**Previous months:**

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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)

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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(31)

Any replacements are listed further down

[700] **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

[699] **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

[698] **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

[697] **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

[696] **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

[695] **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

[694] **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

[693] **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

[692] **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

[691] **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

[690] **viXra:1407.0138 [pdf]**
*submitted on 2014-07-19 02:39:40*

**Authors:** Predrag Terzic

**Comments:** 3 Pages.

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

**Category:** Number Theory

[689] **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

[688] **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

[687] **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

[686] **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

[685] **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

[684] **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

[683] **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

[682] **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

[681] **viXra:1407.0091 [pdf]**
*submitted on 2014-07-13 03:20:13*

**Authors:** Predrag Terzic

**Comments:** 2 Pages.

Primality criterion for specific class of generalized Fermat numbers is introduced .

**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:** Taekyoon park, Yeonsoo Kim

**Comments:** 10 Pages.

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

[655] **viXra:1406.0043 [pdf]**
*submitted on 2014-06-08 03:12:08*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I just enunciate a formula which often leads to primes and products of very few primes and I state five conjectures about the pairs of primes of the form [(q^2 - p^2 – 2*r)/2,(q^2 – p^2 + 2*r)/2], where p, q, r are odd primes.

**Category:** Number Theory

[654] **viXra:1406.0030 [pdf]**
*submitted on 2014-06-05 14:30:24*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make few conjectures about a way to write an odd prime p, id est p = q – r + 1, where q and r are also primes; two of these conjectures can be regarded as generalizations of the twin primes conjecture, which states that there exist an infinity of pairs of twin primes.

**Category:** Number Theory

[653] **viXra:1406.0026 [pdf]**
*submitted on 2014-06-04 15:35:58*

**Authors:** Ramón Ruiz

**Comments:** 34 Pages. This document has been written in Spanish.

Abstract
The Goldbach's Conjecture states: “Every even integer greater than 2 can be expressed as the sum of two primes”.
Initially, to demonstrate this conjecture, it is possible to create two sequences (A and B) different for each even number, with all the natural numbers that can become prime, that added, in pairs, give us the appropriate even number.
The study of the matching process, in general, all the non-prime terms of the sequence A, with terms of the sequence B, or vice versa, to obtain a pair number and observing that some prime number pairs are found, allow us to develop a non-probabilistic formula, to calculate, the approximate quantity of prime number pairs, that will meet the conjecture requirements for a x even number. The result of this formula is always equal or greater than 1, and tends to infinite when x tends to infinite, which allow us to confirm that the Goldbach's Conjecture is true.
The prime numbers theorem by Gauss, the prime numbers theorem in arithmetic progression and some axioms have been used to complete this investigation.

**Category:** Number Theory

[652] **viXra:1406.0025 [pdf]**
*submitted on 2014-06-04 15:53:02*

**Authors:** Ramon Ruiz

**Comments:** 24 Pages. This document has been written in Spanish.

Abstract.
The Twin Primes Conjecture states: “There are infinitely many primes p such that p + 2 is also prime”.
Initially, to demonstrate this conjecture, it is possible to create two sequences (A and B), with all the natural numbers smaller than a number x that can become prime, and being each term of sequence B equal to its pair of sequence A plus 2.
The study of the matching process, in general, all the non-prime terms of the sequence A, with terms of the sequence B, or vice versa, and observing that some prime number pairs are found, allow us to develop a non-probabilistic formula, to calculate, the approximate quantity of primes pairs, p y p + 2, that are smaller 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 Gauss, the prime numbers theorem in arithmetic progression and some axioms have been used to complete this investigation.

**Category:** Number Theory

[651] **viXra:1406.0023 [pdf]**
*submitted on 2014-06-04 13:47:59*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 8 Pages.

We analyze the anatomy of critical line. In this paper, we change the center of coordinate. Hence, we obtain the minimum quantity of the critical line. Meanwhile, we investigate further many characteristic of *T* and σ.

**Category:** Number Theory

[650] **viXra:1406.0013 [pdf]**
*submitted on 2014-06-02 21:07:58*

**Authors:** Germán Paz

**Comments:** 16 Pages. 3 figures, Mathematica code; keywords: Andrica's conjecture, Brocard's conjecture, Legendre's conjecture, Oppermann's conjecture, prime numbers, triangular numbers. This paper (with plots as ancillary files) is also available at arxiv.org/abs/1406.4801.

