Abstract
A well-known open problem asks to show that 2n+5 is composite for almost all values of n. 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 an−b and show that, under the same assumptions, such numbers are k-almost primes for a density 0 of natural numbers n. Furthermore, under the same assumptions we show that ap−b is composite for almost all primes p whenever (a,b)≠(2,1).
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