[26] **viXra:1702.0335 [pdf]**
*submitted on 2017-02-27 15:30:38*

**Authors:** Stephen Marshall

**Comments:** 9 Pages.

This paper presents a complete proof of the Pierpont Primes are infinite, even though only 16 of them have been found as of 21 Feb 2017. 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 m:
m = (p-1)!( + ) + +
We use this proof for d = 2u(n+1)3v(x+1) – 2u(n)3v(x) to prove the infinitude of Pierpont 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 Pierpont Prime Conjecture possible.

**Category:** Number Theory

[25] **viXra:1702.0331 [pdf]**
*replaced on 2017-03-07 17:04:09*

**Authors:** A. A. Frempong

**Comments:** 5 Pages. Copyright © by A. A. Frempong

Using a direct construction approach, the author proved the original Beal conjecture that if A^x + B^y = C^z , where A, B, C, x, y, z are positive integers and x, y, z > 2, then A, B and C have a common prime factor. Two main types of equations were involved, namely, the equation A^x + B^y = C^z and an equation which was called a tester equation. A tester equation has similar properties as A^x + B^y = C^z and was used to determine the properties of A^x + B^y = C^z . Also, two types of tester equations, namely, a literal tester equation and a numerical tester equation were applied. Each side of A^x + B^y = C^z and a tester equation was reduced to unity by division. The non-unity sides were justifiably equated to each other to produce a new equation which was called the master equation. The side of the master equation involving the terms of the tester equation was called the tester side of the master equation. Three versions of the proof were presented. In Version 1 proof, the tester equation was the literal equation G^m + H^n = I^p, but in Versions 2 and 3 proofs, the tester equations were the numerical tester equations, 2^9 + 8^3 = 4^5 and 3^3 + 6^3 = 3^5, respectively. By a comparative analysis, in which the corresponding "terms" on the right and left sides of the master equation were equated to each other, it was determined that if A^x + B^y = C^z , then A, B and C have a common prime factor. The proof is very simple, and occupies a single page, and even, high school students can learn it.

**Category:** Number Theory

[24] **viXra:1702.0323 [pdf]**
*replaced on 2017-02-27 13:42:01*

**Authors:** Stephen Crowley

**Comments:** 3 Pages.

It is proved that the non-trivial roots of the Hardy Z function are simple having multiplicity 1 by showing that the fixed-points N_Z(α)=α of the Newton map N_Z(t)=t-(Z(t))/(Z˙(t)) must have a multiplier λ_(N_Z)(α)=|(N_Z)˙(α)|=|(Z(α)Z¨(α))/(Z˙(α))|=0 and therefore a multiplicity m_Z(α)=1/(1-λ_(N_Z)(α))=1/(1-0)=1.

**Category:** Number Theory

[23] **viXra:1702.0318 [pdf]**
*submitted on 2017-02-26 13:00:05*

**Authors:** Reza Farhadian

**Comments:** 5 Pages.

Let p_n be the nth prime number. We prove that p_(n+1)<〖p_n〗^((n+1)/n ((logp_(n+1))/(logp_n )) ) for every n≥1. This inequality is weaker than the Firoozbakht’s conjecture p_(n+1)<〖p_n〗^((n+1)/n).Afterward we prove that the new inequality is equivalent to Firoozbakht’s conjecture, as n→∞, and hence the Cramér’s conjecture p_(n+1)-p_n=O(log^2 p_n ) to be hold, because the Firoozbakht’s conjecture is stronger than the Cramér’s conjecture, see [5].

**Category:** Number Theory

[22] **viXra:1702.0313 [pdf]**
*submitted on 2017-02-26 02:45:31*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I conjecture that any number of the form 16^n – 4^n + 1, where n is positive integer, is either prime either divisible by a Poulet number (see the sequence A020520 in OEIS for the numbers of this form).

