**Previous months:**

2007 - 0703(3) - 0706(2)

2008 - 0807(1) - 0809(1) - 0810(1) - 0812(2)

2009 - 0901(2) - 0904(2) - 0907(2) - 0908(4) - 0909(1) - 0910(2) - 0911(1) - 0912(3)

2010 - 1001(3) - 1002(1) - 1003(55) - 1004(50) - 1005(36) - 1006(7) - 1007(11) - 1008(16) - 1009(21) - 1010(8) - 1011(7) - 1012(13)

2011 - 1101(14) - 1102(7) - 1103(13) - 1104(3) - 1105(1) - 1106(2) - 1107(1) - 1108(2) - 1109(3) - 1110(5) - 1111(4) - 1112(4)

2012 - 1201(2) - 1202(13) - 1203(7) - 1204(9) - 1205(7) - 1206(6) - 1207(5) - 1208(5) - 1209(11) - 1210(15) - 1211(10) - 1212(4)

2013 - 1301(5) - 1302(10) - 1303(16) - 1304(15) - 1305(12) - 1306(13) - 1307(26) - 1308(12) - 1309(9) - 1310(13) - 1311(16) - 1312(21)

2014 - 1401(20) - 1402(11) - 1403(25) - 1404(12) - 1405(19) - 1406(21) - 1407(36) - 1408(23)

Any replacements are listed further down

[728] **viXra:1408.0134 [pdf]**
*submitted on 2014-08-20 08:04:44*

**Authors:** Predrag Terzic

**Comments:** 4 Pages.

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

**Category:** Number Theory

[727] **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

[726] **viXra:1408.0126 [pdf]**
*submitted on 2014-08-18 15:16:53*

**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

[725] **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

[724] **viXra:1408.0113 [pdf]**
*submitted on 2014-08-18 06:11:05*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make five conjectures about the primes r, t 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

[723] **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

[722] **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

[721] **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

[720] **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

[719] **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

[718] **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

[717] **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

[716] **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

[715] **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

[714] **viXra:1408.0068 [pdf]**
*submitted on 2014-08-11 10:15:02*

**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

[713] **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

[712] **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

[711] **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

[710] **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

[709] **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

[708] **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

[707] **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

[706] **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

[705] **viXra:1407.0224 [pdf]**
*submitted on 2014-07-30 20:04:10*

**Authors:** Russell Letkeman

**Comments:** 3 Pages.

We build a simple recursive model for the prime numbers which at its heart is the prime sieve of Eratosthenes. We also show for prime numbers greater than 3 and their gaps posses a handedness which forbids a large range of possibilities for the choice of intervals in arithmetic progressions.

**Category:** Number Theory

[704] **viXra:1407.0214 [pdf]**
*submitted on 2014-07-29 23:15:32*

**Authors:** Russell Letkeman

**Comments:** 7 Pages.

We show every set modulo the product of a random collection of unique prime numbers has a palindrome in its gaps of length the minimum set minus 1. There is one more gap which is always a twin. Together the count of the gaps equals the count of the minimum modular set. This symmetry not only forces all constellations of gaps to have mirror images existing at exactly the same counts, but it also precisely identifies the center of mass (counts) of the set.

**Category:** Number Theory

[703] **viXra:1407.0209 [pdf]**
*submitted on 2014-07-29 02:38:15*

**Authors:** Pingyuan Zhou

**Comments:** 9 Pages. Author gives an argument for indirect connections between Fermat primes and regular 2^k-sided polygons to make Gauss-Wantzel theorem have general sense in implying connections between Fermat primes and all constructible polygons.

