[50] **viXra:1408.0231 [pdf]**
*submitted on 2014-08-31 12:01:39*

**Authors:** Predrag Terzic

**Comments:** 1 Page.

Conjectured polynomial time primality test for specific class of 13*2^n+1 is introduced .

**Category:** Number Theory

[49] **viXra:1408.0230 [pdf]**
*submitted on 2014-08-31 12:10:44*

**Authors:** Predrag Terzic

**Comments:** 2 Pages.

Conjectured polynomial time compositeness tests for numbers of the form k10^n-c and k10^n+c are introduced .

**Category:** Number Theory

[48] **viXra:1408.0225 [pdf]**
*submitted on 2014-08-31 00:12:58*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make four conjectures on primes, conjectures which involve the sums of distinct unit fractions such as 1/p(1) + 1/p(2) + (...), where p(1), p(2), (...) are distinct primes, more specifically the periods of the rational numbers which are the results of the sums mentioned above.

**Category:** Number Theory

[47] **viXra:1408.0223 [pdf]**
*submitted on 2014-08-31 01:36:10*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I present three formulas, each of them with the following property: starting from a given prime p, are obtained in many cases two other primes, q and r. I met the triplets of primes [p, q, r] obtained with these formulas in the study of Carmichael numbers; the three primes mentioned are often the three prime factors of a 3-Carmichael number.

**Category:** Number Theory

[46] **viXra:1408.0221 [pdf]**
*submitted on 2014-08-31 06:11:45*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make a conjecture which states that any prime greater than or equal to 53 can be written at least in one way as a sum of three odd primes, not necessarily distinct, of the same form from the following four ones: 10k + 1, 10k + 3, 10k + 7 or 10k + 9.

**Category:** Number Theory

[45] **viXra:1408.0220 [pdf]**
*submitted on 2014-08-31 06:41:55*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make a conjecture which states that any square of a prime greater than or equal to 7 can be written at least in one way as a sum of three odd primes, not necessarily distinct, but all three of the form 10k + 3 or all three of the form 10k + 7.

**Category:** Number Theory

[44] **viXra:1408.0218 [pdf]**
*submitted on 2014-08-30 12:33:04*

**Authors:** Predrag Terzic

**Comments:** 1 Page.

Conjectured polynomial time primality test for specific class of 7*2^n+1 is introduced .

**Category:** Number Theory

[43] **viXra:1408.0217 [pdf]**
*submitted on 2014-08-30 12:34:57*

**Authors:** Predrag Terzic

**Comments:** 1 Page.

Conjectured polynomial time primality test for specific class of 11*2^n+1 is introduced .

**Category:** Number Theory

[42] **viXra:1408.0210 [pdf]**
*submitted on 2014-08-29 11:21:12*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I present a formula for generating big primes and products of very few primes, based on the numbers 25 and 906304, formula equally extremely interesting and extremely simple, id est 25^n + 906304. This formula produces for n from 1 to 30 (and for n = 30 is obtained a number p with not less than 42 digits) only primes or products of maximum four prime factors.

**Category:** Number Theory

[41] **viXra:1408.0209 [pdf]**
*submitted on 2014-08-29 12:10:30*

**Authors:** Stephen Marshall

**Comments:** 6 Pages.

In number theory, a prime number p is a Sophie Germain prime if 2p + 1 is also prime. For example, 29 is a Sophie Germain prime because it is a prime and 2 × 29 + 1 = 59, and 59 is also a prime number. These numbers are named after French mathematician Marie-Sophie Germain. We shall prove that there are an infinite number of Sophie Germain primes.

**Category:** Number Theory

[40] **viXra:1408.0208 [pdf]**
*submitted on 2014-08-29 07:28:09*

**Authors:** Predrag Terzic

**Comments:** 1 Page.

Conjectured polynomial time primality test for specific class of 5*2^n+1 is introduced .

**Category:** Number Theory

[39] **viXra:1408.0201 [pdf]**
*submitted on 2014-08-28 15:30:00*

**Authors:** Stephen Marshall

**Comments:** 12 Pages.

This paper presents a complete and exhaustive proof that an Infinite Number of Triplet Primes exist. The approach to this proof uses same logic that Euclid used to prove there are an
infinite number of prime numbers. Then we prove that if p > 1 and d > 0 are integers, that p and p + d are both primes if and only if for integer n (see reference 1 and 2):
n =(p−1)!(1/p+(−1)d(d!)/(p + d)+ 1/(p+1)+ 1/(p+d)
We use this proof and Euclid logic to prove only an infinite number of Triplet Primes exist. However we shall begin by assuming that a finite number of Triplet Primes exist, we shall
prove a contradiction to the assumption of a finite number, which will prove that
an infinite number of Triplet Primes exist.