**Category:** Number Theory

[21] **viXra:1702.0300 [pdf]**
*replaced on 2017-05-13 01:29:59*

**Authors:** Ralf Wüsthofen

**Comments:** 11 Pages. Older versions on http://vixra.org/abs/1403.0083

The present paper shows that a principle known as emergence lies beneath the strong Goldbach conjecture. Whereas the traditional approaches focus on the control over the distribution of the primes by means of circle method and sieve theory, we give a proof of the conjecture that involves the constructive properties of the prime numbers, reflecting their multiplicative character within the natural numbers. With an equivalent but more convenient form of the conjecture in mind, we create a structure on the natural numbers which is based on the prime factorization. Then, we realize that the characteristics of this structure immediately imply the conjecture and, in addition, an even strengthened form of it. Moreover, we can achieve further results by generalizing the structuring. Thus, it turns out that the statement of the strong Goldbach conjecture is the special case of a general principle.

**Category:** Number Theory

[20] **viXra:1702.0299 [pdf]**
*submitted on 2017-02-23 14:27:23*

**Authors:** Stephen Marshall

**Comments:** 7 Pages.

This paper presents a complete proof of the Factorial Primes are infinite, even though only 16 of them have been found as of 21 Feb 2017. 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 m:
m = (p-1)!( 1/p + ((-1)^d(d!))/(p+d)) + 1/p + 1/(p+d)
We use this proof for d = n(n!) to prove the infinitude of Factorial 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 Factorial Prime possible.

**Category:** Number Theory

[19] **viXra:1702.0286 [pdf]**
*submitted on 2017-02-22 16:09:30*

**Authors:** Stephen Marshall

**Comments:** 3 Pages.

In mathematics, and in particular number theory, Grimm's Conjecture (named after Karl Albert Grimm) states that to each element of a set of consecutive composite numbers one can assign a distinct prime that divides it. It was first published in American Mathematical Monthly, 76(1969) 1126-1128.
The Formal statement defining Grimm’s Conjecture, still unproved, is as follows:
Suppose n + 1, n + 2, …, n + k are all composite numbers, then there are k distinct primes pi such that pi divides n + i for 1 ≤ i ≤ k.

**Category:** Number Theory

[18] **viXra:1702.0285 [pdf]**
*submitted on 2017-02-22 16:11:33*

**Authors:** Stephen Marshall

**Comments:** 3 Pages.

In mathematics, Hall's conjecture is an open question, as of 2015, on the differences cube x3 that are not equal must lie a substantial distance apart. This question arose from consideration of the Mordell equation in the theory of integer points on elliptic curves. The original version of Hall's conjecture, formulated by Marshall Hall, Jr. in 1970, says that there is a positive constant C such that for any integers x and y for which y2 ≠ x3,

**Category:** Number Theory

[17] **viXra:1702.0273 [pdf]**
*replaced on 2017-02-28 13:59:01*

**Authors:** Stephen Crowley

**Comments:** 10 Pages.

A sequence of Cauchy sequences which conjecturally converge to the Riemann zeros is constructed and related to the LeClair-França criteria for the Riemann hypothesis.

**Category:** Number Theory

[16] **viXra:1702.0271 [pdf]**
*submitted on 2017-02-21 16:20:17*

**Authors:** Stephen Marshall

**Comments:** 7 Pages.

This paper presents a complete proof of the Cullen Primes are infinite, even though only 16 of them have been found as of 21 Feb 2017. 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 m:
See paper for this equation, as the text in this abstract does not support the mathematical format for this equation.
We use this proof for d = P2 + 1 to prove the infinitude of Cullen 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 Cullen Prime Conjecture possible.

**Category:** Number Theory

[15] **viXra:1702.0265 [pdf]**
*submitted on 2017-02-21 06:04:58*

**Authors:** Rédoane Daoudi

**Comments:** 7 Pages.

In our previous work (The distribution of prime numbers: overview of n.ln(n), (1) and (2)) we defined a new method derived from Rosser's theorem (2) and we used it in order to approximate the nth prime number. In this paper we improve our method to try to determine the next prime number if the previous is known. We use our method with five intervals and two values for n (see Methods and results). Our preliminary results show a reduced difference between the real next prime number and the number given by our algorithm. However long-term studies are required to better estimate the next prime number and to reduce the difference when n tends to infinity. Indeed an efficient algorithm is an algorithm that could be used in practical research to find new prime numbers for instance.