Abstract: Gauss-Wantzel theorem shows that regular n-sided polygons, whose number of sides contains a(distrinct) Fermat prime(s) as odd prime factor(s) of n or number of sides is power of 2, are all constructible with compass and straightedge. But of these caces, the constructibility of all regular 2^k-sided polygons is not related to Fermat primes. We discover the number of so-called root Mersenne primes Mp for p

**Category:** Number Theory

[702] **viXra:1407.0205 [pdf]**
*submitted on 2014-07-27 17:21:20*

**Authors:** JinHua Fei

**Comments:** 7 Pages.

In this paper, we assume that weaker Hardy-Littlewood Conjecture, we got a better upper bound of the exceptional real zero for a class of prime number module.

**Category:** Number Theory

[701] **viXra:1407.0203 [pdf]**
*submitted on 2014-07-27 20:37:59*

**Authors:** Réjean Labrie

**Comments:** 7 Pages.

This article is a demonstration of the existence of at least one prime number between two consecutive squares.

**Category:** Number Theory

[700] **viXra:1407.0201 [pdf]**
*submitted on 2014-07-27 03:06:26*

**Authors:** T.Nakashima

**Comments:** 1 Page.

This Paper is the result Counting the Prime Numbers by using Mathematica 9.

**Category:** Number Theory

[699] **viXra:1407.0193 [pdf]**
*submitted on 2014-07-25 10:25:35*

**Authors:** Predrag Terzic

**Comments:** 5 Pages.

We present deterministic primality test for Fermat
numbers , Fn = 2^2^n+ 1 , where n > 1 . Essentially this test is similar to the Lucas-Lehmer primality test for Mersenne numbers.

**Category:** Number Theory

[698] **viXra:1407.0166 [pdf]**
*submitted on 2014-07-21 18:58:13*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 7 Pages.

The critical line lies on a surface. And the critical line inherits the characteristics from this surface. Then, the location of the critical line can be determined.

**Category:** Number Theory

[697] **viXra:1407.0164 [pdf]**
*submitted on 2014-07-22 01:26:41*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I present two possible infinite sequences of primes, having in common the fact that their formulas contain the powers of the number 2.

**Category:** Number Theory

[696] **viXra:1407.0159 [pdf]**
*submitted on 2014-07-21 04:04:27*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I make a conjecture which states that any prime greater than or equal to 5 can be written in a certain way, in other words that any such prime can be expressed using just two other primes and a power of the number 2.

**Category:** Number Theory

[695] **viXra:1407.0158 [pdf]**
*submitted on 2014-07-21 04:47:52*

**Authors:** Marius Coman

**Comments:** 3 Pages.

These conjectures state that any prime p greater than 60 can be written as a sum of three primes of a certain type from the following four ones: 10k + 1, 10k + 3, 10k + 7 and 10k + 9.

**Category:** Number Theory

[694] **viXra:1407.0157 [pdf]**
*submitted on 2014-07-21 05:43:31*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I present two possible infinite sequences of primes, having in common the fact that their formulas contain the number 360.

**Category:** Number Theory

[693] **viXra:1407.0153 [pdf]**
*submitted on 2014-07-20 23:20:01*

**Authors:** Pingyuan Zhou

**Comments:** 14 Pages. Author presents the strong finiteness of double Mersenne primes and the infinity of root Mersenne primes and near-square primes of Mersenne primes by generalizing conjecture about primality of Mersenne number.

Abstract: In this paper we present the strong finiteness of double Mersenne primes to be a subset of Mersenne primes, the infinity of so-called root Mersenne primes to be also a subset of Mersenne primes and the infinity of so-called near-square primes of Mersenne primes by generalizing our previous conjecture about primality of Mersenne number. These results and our previous results about the strong finiteness of Fermat, double Fermat and Catalan-type Fermat primes [1] give an elementary but complete understanding for the infinity or the strong finiteness of some prime number sequences of the form 2^x±1, which all have a corresponding original continuous natural ( prime ) number sequence. It is interesting that the generalization to near-square primes of Mersenne primes Wp=2(Mp)^2-1 has brought us positive result.