**Category:** Number Theory

[38] **viXra:1408.0197 [pdf]**
*submitted on 2014-08-28 12:50:19*

**Authors:** Anibal Fernando Barral

**Comments:** 24 Pages.

In mathematics, a prime number is a natural number that is divisible only by 1 and itself.
For centuries, the search for an algorithm that could generate the sequence of these numbers became a mystery.
Perhaps the problem arises at the beginning of the enterprise, that is, the search for a single algorithm.
I noticed that all the primes without exception increased by one unit in some cases, or decreased by one unit in the other cases result in a multiple of 6 (six)
Example: 5+1=6 ; 7-1=6 ; 11+1=12 ; 13-1=12 ; 17+1=18 ; 19-1=18 ; 23+1=24 ; 29+1=30 ; 31-1=30 ;
37-1=36 ; 41+1=42 ; 43-1=42 ; 47+1=48 ; and so on.
Then I thought of making it easier to split the problem solving both cases.
So are passed to assume the presence of # 2 complementary families of primes.
To the number 1000, I worked by hand, a job with some effort but great satisfaction.
At this point my algorithms were reliable, but I needed another test.
To get to number 60,000 I leaned in a computational program, which compiled a dear friend. I would have liked to get up to 1,000,000 but the limit of 60,000 has been imposed by the processing time of the data.
At this point I had no more doubts about the reliability of my algorithms that are developed in continuation.

**Category:** Number Theory

[37] **viXra:1408.0195 [pdf]**
*replaced on 2014-09-13 01:16:26*

**Authors:** Matthias Lesch

**Comments:** 3 Pages.

In a recent series of preprints S. Marshall claims to give proofs of several famous conjectures in number theory, among them the twin prime conjecture and
Goldbach’s conjecture. A claimed proof of Beal’s conjecture would even imply an elemen-
tary proof of Fermat’s Last Theorem.
It is the purpose of this note to point out serious errors. It is the opinion of this author
that it is safe to say that the claims of the above mentioned papers are lacking any basis.

**Category:** Number Theory

[36] **viXra:1408.0193 [pdf]**
*submitted on 2014-08-27 18:59:21*

**Authors:** Simon Plouffe

**Comments:** 38 Pages.

I present here a collection of algorithms that permits the expansion into a finite series or sequence from a real number x∈ R, the precision used is 64 decimal digits. The collection of mathematical constants was taken from my own collection and theses sources [1]-[6][9][10]. The goal of this experiment is to find a closed form of the sequence generated by the algorithm. Some new results are presented.

**Category:** Number Theory

[35] **viXra:1408.0190 [pdf]**
*submitted on 2014-08-27 23:33:11*

**Authors:** Francis Thasayyan

**Comments:** 3 Pages.

This document gives an answer to Beal's Conjection.

**Category:** Number Theory

[34] **viXra:1408.0189 [pdf]**
*submitted on 2014-08-28 00:37:39*

**Authors:** Predrag Terzic

**Comments:** 1 Page.

Conjectured polynomial time primality test for specific class of 9*2^n+1 is introduced .

**Category:** Number Theory

[33] **viXra:1408.0184 [pdf]**
*submitted on 2014-08-27 09:13:25*

**Authors:** Predrag Terzic

**Comments:** 1 Page.

Conjectured polynomial time primality test for specific class of numbers of the form k6^n-1 is introduced .

**Category:** Number Theory

[32] **viXra:1408.0183 [pdf]**
*submitted on 2014-08-27 05:41:21*

**Authors:** Predrag Terzic

**Comments:** 1 Page.

Conjectured polynomial time primality test for specific class@@ of numbers of the form kb^n-1 is introduced .

**Category:** Number Theory

[31] **viXra:1408.0181 [pdf]**
*submitted on 2014-08-26 22:22:43*

**Authors:** Simon Plouffe

**Comments:** 9 Pages. The abstract in english and the main text in french

The iteration formula Z_(n+1)=Z_n^2+c of Mandelbrot will give an algebraic number of degree 4 when it converges most of the time. If we take a good look at some of these algebraic numbers: some of them have a very persistent pattern in their binary expansion.
La formule d’itération de Mandelbrot Z_(n+1)=Z_n^2+c converge vers un nombre algébrique de degré 4 si c est un rationnel simple. Mais en regardant de près certains nombres algébriques en binaire on voit apparaître un motif assez évident et très persistant.

