Composite values of shifted exponentials - Järviniemi Olli; Teräväinen Joni (ACADEMIC PRESS INC ELSEVIER SCIENCE, 2025)
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).