**Category:** Number Theory

[692] **viXra:1407.0152 [pdf]**
*submitted on 2014-07-21 02:26:52*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper are stated ten conjectures on primes, more precisely on the infinity of some types of triplets and quadruplets of primes, all of them using the multiples of the number 30 and also all of them met on the study of Carmichael numbers.

**Category:** Number Theory

[691] **viXra:1407.0151 [pdf]**
*submitted on 2014-07-21 02:50:07*

**Authors:** Predrag Terzic

**Comments:** 2 Pages.

Prime number sieve using LCM function is introduced .

**Category:** Number Theory

[690] **viXra:1407.0150 [pdf]**
*submitted on 2014-07-21 03:00:29*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper are stated six conjectures on primes, more precisely on the infinity of some types of pairs of primes, all of them met in the study of 3-Carmichael numbers.

**Category:** Number Theory

[689] **viXra:1407.0143 [pdf]**
*submitted on 2014-07-19 16:20:26*

**Authors:** Isaac Mor

**Comments:** 3 Pages.

Odd Perfect Number = 36k+9
In 1953, Jacques Touchard proved that an odd perfect number must be of the form 12k + 1 or 36k + 9.
(Judy A. Holdener discovered a simpler proof of the theorem of Touchard in 2002)
if I am right then I (isaac mor lol) just showed that an odd perfect number must be of the form 36k+9 (19 july 2014)

**Category:** Number Theory

[688] **viXra:1407.0129 [pdf]**
*submitted on 2014-07-17 21:54:39*

**Authors:** Pingyuan Zhou

**Comments:** 9 Pages. In this paper, author presents the strong finiteness of Fermat primes, double Fermat primes and Catalan-type Fermat primes by generalizing previous conjecture about primality of Fermat numbers to double Fermat and Catalan-type Fermat numbers.

Abstract: In this paper we present that so-called double Fermat numbers are an infinite subset of well-known Fermat numbers and so-called Catalan-type Fermat numbers are also an infinite subset of Fermat numbers as well as double Fermat primes and Catalan-type Fermat primes are all strongly finite as Fermat primes do. From it we get the same result that composite Fermat numbers, composite double Fermat numbers and composite Catalan-type Fermat numbers are all infinite.

**Category:** Number Theory

[687] **viXra:1407.0128 [pdf]**
*submitted on 2014-07-17 13:03:45*

**Authors:** Yilun Shang

**Comments:** 5 Pages.

In this note, we consider some generalizations of the Lucas
sequence, which essentially extend sequences to triangular arrays.
Some new and elegant results are derived.

**Category:** Number Theory

[686] **viXra:1407.0117 [pdf]**
*submitted on 2014-07-15 22:13:12*

**Authors:** Pingyuan Zhou

**Comments:** 4 Pages. Aothor presents a near-sguare number sequence of all Mersenne primes, which seems to be an accptable awy in searching for larger primes by known Mersenne primes themselves than the largest known Mersenne prime M57885161.

Abstract: In this paper we present a conjecture that there is a near-square prime number sequence of Mersenne primes to arise from the near-square number sequence Wp=2(Mp)^2-1 generated from all Mersenne primes Mp, in which every term is larger prime number than corresponding perfect number. The conjecture has been verified for the first few prime terms in the near-square prime number sequence and we may expect appearing of near-square prime numbers of some known Mersenne primes with large p-values will become larger primes to be searched than the largest known Mersenne prime M57885161.

**Category:** Number Theory

[685] **viXra:1407.0111 [pdf]**
*submitted on 2014-07-15 06:26:38*

**Authors:** Choe Ryong Gil

**Comments:** 8 pages, two tables

In this paper we introduce a new function, which would be called a sigma-index of the natural
number, and consider its boundedness. This estimate is effective for the Robin inequality.

**Category:** Number Theory

[684] **viXra:1407.0098 [pdf]**
*submitted on 2014-07-14 05:42:42*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I enunciate five conjectures on primes, based on the study of Fermat pseudoprimes and on the author’s believe in the importance of multiples of 30 in the study of primes.