Let $n\in\mathbb{Z}^+$. In [8] we ask the question whether any sequence of $n$ consecutive integers greater than $n^2$ and smaller than $(n+1)^2$ contains at least one prime number, and we show that this is actually the case for every $n\leq 1,193,806,023$. In addition, we prove that a positive answer to the previous question for all $n$ would imply Legendre's, Brocard's, Andrica's, and Oppermann's conjectures, as well as the assumption that for every $n$ there is always a prime number in the interval $[n,n+2\lfloor\sqrt{n}\rfloor-1]$.

Let $\pi[n+g(n),n+f(n)+g(n)]$ denote the amount of prime numbers in the interval $[n+g(n),n+f(n)+g(n)]$. Here we show that the conjecture described in [8] is equivalent to the statement that

%

$$\pi[n+g(n),n+f(n)+g(n)]\ge 1\text{, }\forall n\in\mathbb{Z}^+\text{,}$$

%

where

%

$$f(n)=\left(\frac{n-\lfloor\sqrt{n}\rfloor^2-\lfloor\sqrt{n}\rfloor-\beta}{|n-\lfloor\sqrt{n}\rfloor^2-\lfloor\sqrt{n}\rfloor-\beta|}\right)(1-\lfloor\sqrt{n}\rfloor)\text{, }g(n)=\left\lfloor1-\sqrt{n}+\lfloor\sqrt{n}\rfloor\right\rfloor\text{,}$$

%

and $\beta$ is any real number such that $1<\beta<2$. We also prove that the conjecture in question is equivalent to the statement that

%

$$\pi[S_n,S_n+\lfloor\sqrt{S_n}\rfloor-1]\ge 1\text{, }\forall n\in\mathbb{Z}^+\text{,}$$

%

where

%

$$S_n=n+\frac{1}{2}\left\lfloor\frac{\sqrt{8n+1}-1}{2}\right\rfloor^2-\frac{1}{2}\left\lfloor\frac{\sqrt{8n+1}-1}{2}\right\rfloor+1\text{.}$$

%

We use this last result in order to create plots of $h(n)=\pi[S_n,S_n+\lfloor\sqrt{S_n}\rfloor-1]$ for many values of $n$.

**Category:** Number Theory

[649] **viXra:1406.0010 [pdf]**
*submitted on 2014-06-02 13:17:10*

**Authors:** Peter Schorer

**Comments:** 43 Pages.

We present several proofs of the 3x + 1 Conjecture, which asserts that repeated iterations of the function C(x) = (3x + 1)/(2^a) always terminate in 1 Here x is an odd, positive integer, and a is the largest positive integer such that the denominator divides the numerator. Our first proofs are based on a structure called “tuple-sets” that represents the 3x + 1 function in the “forward” (as opposed to the inverse) direction. All of our proofs are counter-intuitive, but not for that reason wrong. In “Most Recent Proof of the Conjecture” on page 11, we show that, because the “number” of tuples in each tuple-set is the same, regardless if counterexamples exist or not, and because the set of all non-counterexamples is the same, regardless if counterexampels exist or not, it follows that counterexamples do not exist. In our next two proofs, we show, by a simple inductive argument, that the contents of the set of all tuple-sets is the same, regardless if counterexamples exist or not, and from this we conclude that counterexamples do not exist. “Third Proof” is based on a structure called “recursive ‘spiral’s” that represents the 3x + 1 function in the inverse direction. We show that, because a large number of consecutive odd, positive integers are known, by computer test, to be non-counterexamples, it follows, by an inductive argument based on certain fundamental properties of recursive “spiral”s, that the set of all tuples in each infinite set of recursive “spiral”s is the same regardless if counterexamples exist or not. We infer from this that counterexamples do not exist.
As far as we have been able to determine, our approach to a solution of the Problem is original.

**Category:** Number Theory

[648] **viXra:1405.0348 [pdf]**
*submitted on 2014-05-28 14:44:28*

**Authors:** Raffaele Cogoni

**Comments:** 4 Pages. The Sieve of Eratosthenes a simplified method to find the prime numbers, useful for middle school students and supperiori

Abstract
It is a procedure to find the prime numbers, in practice it is a simplification
the sieve of Eratosthenes

**Category:** Number Theory

[647] **viXra:1405.0336 [pdf]**
*submitted on 2014-05-28 05:50:51*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state two conjectures about the sum of a prime and a factorial.

