Prime numbers that avoid the Mersenne property with respect to all other primes

David Easdown

Abstract

An integer \(m\ge 3\) is said to be Mersenne with respect to \(n\), where \(n\ge 2\), if \(m = 1 + n +\ldots + n^k\) for some \(k\ge 1\). This generalises the notion of a Mersenne prime number, since if \(m\) is a prime number Mersenne with respect to \(2\), then \(m\) is a usual Mersenne prime. For example, \(31\) is a usual Mersenne prime, but also Mersenne with respect to \(5\). By contrast, \(13\) is Mersenne with respect to \(3\) but not \(2\), and \(5\) and \(11\) are not Mersenne with respect to any prime. In this short note, we prove that there are infinitely many prime numbers that are not Mersenne with respect to any prime number. The first, more elementary, proof relies on a lower bound for \(\pi(x)-\pi(x/2)\), established by Ramanujan (1919), where \(\pi(x)\) is the number of primes not exceeding a given integer \(x\). The second proof uses the full force of the Prime Number Theorem to deduce that \(\mu(x)=O(\sqrt{x})\) and \(\pi(x)-\mu(x)\) is asympotically equivalent to \(x/\log x\), where \(\mu(x)\) denotes the number of primes not exceeding \(x\) that are Mersenne with respect to some prime.

Keywords: primes, Mersenne numbers.

AMS Subject Classification: Primary 11A41.