Computer experiments with Mersenne primes

TL;DR

This paper computes the sum of reciprocals of known Mersenne primes with high precision, analyzes its continued fraction expansion, and provides evidence supporting the conjecture of infinitely many Mersenne primes and the transcendental nature of a related continued fraction.

Contribution

It introduces high-precision calculations of the reciprocal sum, analyzes its continued fraction, and explores the properties of a continued fraction with Mersenne primes as partial quotients, suggesting irrationality and transcendence.

Findings

Sum of reciprocals converges to constants related to Khinchin's and Lévy's constants.

Evidence supporting the infinitude of Mersenne primes.

Proposed that a continued fraction with Mersenne primes as partial quotients is transcendental.

Abstract

We have calculated on the computer the sum $\bar{\BB}_M$ of reciprocals of all 47 known Mersenne primes with the accuracy of over 12000000 decimal digits. Next we developed $\bar{\BB}_M$ into the continued fraction and calculated geometrical means of the partial denominators of the continued fraction expansion of $\bar{\BB}_M$. We get values converging to the Khinchin's constant. Next we calculated the $n$-th square roots of the denominators of the $n$-th convergents of these continued fractions obtaining values approaching the Khinchin-L{\`e}vy constant. These two results suggests that the sum of reciprocals of all Mersenne primes is irrational, supporting the common believe that there is an infinity of the Mersenne primes. For comparison we have done the same procedures with slightly modified set of 47 numbers obtaining quite different results. Next we investigated the continued fraction whose partial quotients are Mersenne primes and we argue that it should be transcendental.