Composite values of shifted exponentials

Abstract

A well-known open problem asks to show that $2^n+5$ is composite for almost all values of $n$. This was proposed by Gil Kalai as a possible Polymath project, and was posed originally by Christopher Hooley. We show that, assuming GRH and a form of the pair correlation conjecture, the answer to this problem is affirmative. We in fact do not need the full power of the pair correlation conjecture, and it suffices to assume a generalization of the Brun-Titchmarsh inequality for the Chebotarev density theorem that is implied by it. Our methods apply to any shifted exponential sequence of the form $a^n-b$ and show that, under the same assumptions, such numbers are $k$-almost primes for a density $0$ of natural numbers $n$. Furthermore, we show that $a^p-b$ is composite for almost all primes $p$ whenever $(a, b) \neq (2, 1)$.