Mersenne numbers and the doubling map

Abstract

We study the connection between the Mersenne numbers $M(n) = 2^n-1$ and the dynamics of the angle-doubling map. Within this framework, we develop an algorithm to compute divisors of Mersenne numbers without explicitly evaluating $M(n)$. Determining whether $M(n)$ is prime for a prime $n$ (and knowing if there are infinitely many of them), is a central problem, traditionally addressed with the help of the Lucas-Lehmer test. We provide an alternative approach based on dynamical methods. As an application, we prove that $M(2{,}199{,}023{,}254{,}451)$ (with approximately $6.6 \times 10^{11}$ digits) is composite by exhibiting a non-trivial divisor.