Global Generalized Mersenne Numbers: Definition, Decomposition, and Generalized Theorems

Abstract

A new generalized definition of Mersenne numbers is proposed of the form an−a−1n, called global generalized Mersenne numbers and noted GMa,n with base a and exponent n positive integers. The properties are investigated for prime n and several theorems on Mersenne numbers regarding their congruence properties are generalized and demonstrated. It is found that for any a, GMa,n−1 is even and divisible by n, a and a−1 for any prime n>2, and by aa−1+1 for any prime n>5. The remaining factor is a function of triangular numbers of a−1, specific for each prime n. Four theorems on Mersenne numbers are generalized and four new theorems are demonstrated, showing first that GMa,n≡1or7mod12 depending on the congruence of amod4; second, that GMa,n−1 are divisible by 10 if n≡1mod4 and, if n≡3mod4, GMa,n≡1or7or9mod10, depending on the congruence of amod5; third, that all factors ci of GMa,n are of the form 2nfi+1 such that ci is either prime or the product of primes of the form 2nj+1, with fi,j natural integers; fourth, that for prime n>2, all GMa,n are periodically congruent to ±1or±3mod8 depending on the congruence of amod8; and fifth, that the factors of a composite GMa,n are of the form 2nfi+1 with fi≡umod4 with u=0, 1, 2 or 3 depending on the congruences of nmod4 and of amod8. The potential use of generalized Mersenne primes in cryptography is shortly addressed.