Author(s)

Ramon Carbó-Dorca

Institut de Química Computacional i Catàlisi, Universitat de Girona, Campus de Montilivi, Girona (Catalonia), Spain.

A simple recursive algorithm to generate the set of natural numbers, based on Mersenne numbers: M_N = 2^N – 1, is used to count the number of prime numbers within the precise Mersenne natural number intervals: [0; M_N]. This permits the formulation of an extended twin prime conjecture. Moreover, it is found that the prime numbers subsets contained in Mersenne intervals have cardinalities strongly correlated with the corresponding Mersenne numbers.

Mersenne Numbers, Recursive Generation of Natural Numbers, Mersenne Natural Number Intervals, Counting the Number of Prime Numbers in Mersenne Natural Intervals, Correlation between Prime Number Set Cardinalities and Mersenne Numbers, Extended Twin Prime Number Conjecture

Carbó-Dorca, R. (2022) Mersenne Numbers, Recursive Generation of Natural Numbers, and Counting the Number of Prime Numbers. Applied Mathematics, 13, 538-543. doi: 10.4236/am.2022.136034.