**Category:** Number Theory

[683] **viXra:1407.0096 [pdf]**
*submitted on 2014-07-14 02:57:45*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I enunciate nine conjectures on primes, all of them on the infinity of certain sequences of primes.

**Category:** Number Theory

[682] **viXra:1407.0095 [pdf]**
*submitted on 2014-07-13 12:16:35*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make a conjecture on the squares of primes of the form 6k + 1, conjecture that states that by a certain deconcatenation of those numbers (each one in other two numbers) it will be obtained similar results.

**Category:** Number Theory

[681] **viXra:1407.0093 [pdf]**
*submitted on 2014-07-13 04:24:29*

**Authors:** Pingyuan Zhou

**Comments:** 4 Pages. Author presents a new and equivalent statement of Fermat's little theorem for Fermat numbers by using double Fermat number formula to give a very simple explanation for all composite Fermat numbers to be pseudoprimes.

Abstract: In this paper we present a new and equivalent statement of Fermat's little theorem for Fermat numbers by introducing double Fermat number formula and give a very simple and accptable explanation for all composite Fermat numbers to be pseudoprimes.

**Category:** Number Theory

[680] **viXra:1407.0083 [pdf]**
*submitted on 2014-07-12 02:11:36*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make a conjecture on the squares of primes of the form 6k – 1, conjecture that states that by a certain deconcatenation of those numbers (each one in other two numbers) it will be obtained similar results.

**Category:** Number Theory

[679] **viXra:1407.0081 [pdf]**
*submitted on 2014-07-11 03:55:10*

**Authors:** Pingyuan Zhou

**Comments:** 8 Pages. Author presents two symmetric conjectures related to Mersenne and Fermat primes themselves. It may imply that Mersenne primes are infinite but Fermat primes are finite.

Abstract: From existence of the intersection of the set of Mersenne primes and the set of Fermat primes being a set to contain only one element 3 to be the first Mersenne prime and also the first Fermat prime we fell there are connections between Mersenne and Fermat primes. In this paper, it is presented that two symmetric conjectures related to Mersenne and Fermat primes themselves will lead us to expect Mersenne primes to be infinite but Fermat primes to be finite.

**Category:** Number Theory

[678] **viXra:1407.0080 [pdf]**
*submitted on 2014-07-11 05:52:35*

**Authors:** Jinhua Fei

**Comments:** 9 Pages.

This paper use Nevanlinna's Second Main Theorem of the value distribution theory, we got an important conclusion by Riemann hypothesis.Thus, we launch a contradiction.

**Category:** Number Theory

[677] **viXra:1407.0077 [pdf]**
*submitted on 2014-07-11 03:03:25*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I will define four sequences of numbers obtained through concatenation, definitions which also use the notion of “sum of the digits of a number”, sequences that have the property to produce many primes, semiprimes and products of very few prime factors.

**Category:** Number Theory

[676] **viXra:1407.0061 [pdf]**
*submitted on 2014-07-08 13:20:13*

**Authors:** Carlos Giraldo Ospina

**Comments:** 19 Pages.

En este documento se demuestra la existencia de ciclo único para el Algoritmo de Collatz, con ello la conjetura correspondiente queda en firme; de otra parte, sin necesidad de demostrar la existencia de ciclo único, se puede emplear la inducción completa mediante el Teorema de Wailly.

El presente es un documento de lectura lenta y atenta, no significa que sea difícil… en ABCdatos aparecen archivos preliminares acerca del Algoritmo y Conjetura de Collatz; los referidos documentos, criticables en algún aspecto y subsanables con la demostración de ciclo único, son el andamiaje que hizo posible la demostración de la Conjetura de Collatz… plasman aciertos y errores normales en el terreno investigativo… además, muestran las innumerables bellezas del algoritmo… ellos quedarán como legado en la Historia de las Matemáticas…

¡Bienvenido a la ansiada demostración de la Conjetura de Collatz!