**Category:** Number Theory

[30] **viXra:1408.0180 [pdf]**
*submitted on 2014-08-26 22:24:59*

**Authors:** Simon Plouffe

**Comments:** 13 Pages. The abstract in english and the main text in french

An analysis of the function 1/π Arg ζ((1/2)+in) is presented. This analysis permits to find a general expression for that function using elementary functions of floor and fractional part. These formulas bring light to a remark from Freeman Dyson which relates the values of the ζfunction to quasi-crystals. We find these same values for another function which is very similar, namely 1/π Arg Γ((1/4)+in/2). These 2 sets of formula have a definite pattern, the n’th term is related to values like π,ln(π),ln(2),…,log(p), where p is a prime number. The coefficients are closed related to a certain sequence of numbers which counts the number of 0’s from the right in the binary representation of n. These approximations are regular enough to deduce an asymptotic and precise formula. All results presented here are empirical.

**Category:** Number Theory

[29] **viXra:1408.0176 [pdf]**
*replaced on 2015-03-16 10:41:12*

**Authors:** Ramón Ruiz

**Comments:** 34 Pages. This research is based on an approach developed solely to demonstrate the binary Goldbach Conjecture and the Twin Primes Conjecture.

Goldbach's Conjecture statement: “Every even integer greater than 2 can be expressed as the sum of two primes”.
Initially, to prove this conjecture, we can form two arithmetic sequences (A and B) different for each even number, with all the natural numbers that can be primes, that can added, in pairs, result in the corresponding even number.
By analyzing the pairing process, in general, between all non-prime numbers of sequence A with terms of sequence B, or vice versa, to obtain the even number, we note that some pairs of primes are always formed. This allow us to develop a non-probabilistic formula to calculate the approximate number of pairs of primes that meet the conjecture for an even number x.
The result of this formula is always equal or greater than 1, and it tends to infinite when x tends to infinite, which allow us to confirm that Goldbach's Conjecture is true.
The prime numbers theorem by Carl Friedrich Gauss, the prime numbers theorem in arithmetic progressions and some axioms have been used to complete this investigation.

**Category:** Number Theory

[28] **viXra:1408.0175 [pdf]**
*replaced on 2015-03-16 10:57:20*

**Authors:** Ramón Ruiz

**Comments:** 24 Pages. This research is based on an approach developed solely to demonstrate the Twin Primes Conjecture and the binary Goldbach Conjecture.

Twin Primes Conjecture statement: “There are infinitely many primes p such that (p + 2) is also prime”.
Initially, to prove this conjecture, we can form two arithmetic sequences (A and B) with all the natural numbers, lesser than a number x, that can be primes and being each term of sequence B equal to its partner of sequence A plus 2.
By analyzing the pairing process, in general, between all non-prime numbers of sequence A with terms of sequence B, or vice versa, we note that some pairs of primes are always formed. This allow us to develop a non-probabilistic formula to calculate the approximate number of pairs of primes, p and (p + 2), that are lesser than x.
The result of this formula tends to infinite when x tends to infinite, which allow us to confirm that the Twin Primes Conjecture is true.
The prime numbers theorem by Carl Friedrich Gauss, the prime numbers theorem in arithmetic progressions and some axioms have been used to complete this investigation.

**Category:** Number Theory

[27] **viXra:1408.0174 [pdf]**
*replaced on 2014-10-07 09:10:22*

**Authors:** Stephen Marshall

**Comments:** 10 Pages. This is an updated proof by the author.

This paper presents a complete and exhaustive proof of the Polignac Prime Conjecture. The approach to this proof uses same logic that Euclid used to prove there are an infinite number of prime numbers. Then we use a proof found in Reference 1, that if p > 1 and d > 0 are integers, that p and p + d are both primes if and only if for integer n:
n = (p−1)!(1/p+(−1)d(d!)/(p + d)+ 1/(p+1)+ 1/(p+d)
We use this proof for d = 2k to prove the infinitude of Polignac prime numbers.
The author would like to give many thanks to the authors of 1001 Problems in Classical Number Theory, Jean-Marie De Koninck and Armel Mercier, 2004, Exercise Number 161 (see Reference 1). The proof provided in Exercise 6 is the key to making this paper on the Polignac Prime Conjecture possible.
Additionally, our proof of the Polignac Prime Conjecture leads to proofs of several other significant number theory conjectures such as the Goldbach Conjecture, Twin Prime Conjecture, Cousin Prime Conjecture, and Sexy Prime Conjecture. Our proof of Polignac’s Prime Conjecture provides significant accomplishments to Number Theory, yielding proofs to several conjectures in number theory that has gone unproven for hundreds of years.