**Category:** Number Theory

[14] **viXra:1702.0264 [pdf]**
*submitted on 2017-02-21 01:57:28*

**Authors:** Marius Coman

**Comments:** 2 Pages.

The Woodall numbers are defined by the formula W(n) = n*2^n – 1 (see the sequence A003261 in OEIS). In this paper I conjecture that any Woodall number of the form 2^k*2^(2^k) – 1, where k ≥ 3, is either prime either divisible by a Poulet number.

**Category:** Number Theory

[13] **viXra:1702.0259 [pdf]**
*submitted on 2017-02-20 10:38:56*

**Authors:** Marius Coman

**Comments:** 3 Pages.

The Poulet numbers (or the Fermat pseudoprimes to base 2) are defined by the fact that are the only composites n for which 2^(n – 1) – 1 is divisible by n (so, of course, all Mersenne numbers 2^(n - 1) – 1 are divisible by Poulet numbers if n is a Poulet number; but these are not the numbers I consider in this paper). In a previous paper I conjectured that any composite Mersenne number of the form 2^m – 1 with odd exponent m is divisible by a 2-Poulet number but seems that the conjecture was infirmed for m = 49. In this paper I conjecture that any Mersenne number (with even exponent) 2^(p – 1) – 1 is divisible by at least a Poulet number for any p prime, p ≥ 11, p ≠ 13.

**Category:** Number Theory

[12] **viXra:1702.0253 [pdf]**
*submitted on 2017-02-20 09:23:51*

**Authors:** Rédoane Daoudi

**Comments:** 12 Pages.

The empirical formula giving the nth prime number p(n) is p(n) = n.ln(n) (from ROSSER (2)). Other studies have been performed (from DUSART for example (1)) in order to better estimate the nth prime number. Unfortunately these formulas don't work since there is a significant difference between the real nth prime number and the number given by the formulas. Here we propose a new model in which the difference is effectively reduced compared to the empirical formula. We discuss about the results and hypothesize that p(n) can be approximated with a constant defined in this work. As prime numbers are important to cryptography and other fields, a better knowledge of the distribution of prime numbers would be very useful. Further investigations are needed to understand the behavior of this constant and therefore to determine the nth prime number with a basic formula that could be used in both theoretical and practical research.

**Category:** Number Theory

[11] **viXra:1702.0226 [pdf]**
*submitted on 2017-02-17 04:27:18*

**Authors:** Predrag Terzic

**Comments:** 3 Pages.

Polynomial time probable prime test for specific class of N=k*b^n-1 is introduced .

**Category:** Number Theory

[10] **viXra:1702.0191 [pdf]**
*submitted on 2017-02-16 10:26:00*

**Authors:** Zeraoulia Elhadj

**Comments:** 8 Pages.

This note is concerned with presenting sufficient conditions to proves that the number of elements of certain real sequences is infinite.

**Category:** Number Theory

[9] **viXra:1702.0166 [pdf]**
*submitted on 2017-02-14 10:18:35*

**Authors:** Chongjunhuang

**Comments:** 10 Pages.

Prime density formula

**Category:** Number Theory

[8] **viXra:1702.0162 [pdf]**
*submitted on 2017-02-14 08:01:15*

[7] **viXra:1702.0160 [pdf]**
*submitted on 2017-02-13 16:00:14*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following conjecture: If F(2*p) is a Fibonacci number with an index equal to 2*p, where p is prime, p ≥ 5, then there exist a prime or a product of primes q1 and a prime or a product of primes q2 such that F(2*p) = q1*q2 having the property that q2 – 2*q1 is also a Fibonacci number with an index equal to 2^n*r, where r is prime or the unit and n natural. Also I observe that the ratio q2/q1 seems to be a constant k with values between 2.2 and 2.237; in fact, for p ≥ 17, the value of k seems to be 2.236067(...).