[675] **viXra:1407.0057 [pdf]**
*submitted on 2014-07-08 04:15:50*

**Authors:** Dhananjay P. Mehendale

**Comments:** 7 pages

Goldbach conjecture asserts that every even integer greater than 4 is sum of two odd primes. Stated in a letter to Leonard Euler by Christian Goldbach in 1842, this is still an enduring unsolved problem. In this paper we develop a new simple strategy to settle this most easy to state problem which has baffled mathematical community for so long. We show that the existence of two odd primes for every even number greater than 4 to express it as their sum follows from the well known Chinese remainder theorem. We further develop a method to actually determine a pair of primes for any given even number to express it as their sum using remainders modulo all primes up to square root of that given even number.

**Category:** Number Theory

[674] **viXra:1407.0056 [pdf]**
*submitted on 2014-07-07 22:15:58*

**Authors:** Taekyyon Park, Yeonsoo Kim, Jong Min Lee

**Comments:** 10 Pages. To prove Goldbach's conjecture, we developed our own dynamic model for Goldbach partition. This is the first step of our study.

There have been various approach to prove Goldbach's conjecture using analytical number theory. We go back to the starting point of this famous probelm and are able to show that the number of Goldbach partition is related to that of ordered pairs of non-primes. This proof is based on the world's first dynamic model of primes and can be a key to identify the structure of prime numbers.

**Category:** Number Theory

[673] **viXra:1407.0045 [pdf]**
*submitted on 2014-07-05 22:49:54*

**Authors:** Pingyuan Zhou

**Comments:** 9 Pages. Author presents a conjecture called the simple Mersenne conjecture, which may imply there are no more double Mersenne primes.

Abstract: In this paper we conjecture that there is no Mersenne number M(p)=2^p-1 to be prime for p=2^k±1,±3 when k>7, where p is positive integer and k is natural number. It is called the simple Mersenne conjecture and holds till p≤30402457 from status of this conjecture. If the conjecture is true then there are no more double Mersenne primes besides known double Mersenne primes MM(2), MM(3), MM(5), MM(7).

**Category:** Number Theory

[672] **viXra:1407.0031 [pdf]**
*submitted on 2014-07-03 22:53:13*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I present a very simple formula which conducts often to primes or composites with very few prime factors; for instance, for the first 27 consecutive values introduced as “input” in this formula were obtained 10 primes, 4 squares of primes and 12 semiprimes; just 2 from the numbers obtained have three prime factors; but the most interesting thing is that the composites obtained have a special property that make them form a class of numbers themselves.

**Category:** Number Theory

[671] **viXra:1407.0028 [pdf]**
*submitted on 2014-07-03 11:56:14*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In a previous paper I made a generalization of de Polignac’s conjecture. In this paper I extend that generalization as much as is possible.

**Category:** Number Theory

[670] **viXra:1407.0026 [pdf]**
*submitted on 2014-07-03 09:09:42*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I show a set of Poulet numbers, each one of them having the same interesting relation between its prime factors, and I make four conjectures, one about the infinity of this set, one about the infinity of a certain type of duplets respectively triplets respectively quadruplets and so on of primes and finally two generalizations, of the twin primes conjecture respectively of de Polignac’s conjecture.

**Category:** Number Theory

[669] **viXra:1406.0182 [pdf]**
*submitted on 2014-06-30 01:00:00*

**Authors:** Pingyuan Zhou

**Comments:** 5 Pages. Auther presents a conjecture related to distribution of a kind of special prime factors of Fermat numbers, which may imply existence of infinitely many primes of the form x^2+1.