**Category:** Number Theory

[26] **viXra:1408.0173 [pdf]**
*replaced on 2014-12-11 14:56:10*

**Authors:** Stephen Marshall

**Comments:** 7 Pages. Updated some wording

This paper presents a complete and exhaustive proof of the Beal Conjecture. The approach to this proof uses the Fundamental Theorem of Arithmetic as the basis for the proof of the Beal Conjecture. The Fundamental Theorem of Arithmetic states that every number greater than 1 is either prime itself or is unique product of prime numbers. The prime factorization of every number greater than 1 is used throughout every section of the proof of the Beal Conjecture. Without the Fundamental Theorem of Arithmetic, this approach to proving the Beal Conjecture would not be possible.

**Category:** Number Theory

[25] **viXra:1408.0169 [pdf]**
*replaced on 2014-12-11 09:40:54*

**Authors:** Stephen Marshall

**Comments:** 8 Pages. Replaces original paper with some corrections

This paper presents a complete and exhaustive proof of that an infinite number of Fibonacci Primes exist . The approach to this proof uses same logic that Euclid used to prove there are an infinite number of prime numbers. Then we prove that if p > 1 and d > 0 are integers, that p and p + d are both primes if and only if for integer n (see reference 1 and 2):
n=(p-1)!(1/p+(-1)dd!/(p + d))+1/p+1/(p+d)
We use this proof for p = Fy-1 and d = Fy-2 to prove the infinitude of Fibonacci prime numbers.
The author would like to give many thanks to the authors of 1001 Problems in Classical Number Theory, Jean-Marie De Koninck and Armel Mercier, 2004, Exercise Number 161 (see Reference 1). The proof provided in Exercise 6 is the key to making this paper on the finitude or Fibonacci Primes possible.

**Category:** Number Theory

[24] **viXra:1408.0166 [pdf]**
*replaced on 2014-08-27 05:29:01*

**Authors:** Predrag Terzic

**Comments:** 1 Page.

Conjectured polynomial time primality test for specific class of 3*2^n+1 is introduced .

**Category:** Number Theory

[23] **viXra:1408.0134 [pdf]**
*replaced on 2014-09-18 08:45:00*

**Authors:** Predrag Terzic

**Comments:** 7 Pages.

Conjectured polynomial time primality and compositeness tests for numbers of special forms are introduced .

**Category:** Number Theory

[22] **viXra:1408.0128 [pdf]**
*submitted on 2014-08-19 05:07:11*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 2 Pages.

We use the positivity axiom of inner product spaces to prove the equivalent statement of the Riemann hypothesis.

**Category:** Number Theory

[21] **viXra:1408.0126 [pdf]**
*replaced on 2014-08-27 05:23:44*

**Authors:** Predrag Terzic

**Comments:** 2 Pages.

Conjectured polynomial time primality tests for specific classes of numbers of the form kb^n-1 are introduced .

**Category:** Number Theory

[20] **viXra:1408.0119 [pdf]**
*submitted on 2014-08-18 09:49:52*

**Authors:** Predrag Terzic

**Comments:** 2 Pages.

Conjectured polynomial time primality test for specific class of numbers of the form 9b^n-1 is introduced .

**Category:** Number Theory

[19] **viXra:1408.0113 [pdf]**
*replaced on 2014-08-18 06:42:15*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make five conjectures about the primes q, r and the square of prime p^2, which appears as solutions in the diophantine equation 120*n*q*r + 1 = p^2, where n is non-null positive integer.

**Category:** Number Theory

[18] **viXra:1408.0111 [pdf]**
*submitted on 2014-08-18 02:11:31*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make two conjectures abut the pairs of primes [p1, q1], where the difference between p1 and q1 is a certain even number d. I state that any such pair has at least one other corresponding, in a specified manner, pair of primes [p2, q2], such that the difference between p2 and q2 is also equal to d.

