## 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.