, where [p, q] is a pair of twin primes: (I) there exist an infinity of n primes; (II) there exist an infinity of n semiprimes of the form (10k + 1)*(10h + 1); (II) there exist an infinity of n semiprimes of the form (10k + 9)*(10h + 9); (II) there exist an infinity of n semiprimes of the form (10k + 3)*(10h + 7). Note that for 40 from the first 43 pairs of twin primes the number n belongs to one of the four sequences considered by the conjectures above.

**Category:** Number Theory

[20] **viXra:1603.0136 [pdf]**
*submitted on 2016-03-08 21:13:32*

**Authors:** Quang Nguyen Van

**Comments:** 1 Page.

We give a conjecture on sequence of consecutive natural numbers.

**Category:** Number Theory

[19] **viXra:1603.0130 [pdf]**
*submitted on 2016-03-08 12:31:05*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 19 Pages. Version corrected to be submitted to Journal of Number Theory. in French.

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. Three numerical examples are also given.

**Category:** Number Theory

[18] **viXra:1603.0110 [pdf]**
*submitted on 2016-03-07 05:38:03*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I state the following conjecture: there exist an infinity of primes obtained deconcatenating the numbers of the form (30*k – 1)*(30*k + 1) to the right with digit 9; example: 449*451 = 202499 and 20249 is a prime.

**Category:** Number Theory

[17] **viXra:1603.0109 [pdf]**
*submitted on 2016-03-07 05:39:13*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I state the following three conjectures on the numbers obtained concatenating to the left the odd numbers with 1234: (I) There exist an infinity of primes obtained concatenating to the left odd numbers with 1234; (II) There exist an infinity of primes obtained concatenating to the left prime numbers with 1234; (III) There exist an infinity of primes obtained concatenating to the left Poulet numbers with 1234.

**Category:** Number Theory

[16] **viXra:1603.0108 [pdf]**
*submitted on 2016-03-07 05:40:25*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following four conjectures: (I) There exist an infinity of primes p which, concatenated to the right with the digit 9, form also prime numbers; (II) There exist an infinity of primes obtained concatenating the reversal of p as is defined in Conjecture I to the right with the digit 9; (III) There exist an infinity of semiprimes obtained concatenating primes to the right with the digit 9, semiprimes m*n having the property that n – m + 1 is prime; (IV) There exist an infinity of semiprimes obtained concatenating the reversal of p as is defined in Conjecture I to the right with the digit 9, semiprimes m*n having the property that n – m + 1 is prime.

**Category:** Number Theory

[15] **viXra:1603.0096 [pdf]**
*submitted on 2016-03-07 03:14:50*

**Authors:** Marius Coman

**Comments:** 6 Pages.

In a previous paper, “Fourteen Smarandache-Coman sequences of primes”, I defined the “Smarandache-Coman sequences” as “all the sequences of primes obtained from the Smarandache concatenated sequences using basic arithmetical operations between the terms of such a sequence, like for instance the sum or the difference between two consecutive terms plus or minus a fixed positive integer, the partial sums, any other possible basic operations between terms like a(n) + a(n+2) – a(n+1), or on a term like a(n) + S(a(n)), where S(a(n)) is the sum of the digits of the term a(n) etc.” In this paper I extend the notion to the sequences of primes obtained from the Smarandache concatenated sequences using any arithmetical operation and I present seven sequences obtained from the Smarandache concatenated sequences using concatenation between the terms of the sequence and other numbers and also fourteen conjectures on them.

**Category:** Number Theory

[14] **viXra:1603.0089 [pdf]**
*submitted on 2016-03-06 09:10:32*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following conjecture: there exist an infinity of primes of the form 3*p*(q – 1) – 1, where p and q are primes and p = q + 6. Note that from the first terms of the sequence of sexy primes we have a chain of consecutive 9 primes: 131, 233, 509, 683, 1103, 1913, 3329, 4643, 5639 (for q = 5, 7, 11, 13, 17, 23, 31, 37, 41).

**Category:** Number Theory

[13] **viXra:1603.0088 [pdf]**
*submitted on 2016-03-06 10:27:01*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I state the following conjecture: for any p prime there exist at least a value of n, different from p, for which the number (p^2 – n)/(n – 1) is prime.

**Category:** Number Theory

[12] **viXra:1603.0075 [pdf]**
*submitted on 2016-03-04 17:51:23*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I share a very interesting discovery made more or less by accident: taking a number having just even digits, like for instance 224866802226608 (I have chosen this randomly right now when I am writing the Abstract) and concatenating it three times with itself and then to the right with the digit 1 (like in the example taken 2248668022266082248668022266082248668022266081) seems that are great chances to obtain a number with very few prime factors (in the case taken just 4 prime factors).

**Category:** Number Theory

[11] **viXra:1603.0066 [pdf]**
*submitted on 2016-03-04 15:25:15*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I show an interesting sequence of numbers created concatenating to the right the digit 1, twice, with a prime of the form 6*k – 1 (example of such numbers, terms of this sequence: 12929 and 15353), sequence that has, from the first 50 terms, 21 terms that are primes and 22 that are semiprimes.

**Category:** Number Theory

[10] **viXra:1603.0063 [pdf]**
*submitted on 2016-03-04 11:28:43*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I make the following observation: the formula 529 + 60*10^k, where k positive integer, seems to generate a large amount of big primes and semiprimes. Indeed, up to k = 32, this formula generates 11 primes and 11 semiprimes!