It is well known that there are infinitely many prime factors of Fermat numbers, because prime factor of a Fermat prime is the Fermat prime itself but a composite Fermat number has at least two prime factors and Fermat numbers are pairwise relatively prime. Hence we conjecture that there is at least one prime factor (k^(1/2)*2^(a/2))^2+1 of Fermat number for F(n)-1≤a<F(n+1)-1 (n=0,1,2,3,…), where k^(1/2)is odd posotive integer, a is even positive integer and F(n) is Fermat number. The conjecture holds till a<F(4+1)-1=4294967296 from known evidences. Two corollaries of the conjecture imply existence of infinitely many primes of the form x^2+1, which is one of four basic problems about primes mentioned by Landau at ICM 1912.

**Category:** Number Theory

[668] **viXra:1406.0181 [pdf]**
*submitted on 2014-06-30 02:05:49*

**Authors:** Pingyuan Zhou

**Comments:** 13 Pages. Author presents a conjecture on composite terms in so-called generilized Catalan-Mersenne number sequence, and tries to find a new way to imply existence of infinitely many composite Mersenne numbers whose exponets are primes.

We conjecture that there is at least one composite term in sequence generated from Mersenne-type recurrence relations. Hence we may expect that all terms are composite besides the first few continuous prime terms in Catalan-Mersenne number sequence and composite Mersenne numbers with exponets restricted to prime values are infinite.

**Category:** Number Theory

[667] **viXra:1406.0161 [pdf]**
*submitted on 2014-06-25 16:47:07*

**Authors:** Isaac Mor

**Comments:** 3 Pages. I got rid of the power of p when n=P*Q^2 with a simple proof

if n is an Odd Perfect Number then n=P*Q^2
I got rid of the power of P with a simple proof

**Category:** Number Theory

[666] **viXra:1406.0155 [pdf]**
*submitted on 2014-06-25 09:04:18*

**Authors:** Arnaud Dhallewyn

**Comments:** 5 Pages. Tout droit réservé

Différente démonstration du postulat de Bertrand

**Category:** Number Theory

[665] **viXra:1406.0147 [pdf]**
*submitted on 2014-06-24 03:05:09*

**Authors:** Andrey Loshinin

**Comments:** 75 Pages.

Collected back formulas of the solutions of certain Diophantine equations and their systems. These decisions were not known earlier.

**Category:** Number Theory

[664] **viXra:1406.0142 [pdf]**
*submitted on 2014-06-23 04:05:36*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I combine two of my objects of study, the Poulet numbers and the different types of pairs of primes and I state two conjectures about few ways in which types of Poulet numbers could be associated with types of pairs of primes.

**Category:** Number Theory

[663] **viXra:1406.0131 [pdf]**
*submitted on 2014-06-20 18:19:34*

**Authors:** Allan Cacdac

**Comments:** 2 Pages.

Using a functional equation and different proofs for its existence, we are able to prove and show that A,B and C will always have a common prime factor.

**Category:** Number Theory

[662] **viXra:1406.0116 [pdf]**
*submitted on 2014-06-18 11:23:35*

**Authors:** Michael Pogorsky

**Comments:** 2 pages

Any odd perfect number is unknown. Simple analysis valid almost for all combinations of odd prime divisors proves that odd numbers constituted of them cannot be perfect.

**Category:** Number Theory

[661] **viXra:1406.0114 [pdf]**
*submitted on 2014-06-18 04:35:50*

**Authors:** Andrey Loshinin

**Comments:** 46 Pages.

Collected formula of the solution of Diophantine equations. All the formulas given to me. There are solutions of the equations in General form.

**Category:** Number Theory

[660] **viXra:1406.0112 [pdf]**
*submitted on 2014-06-18 05:16:00*

**Authors:** Xu Feng

**Comments:** 1 Page.

The Best Formula on the Prime Numbers is awesome.