**Category:** Number Theory

[17] **viXra:1408.0110 [pdf]**
*submitted on 2014-08-18 00:02:36*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make a conjecture which states that any odd prime can be written in a certain way, in other words that any such prime can be expressed using just another prime and the powers of the numbers 2, 3 and 5. I also make a related conjecture about twin primes.

**Category:** Number Theory

[16] **viXra:1408.0098 [pdf]**
*submitted on 2014-08-16 08:37:00*

**Authors:** Predrag Terzic

**Comments:** 2 Pages.

Compositeness criteria for specific classes of numbers of the form b^n+b+1 and b^n-b-1 are introduced .

**Category:** Number Theory

[15] **viXra:1408.0095 [pdf]**
*submitted on 2014-08-16 05:39:21*

**Authors:** Predrag Terzic

**Comments:** 2 Pages.

Conjectured polynomial time primality test for specific class of numbers of the form 3b^n-1 is introduced .

**Category:** Number Theory

[14] **viXra:1408.0087 [pdf]**
*submitted on 2014-08-14 07:34:55*

**Authors:** William Maclachlan

**Comments:** 11 Pages.

The aim of my "experiment" was to gather some curious information about the understanding of primes- to my understanding I seemed to have created a system that can find primes considerably quicker in contrast to merely searching through all the given number's factors.
I am not a professional, but it would be nice if I could get some form of a reply from someone with experience to explain the irrelevancy of my findings.

**Category:** Number Theory

[13] **viXra:1408.0085 [pdf]**
*submitted on 2014-08-14 03:16:30*

**Authors:** Pingyuan Zhou

**Comments:** 5 Pages. Author gives an argument for the infinity of primes of the form 2x^2-1 by the infinity of near-square primes of Mersenne primes to arise from a corresponding Fermat prime criterion.

Abstract: In this paper we consider primes of the form 2x^2-1 and discover there is a very great probability for appearing of such primes, and give an argument for the infinity of primes of the form 2x^2-1 by the infinity of near-square primes of Mersenne primes.

**Category:** Number Theory

[12] **viXra:1408.0083 [pdf]**
*submitted on 2014-08-14 00:17:08*

**Authors:** Predrag Terzic

**Comments:** 1 Page.

Conjectured polynomial time primality test for@@ specific class of numbers of the form k6^n-1 is introduced .

**Category:** Number Theory

[11] **viXra:1408.0079 [pdf]**
*submitted on 2014-08-13 07:26:37*

**Authors:** Predrag Terzic

**Comments:** 2 Pages.

Conjectured polynomial time compositeness tests for numbers of the form k2^n-c and k2^n+c are introduced .

**Category:** Number Theory

[10] **viXra:1408.0071 [pdf]**
*submitted on 2014-08-12 02:52:56*

**Authors:** Predrag Terzic

**Comments:** 1 Page.

Conjectured polynomial time primality test for specific class of generalized Fermat numbers is introduced .

**Category:** Number Theory

[9] **viXra:1408.0068 [pdf]**
*replaced on 2014-08-12 06:36:01*

**Authors:** Predrag Terzic

**Comments:** 2 Pages.

Conjectured polynomial time compositeness tests for numbers of the form k2^n-1 and k2^n+1 are introduced .

**Category:** Number Theory

[8] **viXra:1408.0050 [pdf]**
*submitted on 2014-08-08 18:22:49*

**Authors:** Oh Jung Uk

**Comments:** 20 Pages. I don't know how to show abstract well

If π_g (N) is the number of cases that even number N could be expressed as the sum of the two primes of 6n±1 type then the formula of π_g (N) is below
π_g (6n+0)=n-1- 2/3 ∑_(k=1)^(n-1)▒((〖πβ〗_g (6k-1))/(πβ_g (6k-1)-1)) -2/3π ∑_(k=1)^(n-1)▒∑_(m=1)^∞▒sin((2〖mπ〗^2 β_g (6k-1))/(πβ_g (6k-1)-1))/m
where,β_g (6k-1)=τ(6k-1)-2+τ(6(n-k)+1)-2,…
But,the formula of π_g (6n+2),π_g (6n-2) is omitted in abstract.

