Abstract

In a previous paper, a new generalized definition of Mersenne numbers was proposed of the form (aⁿ – (a – 1)ⁿ) called Global Generalized Mersenne numbers, or Generalized Mersenne numbers in short. For prime exponents n, Generalized Mersenne primes and composites are generated. In this paper, the properties and distributions of Generalized Mersenne composites are investigated. It is found that the distribution of composite Generalized Mersenne numbers follow simple laws demonstrated in three theorems, as composite GMa,n appear periodically in an infinite number of groups of pairs of solutions in a, embedded into each others. It is remarkable that the distribution of composite GM_a,n is completely characterized once the first values of a yielding composite GM_a,n are found, as composite GM_a,n are spaced regularly, separated by intervals of values depending on their factors c₁ = 2nf₁ + 1. Three methods are presented to calculate composite GM_a,n and applied for the first six prime exponents n from 3 to 17.

Keywords: Mersenne numbers, Generalized mersenne numbers, Distribution of generalized mersenne composites