**Category:** Number Theory

[659] **viXra:1406.0101 [pdf]**
*submitted on 2014-06-16 04:44:43*

**Authors:** Tatenda Kubalalika

**Comments:** 6 Pages.

In a paper published in 1997, Xian Jin Li showed that the Riemann Hypothesis holds if and only if a certain sequence of real numbers is nonnegative. In this note, we give an unconditional proof of Li's criterion for the Riemann Hypothesis.

**Category:** Number Theory

[658] **viXra:1406.0088 [pdf]**
*submitted on 2014-06-14 11:50:07*

**Authors:** Arnaud Dhallewyn

**Comments:** 102 Pages. Tout droit réservé

Présentation globale de la fonction zêta de Riemann

**Category:** Number Theory

[657] **viXra:1406.0079 [pdf]**
*submitted on 2014-06-13 15:03:54*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make three conjectures about a type of triplets of primes related in a certain way, i.e. the triplets of primes [p, q, r], where 2*p^2 – 1 = q*r and I raise an open problem about the primes of the form q = (2*p^2 – 1)/r, where p, r are also primes.

**Category:** Number Theory

[656] **viXra:1406.0066 [pdf]**
*submitted on 2014-06-11 06:18:07*

**Authors:** Diego Marin

**Comments:** 16 Pages.

We define an infinite summation which is proportional to the reverse of Riemann Zeta function \zeta(s). Then we demonstrate that such function can have singularities only for Re s = 1/n with n in N\0. Finally, using the functional equation, we reduce these possibilities to the only Re s = 1/2.

**Category:** Number Theory

[321] **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

[320] **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

[319] **viXra:1407.0214 [pdf]**
*replaced on 2014-07-31 22:10:45*

**Authors:** Russell Letkeman

**Comments:** 6 Pages.

We introduce a fundamental theorem of prime sieving (FTPS) and show how it illuminates structure on numbers co-prime to a random product of unique prime numbers. This theorem operates on the transition between the set of numbers co-prime to any product of unique prime numbers and the new set when another prime number is introduced in the product.

**Category:** Number Theory

[318] **viXra:1407.0166 [pdf]**
*replaced on 2014-07-27 13:08:59*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 7 Pages.

The critical line lies on a surface. And the critical line inherits the characteristics from this surface. Then, the location of the critical line can be determined.

**Category:** Number Theory

[317] **viXra:1407.0083 [pdf]**
*replaced on 2014-07-13 10:45:58*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make a conjecture on the squares of primes of the form 6k – 1, conjecture that states that by a certain deconcatenation of those numbers (each one in other two numbers) it will be obtained similar results.

**Category:** Number Theory

[316] **viXra:1407.0057 [pdf]**
*replaced on 2014-08-18 03:52:29*

**Authors:** Dhananjay P. Mehendale

**Comments:** 9 pages.

Goldbach conjecture asserts that every even integer greater than 4 is sum of two odd primes. Stated in a letter to Leonard Euler by Christian Goldbach in 1842, this is still an enduring unsolved problem. In this paper we develop a new simple strategy to settle this most easy to state problem which has baffled mathematical community for so long. We show that the existence of two odd primes for every even number greater than 4 to express it as their sum follows from the well known Chinese remainder theorem. We develop a method to actually determine a pair (and subsequently all pairs) of primes for any given even number to express it as their sum. For proof sake we will be using an easy equivalent of Goldbach conjecture. This easy equivalent leads to a congruence system and existence of solution for this congruence system is assured by Chinese remainder theorem. Each such solution actually provides a pair of primes to express given even number as their sum. We also discuss how twin prime conjecture follows from existence of certain x as a solution of certain congruence system.