**Category:** Number Theory

[7] **viXra:1408.0046 [pdf]**
*submitted on 2014-08-08 08:35:19*

**Authors:** Th. Guyer

**Comments:** 1 Page.

A briefly olympic idea about P = NP
(include the Prime_Twin_Conjecture)
Whoever is able to(o) kicks out m(e?

**Category:** Number Theory

[6] **viXra:1408.0044 [pdf]**
*submitted on 2014-08-08 04:07:06*

**Authors:** Oh Jung Uk

**Comments:** 21 Pages. I don't know how to show abstract well

If π_t (6n+1) is the number of twin prime of 6n+1 or less then the formula of π_t (6n+1) is described below.
π_t (6n+1)=n+1-2/3 ∑_(k=1)^n▒((πβ_t (6k))/(πβ_t (6k)-1)) -2/3π ∑_(k=1)^n▒∑_(m=1)^∞▒sin((2〖mπ〗^2 β_t (6k))/(πβ_t (6k)-1))/m
where,β_t (6k)={τ(6k-1)-2}+{τ(6k+1)-2},…

**Category:** Number Theory

[5] **viXra:1408.0043 [pdf]**
*submitted on 2014-08-08 04:11:24*

**Authors:** Oh Jung Uk

**Comments:** 16 Pages. I don't know how to show abstract well

For Mersenne prime of 2^(6n+1)-1 type, if a Mersenne prime is 2^(6p+1)-1, just next Mersenne prime is 2^(6x+1)-1 then the following equation is satisfied.
x =p+3/2+1/2 ∑_(k=p+1)^x▒〖(πβ(2^(6k+1)-1)+1)/(πβ(2^(6k+1)-1)-1)+1/π ∑_(k=p+1)^x▒∑_(m=1)^∞▒sin((2mπ^2 β(2^(6k+1)-1))/(πβ(2^(6k+1)-1)-1))/m〗
where,β(2^(6k+1)-1)=τ(2^(6k+1)-1)-2,…
Mersenne prime of 2^(6n-1)-1 type is omitted in abstract.

**Category:** Number Theory

[4] **viXra:1408.0042 [pdf]**
*submitted on 2014-08-08 04:16:18*

**Authors:** Oh Jung Uk

**Comments:** 12 Pages. I don't know how to show abstract well

A number of 6n-1 type is not odd perfect number, Fermat number is not also odd perfect number.
And, if Fermat number is composite number then Fermat number is factorized as below
when n is odd number,2^(2^n )+1=(2^(n+1) (3k+1)+1)(2^(n+1) (3m)+1)
when n is even number,2^(2^n )+1=(2^(n+1) ((3k+1)/2)+1)(2^(n+1) (3m)+1)
And, all Fermat number for n≥5 is composite number.

**Category:** Number Theory

[3] **viXra:1408.0041 [pdf]**
*submitted on 2014-08-07 22:23:12*

**Authors:** Oh Jung Uk

**Comments:** 34 Pages. I don't know how I can fix the abstract

The formula of prime-counting function π(N=6n+3) is described below.
π(N=6n+3)=2n+2-2/3 ∑_(k=1)^n▒{πβ(6k-1)/(πβ(6k-1)-1)+πβ(6k+1)/(πβ(6k+1)-1)} -2/3π ∑_(k=1)^n▒∑_(m=1)^∞▒{(sin((2mπ^2 β(6k-1))/(πβ(6k-1)-1))+sin((2mπ^2 β(6k+1))/(πβ(6k+1)-1)))/m}
where,β(6k-1)=τ(6k-1)-2,β(6k+1)=τ(6k+1)-2,…

**Category:** Number Theory

[2] **viXra:1408.0003 [pdf]**
*submitted on 2014-08-02 01:48:13*

**Authors:** Russell Letkeman

**Comments:** 4 Pages.

We study the spacings of numbers co-prime to an even consecutive product of primes, P_m\# and its structure exposed by the fundamental theorem of prime sieving (FTPS). We extend this to prove some parts of the Hardy-Littlewood general prime density conjecture for all finite multiplicative groups modulo a primorial. We then use the FTPS to prove such groups have gap spacings which form arithmetic progressions as long as we wish. We also establish their densities and provide prescriptions to find them.

**Category:** Number Theory

[1] **viXra:1408.0001 [pdf]**
*submitted on 2014-08-01 05:16:54*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make two conjectures, one about how could be expressed a prime of the form 6k + 1 and one about how could be expressed a prime of the form 6k – 1.

**Category:** Number Theory