**Category:** Number Theory

[646] **viXra:1405.0284 [pdf]**
*submitted on 2014-05-21 20:06:40*

**Authors:** Sbiis Saibian

**Comments:** 32 Pages.

In 1976 Donald Knuth introduced the world to his so called "Up-arrow notation" for very large positive integers. In this paper I prove a simple theorem which allows one to easily compare expressions involving Up-arrows.

**Category:** Number Theory

[645] **viXra:1405.0278 [pdf]**
*submitted on 2014-05-20 16:14:43*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In a previous paper I stated a conjecture on primes involving the pairs of sexy primes, which are the two primes that differ from each other by six. In this paper I extend that conjecture on the pairs of primes [p, q], where q is of the form p + p(n)#, where p(n)# is a primorial number, which means the product of first n primes.

**Category:** Number Theory

[644] **viXra:1405.0243 [pdf]**
*submitted on 2014-05-15 02:15:31*

**Authors:** Marius Coman

**Comments:** 1 Page.

This paper states a conjecture on primes involving two types of pairs of primes: the pairs of sexy primes, which are the two primes that differ from each other by six and the pairs of primes of the form [p, q], where q = p + 6*r, where r is positive integer.

**Category:** Number Theory

[643] **viXra:1405.0242 [pdf]**
*submitted on 2014-05-14 09:45:59*

**Authors:** Denise Vella-Chemla

**Comments:** 16 Pages.

We propose a demonstration of Goldbach's conjecture based on an approach using a four letters language.

**Category:** Number Theory

[642] **viXra:1405.0239 [pdf]**
*submitted on 2014-05-14 08:30:17*

**Authors:** Marius Coman

**Comments:** 8 Pages.

In two of my previous papers I treated quadratic polynomials which have the property to produce many primes in a row: in one of them I listed forty-two such polynomials which generate more than twenty-three primes in a row and in another one I listed few generic formulas which may conduct to find such prime-producing quadratic polynomials. In this paper I will present ten such polynomials which I discovered and posted in OEIS, each accompanied by its first fifty terms and some comments about it.

**Category:** Number Theory

[641] **viXra:1405.0238 [pdf]**
*submitted on 2014-05-14 04:02:08*

**Authors:** Marius Coman

**Comments:** 6 Pages.

In one of my previous papers I listed forty-two quadratic polynomials which generate more than twenty-three primes in a row, from which ten were already known from the articles available on Internet and thirty-two were discovered by me. In this paper I list few generic formulas which may conduct to find such prime-producing quadratic polynomials.

**Category:** Number Theory

[640] **viXra:1405.0221 [pdf]**
*submitted on 2014-05-12 20:02:57*

**Authors:** Marouane Rhafli

**Comments:** 5 Pages. A complementary file to my pdf on http://vixra.org/pdf/1405.0011v1.pdf

In this paper I’ll give another proof to the infinity of twin primes , I’ll prove that Lim Inf (Pn+1 – Pn ) =2

**Category:** Number Theory

[320] **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

[319] **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

[318] **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

[317] **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

[316] **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

[315] **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

[314] **viXra:1406.0023 [pdf]**
*replaced on 2014-06-08 18:05:07*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 8 Pages.

We analyze the anatomy of critical line. In this paper, we change the center of coordinate. Hence, we obtain the minimum quantity of the critical line. Meanwhile, we investigate further many characteristics of *T* and σ.

**Category:** Number Theory

[313] **viXra:1405.0242 [pdf]**
*replaced on 2014-05-17 11:01:04*

**Authors:** Denise Vella-Chemla

**Comments:** 16 Pages.

We propose a demonstration of Goldbach's conjecture based on an approach using a four letters language.

**Category:** Number Theory

[312] **viXra:1405.0242 [pdf]**
*replaced on 2014-05-17 08:37:03*

**Authors:** Denise Vella-Chemla

**Comments:** 16 Pages.

We propose a demonstration of Goldbach's conjecture based on an approach using a four letters language.

**Category:** Number Theory