On computing the density of integers of the form 2ⁿ+𝑝

Del Corso, Gianna; Del Corso, Ilaria; Dvornicich, Roberto; Romani, Francesco

doi:10.1090/mcom/3537

On computing the density of integers of the form $2^n+p$
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by Gianna M. Del Corso, Ilaria Del Corso, Roberto Dvornicich and Francesco Romani;

Math. Comp. 89 (2020), 2365-2386

DOI: https://doi.org/10.1090/mcom/3537

Published electronically: April 28, 2020

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Abstract:

The problem of finding the density of odd integers which can be expressed as the sum of a prime and a power of two is a classical one. In this paper we tackle the problem both with a direct approach and with a theoretical approach, suggested by Bombieri. These approaches were already introduced by Romani in [Calcolo 20 (1983), no. 3, pp. 319–336], but here the methods are extended and enriched with statistical and numerical methodologies. Moreover, we give a proof, under standard heuristic hypotheses, of the formulas claimed by Bombieri, on which the theoretical approach is based. The different techniques produce estimates of the densities which coincide up to the first three digits.

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