Evaluating Mersenne Primes Using a Single Quadrant Expanding Square

Abstract

AbstractBy forming a table of sequential odd numbers using a single quadrant expanding square pattern it was observed that multiple Mersenne primes fall into the first column. We prove that the Mersenne primes which fall into the first column will be of the form p = 3 mod 4, where p is the Mersenne prime exponent. Focusing the search of primes to those which fall into the first column of this square may be a means to increase the speed at which large primes are discovered.Keywords: Algorithms; Mersenne Primes; Primes(ProQuest: ... denotes formulae omitted.)IntroductionA Mersenne prime is a prime of the form 2P - 1. For example, the first Mersenne primes are 2, 3, 7, 31, and 127 which corresponds to ρ values of 1, 2, 3, 5, and 7. Currently only 47 Mersenne primes have been discovered.1 The first attempt to compile the primes was performed in the 17th century by the French scholar Marin Mersenne. The search for these^rimes intensified with the advent of digital computing. As with all primes, as the numbers become larger, the primes become increasingly more remote, making an exhaustive search labor intensive. The aim of this paper is to explore a potential means to limit the set of the eligible numbers to those most likely to be a Mersenne Prime. To this end, sequential odd numbers are arranged in the expanding square pattern described below.Single Quadrant Expanding Square PatternThis is defined as expansion of a square limited to one quadrant (Figure 1]. With odd numbers placed along the lines of expansion a limitless grid of numbers is formed. This will be referred to as an Odd Number Single Quadrant Expanding Square, ONSQES, and is demonstrated in Figure 2 (Mersenne primes are bold). The numbers in the first column will be referred to as First Column Odd Number Single Quadrant Expanding Square Integers, FCONSQESI.Proof: Given that the numbers of interest in the first column of the ONSQES follow the equation:......Thus there now exists ρ = 3mod4 and p = ±5 mod8.This creates two possibilities:Case 1: p= 3mod4 and /? = -5mod8 =3mod8p= 3mod4 = 3,7mod8so p= 3mod4becomes 3mod8Case 2: p= 3mod4and ρ = 5 mod8 ξ 5 mod8This cannot occur becausep = 3 mod4 = 3,7 mod8 and then ρ * 5 mod8...RemarksFundamentally there are two ways to increase the speed of the search for Mersenne primes. One is through increased speed of assessment of the individual candidates, such as with faster processors or with more efficient verification of the individual candidates. …