About some Mersenne exponents: Acerca de algunos exponentes de Mersenne - Miramontes de León, Gerardo (2024)
Mersenne primes grow rapidly and become intractable with current computing tools. In this work, Mersenne chains are briefly reviewed to show how this exponential growth imposes a limit on its calculation. Subsequently, the following approach is proposed: “Given any prime number q it is possible find its associated Mersenne exponent, as long as it holds that log2(q + 1) is exactly an integer, where log2() is the base 2 logarithm”. Furthermore, a way is proposed to lighten, to a certain extent, the computational load when calculating log2(q+1) of gigantic quantities. For this, a scaling to q+1 is proposed, since without it the numerical capacity of computing machines cannot differentiate one number from another that has some extremely small decimal value. This simple approach, which might surprise in its simplicity, represents an alternative to verify whether a prime q is a Mersenne prime.