Dimorphic Mersenne numbers and their applications

TL;DR

This paper introduces dimorphic Mersenne numbers, exploring their properties and connections with degenerate Bernoulli and Bell polynomials, expanding the understanding of Mersenne number variants.

Contribution

It presents the novel concept of dimorphic Mersenne numbers and investigates their mathematical properties and relationships with special polynomials.

Findings

Defined dimorphic Mersenne numbers as a degenerate version of Mersenne primes

Established connections with degenerate Bernoulli and incomplete Bell polynomials

Provided initial properties and potential applications of these new numbers

Abstract

The Mersenne primes are primes which can be written as some prime power of 2 minus 1. These primes were studied from antiquity in that their close connection with perfect numbers and even to present day in that their easiness for primality test. In this paper, we introduce the dimorphic Mersenne numbers as a degenerate version of the Mersenne numbers and investigate some of their properties in connection with the degenerate Bernoulli polynomials and the incomplete Bell polynomials.