A new explicit expansion approach to Mersenne primes

TL;DR

This paper introduces a novel explicit expansion method for the Lucas-Lehmer test to identify Mersenne primes, along with a new compositeness criterion and four combinatorial identities, advancing primality testing techniques.

Contribution

It presents a new explicit expansion approach to the Lucas-Lehmer test and a criterion for Mersenne compositeness, along with four proven combinatorial identities.

Findings

New explicit expansion approach for Lucas-Lehmer test

A new criterion for Mersenne compositeness

Proof of four new combinatorial identities

Abstract

This paper first proves what the author called the Eight Levels Theorem and then highlights a new explicit expansion approach to Lucas-Lehmer primality test for Mersenne primes and gives a new criterion for Mersenne compositeness. Also, we prove four new combinatorial identities.