The Pattern of Prime Numbers

Abstract

The prime numbers 5 p ≥ obey a pattern that can be described by two formscilitates obtaining them sequentially, being possible also to calculate the quantity of primes that are in the geometric progressions 6 1 k k + ± → ∈ as it is described in this document.

On Infinite Number of Solutions for one type of Non-Linear Diophantine Equations

“…+ 1 has infinitely many prime solutions, where 's are any primes different from 2. Note 3.1: Corollary 3.3.1 to Theorem 3.3 thus settles a more general form of the conjecture mentioned by Smarandache in [3].…”

“…We adopted the notion of periodic sequences for proving the result. Thus, we settled the conjecture raised in [3] regarding the primes of the required type. Also, we discussed certain results concerning the Euler-Muller sequence.…”

“…Similar to this, there were a number of prime number patterns such as primorial primes, factorial primes, etc. Recently [3] calculated the quantity of primes that are in geometric progressions. In this article, we discuss about primes in some special pattern conjectured in [4].…”

“…The questions raised in [3], is a starting point for our interest in this work, and they are as follows. A.…”

An Innovative Approach to the Finite Sequences of Prime Numbers

“…The issue of prime numbers in number theory has always been a challenge to face and still nowadays it remains one of the major open problems notwithstanding the many theoretical successes achieved both historically and recently [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] owing also to the great importance of the issue and its strict relations to physics [16][17][18][19][20][21][22][23][24][25][26]. There are many physical and even biological phenomena [17] which imitate the behaviour of prime numbers so that prime numbers display a paramount importance both in physics and in mathematics.…”