Composite values of shifted exponentials

Abstract

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