**Category:** Number Theory

[9] **viXra:1603.0062 [pdf]**
*submitted on 2016-03-04 11:30:05*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following observation: there are many primes among the numbers of the form p*q + 10^k, where p and q are emirps (reversible primes but different one from the other) and k is a positive integer; to highlight the observation I will search the least k for which the number p*q + 10^k is prime, for few pairs of emirps [p, q].

**Category:** Number Theory

[8] **viXra:1603.0055 [pdf]**
*replaced on 2016-03-05 12:15:30*

**Authors:** Ilija Barukčić

**Comments:** 7 Pages.

Modern scientist come close to Einstein, the most prominent physicist of the twentieth century and may be of all time. Still, the question is justified, can there be ever another Einstein? Less well known, though of fundamental importance, are Einstein's contributions to the philosophy of science. Einstein's is well known for his conviction that scientist should trust simplicity. Einstein proclaimed that we can discover the true laws of nature or proof theorems by seeking those with the most simple mathematical formulation. Einstein's faith in the supreme power of mathematical simplicity was strong. Such an approach to scientific investigation is of strategic use, since by time the hypotheses from which scientists starts become ever more abstract and more remote from experience. Especially, under conditions where scientific investigations are moving steadily into domains ever further removed from direct contact with an experiment or observation, the starting point should be as simple as possible. This point of view, whose exact formulation while investi-gating the problem of the division of zero by zero may meet with great difficulties, may justify our trust that the problem of the division of zero by zero is solved.

**Category:** Number Theory

[7] **viXra:1603.0053 [pdf]**
*submitted on 2016-03-04 04:11:55*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I make the following conjecture: Any square of a prime larger than 11 can be written as 60*n^2 + 90*n + p, where p prime or power of prime and n positive integer.

**Category:** Number Theory

[6] **viXra:1603.0051 [pdf]**
*submitted on 2016-03-04 04:54:06*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I make the following conjecture: Any square of a prime larger than 7 can be written as 30*n^2 + 60*n + p, where p prime or power of prime and n positive integer.

**Category:** Number Theory

[5] **viXra:1603.0040 [pdf]**
*submitted on 2016-03-03 10:23:06*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following three conjectures on the numbers of the form n = 6*p*q + 1, where p and q are primes and q = 2*p – 1: (I) There exist an infinity of n primes; (II) There exist an infinity of n semiprimes; (III) There exist an infinity of n composites with three or more prime factors, 7 being one of them. Note that for all the first 46 pairs of primes [p, q] with the property mentioned (see the sequence A005382 in OEIS for these primes) the number n obtained belongs to one of the three sequences considered by the three conjectures above.

**Category:** Number Theory

[4] **viXra:1603.0038 [pdf]**
*submitted on 2016-03-03 10:54:20*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I make the following conjecture on the numbers of the form n = 6*p*q + 1, where p and q are primes and q = k*p – k + 1: There exist an infinity of n primes for any k positive integer, k > 1. Note that the conjecture implies that there exist an infinity of pairs of primes [p, q] such that q = k*p – k + 1, for any k positive integer, k > 1, which I already conjectured in previous papers, as well as that there exist an infinity of pairs of primes [p, q] such that q = k*p + k - 1, for any k positive integer, k > 1.

**Category:** Number Theory

[3] **viXra:1603.0036 [pdf]**
*submitted on 2016-03-03 04:42:01*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following observation: let n1, n2,..., ni be the ordered set of the odd semiprimes not divisible by 3; then the period of the rational number which is the sum 1/(n1 – 1) + 1/(n2 – 1) +...+ 1/(ni - 1) seems to be always (for any i) divisible by 48. This is not the fact when the semiprimes n1, n2,..., ni are not the ordered set of such semiprimes but few randomly taken (even consecutive) such semiprimes.

**Category:** Number Theory

[2] **viXra:1603.0034 [pdf]**
*submitted on 2016-03-03 05:53:11*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following observation: let p1, p2,..., pi be the ordered set of the 2-Poulet numbers; then the length of the period of the rational number which is the sum 1/(p1 – 1) + 1/(p2 – 1) +...+ 1/(ni - 1) seems to be always (for any i > 2) divisible by 240. This is not the fact when the numbers p1, p2,..., pi are not the ordered set of 2-Poulet numbers but few randomly taken (even consecutive) 2-Poulet numbers. For a related topic see my previous paper “A pattern that relates Carmichael numbers to the number 66” where I noticed that the length of the period of the rational number which is the sum 1/(c1 – 1) + 1/(c2 – 1) +...+ 1/( ci - 1), where c1, c2, ..., ci is the ordered set of Carmichael numbers, seems to be always divisible by 66.

**Category:** Number Theory

[1] **viXra:1603.0022 [pdf]**
*submitted on 2016-03-03 02:55:42*

**Authors:** Orgest ZAKA

**Comments:** 3 Pages.

In this paper i have presented a partial solution of open problem so called Goldbach’s
conjecture in number theory, which consists in the fact that: “Every even number can be
expressed as a sum of two prime numbers”. In this paper I have set up a hypothesis, which helps
in my line of proof of Goldbach hypothesis. I very much hope that this hypothesis be easier to
prove.

**Category:** Number Theory