**Category:** Number Theory

[315] **viXra:1407.0057 [pdf]**
*replaced on 2014-07-17 12:56:15*

**Authors:** Dhananjay P. Mehendale

**Comments:** 8 pages.

Goldbach conjecture asserts that every even integer greater than 4 is sum of two odd primes. Stated in a letter to Leonard Euler by Christian Goldbach in 1842, this is still an enduring unsolved problem. In this paper we develop a new simple strategy to settle this most easy to state problem which has baffled mathematical community for so long. We show that the existence of two odd primes for every even number greater than 4 to express it as their sum follows from the well known Chinese remainder theorem. We develop a method to actually determine a pair (and subsequently all pairs) of primes for any given even number to express it as their sum. For proof sake we will be using an easy equivalent of Goldbach conjecture. This easy equivalent leads to a congruence system and existence of solution for this congruence system is assured by Chinese remainder theorem. Each such solution actually provides a pair of primes to express given even number as their sum.

**Category:** Number Theory

[314] **viXra:1407.0056 [pdf]**
*replaced on 2014-08-07 00:42:10*

**Authors:** Taekyoon park, Yeonsoo Kim, Jong Min Lee

**Comments:** 10 Pages. To prove Goldbach's conjecture, we developed our own dynamic model for Goldbach partition. This is the first step of our study.

There have been various approach to prove Goldbach's conjecture using analytical number theory. We go back to the starting point of this famous probelm and are able to show that the number of Goldbach partition is related to that of ordered pairs of non-primes. This proof is based on the world's first dynamic model of primes and can be a key to identify the structure of prime numbers.

**Category:** Number Theory

[313] **viXra:1407.0056 [pdf]**
*replaced on 2014-08-05 21:50:18*

**Authors:** Taekyoon park, Yeonsoo Kim, Jong Min Lee

**Comments:** 10 Pages. To prove Goldbach's conjecture, we developed our own dynamic model for Goldbach partition. This is the first step of our study.

There have been various approach to prove Goldbach's conjecture using analytical number theory. We go back to the starting point of this famous probelm and are able to show that the number of Goldbach partition is related to that of ordered pairs of non-primes. This proof is based on the world's first dynamic model of primes and can be a key to identify the structure of prime numbers.

**Category:** Number Theory

[312] **viXra:1407.0026 [pdf]**
*replaced on 2014-07-03 12:54:29*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I show a set of Poulet numbers, each one of them having the same interesting relation between its prime factors, and I make four conjectures, one about the infinity of this set, one about the infinity of a certain type of duplets respectively triplets respectively quadruplets and so on of primes and finally two generalizations, of the twin primes conjecture respectively of de Polignac’s conjecture.

**Category:** Number Theory

[311] **viXra:1406.0131 [pdf]**
*replaced on 2014-07-02 17:51:03*

**Authors:** Allan Cacdac

**Comments:** 4 Pages. Revised.

Using visualization of the pattern by providing examples and an elementary proof, we are able to prove and show that A,B and C will always have a common prime factor.

**Category:** Number Theory

[310] **viXra:1406.0131 [pdf]**
*replaced on 2014-06-21 05:28:56*

**Authors:** Allan Cacdac

**Comments:** 2 Pages. Replacing because of a typo error

Using a functional equation and different proofs for its existence, we are able to prove
and show that A,B and C will always have a common prime factor.

**Category:** Number Theory

[309] **viXra:1406.0101 [pdf]**
*replaced on 2014-07-31 09:03:09*

**Authors:** Tatenda Kubalalika

**Comments:** 5 Pages.

In a paper published in 1997, Xian Jin Li showed that the Riemann Hypothesis is completely equivalent to the nonnegativity of a certain sequence of real numbers.In this note, we unconditionally prove Li's criterion for the Riemann Hypothesis.

**Category:** Number Theory

[308] **viXra:1406.0101 [pdf]**
*replaced on 2014-07-22 09:30:50*

**Authors:** Tatenda Kubalalika

**Comments:** 5 Pages.

In a paper published in 1997, Xian-Jin Li showed that the Riemann Hypothesis is completely equivalent to the non-negativity of a certain sequence of real numbers. In this note,we give an an unconditional proof of Li's criterion for the Riemann Hypothesis.

**Category:** Number Theory