**Category:** Number Theory

[6] **viXra:1702.0157 [pdf]**
*replaced on 2017-05-11 09:42:42*

**Authors:** Chongxi Yu

**Comments:** 15 Pages.

Prime numbers are the basic numbers and are crucial important. There are many conjectures concerning primes are challenging mathematicians for hundreds of years and many “advanced mathematics tools” are used to solve them, but they are still unsolved. A kaleidoscope can produce an endless variety of colorful patterns and it looks like magic, but when you open one and examine it, it contains only very simple, loose, colored objects such as beads or pebbles and bits of glass. Humans are very easily cheated by 2 words, infinite and anything, because we never see infinite and anything, and so we always make a simple thing complex. The pattern of prime numbers similar to a “kaleidoscope” of numbers, if we divide primes into 4 groups, twin primes conjecture becomes much simpler. Based on the fundamental theorem of arithmetic and Euclid’s proof of endless prime numbers, we have proved there are infinitely many twin primes.

**Category:** Number Theory

[5] **viXra:1702.0150 [pdf]**
*submitted on 2017-02-13 14:43:06*

**Authors:** Stephen Marshall

**Comments:** 4 Pages.

Christian Goldbach (March 18, 1690 – November 20, 1764) was a German mathematician. He is remembered today for Goldbach's conjecture.
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states: Every even integer greater than 2 can be expressed as the sum of two primes.
On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII) in which he proposed the following conjecture:
Every even integer which can be written as the sum of two primes (the strong conjecture)
He then proposed a second conjecture in the margin of his letter:
Every odd integer greater than 7 can be written as the sum of three primes (the weak conjecture).
A Goldbach number is a positive even integer that can be expressed as the sum of two odd primes. Since four is the only even number greater than two that requires the even prime 2 in order to be written as the sum of two primes, another form of the statement of Goldbach's conjecture is that all even integers greater than 4 are Goldbach numbers.
The “strong” conjecture has been shown to hold up through 4 × 1018, but remains unproven for almost 300 years despite considerable effort by many mathematicians throughout history.
In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that
Every odd number greater than 5 can be expressed as the sum of three primes. (A prime may be used more than once in the same sum). In 2013, Harald Helfgott proved Goldbach's weak conjecture.
The author would like to give many thanks to Helfgott’s proof of the weak conjecture, because this proof of the strong conjecture is completely dependent on Helfgott’s proof. Without Helfgott’s proof, this elementary proof would not be possible.

**Category:** Number Theory

[4] **viXra:1702.0136 [pdf]**
*replaced on 2017-02-15 03:23:14*

**Authors:** Predrag Terzic

**Comments:** 2 Pages.

Polynomial time primality test for safe primes is introduced .

**Category:** Number Theory

[3] **viXra:1702.0090 [pdf]**
*replaced on 2017-02-22 08:26:54*

**Authors:** Chongxi Yu

**Comments:** 33 Pages.

Prime numbers are the basic numbers and are crucial important. There are many conjectures concerning primes are challenging mathematicians for hundreds of years. Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics. A kaleidoscope can produce an endless variety of colorful patterns and it looks like a magic, but when you open it, it contains only very simple, loose, colored objects such as beads or pebbles and bits of glass. Goldbach’s conjecture is about all numbers, the pattern of prime numbers likes a “kaleidoscope” of numbers, we divided any even numbers into 10 groups and primes into 4 groups, Goldbach’s conjecture becomes much simpler. Here we give a clear proof for Goldbach's conjecture based on the fundamental theorem of arithmetic and Euclid's proof that the set of prime numbers is endless.

**Category:** Number Theory

[2] **viXra:1702.0030 [pdf]**
*submitted on 2017-02-02 11:56:36*

**Authors:** Stephen Marshall

**Comments:** 8 Pages. This is an update to my proff subitted in 2014, I have simpified the submission by removing uneccessary material from the proof.

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-10!(1/p + ((-1)^d(d!))/(p+d)) + 1/p + 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

[1] **viXra:1702.0027 [pdf]**
*replaced on 2017-02-09 15:34:07*

**Authors:** Dragan Turanyanin

**Comments:** 3 Pages.

Three real numbers are introduced via related infinite series. With e, together they complete a quadruplet.

**Category:** Number Theory