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


Authors: Gianna M. Del Corso, Ilaria Del Corso, Roberto Dvornicich and Francesco Romani
Journal: Math. Comp. 89 (2020), 2365-2386
MSC (2010): Primary 11P32; Secondary 11-04, 65C60
DOI: https://doi.org/10.1090/mcom/3537
Published electronically: April 28, 2020
MathSciNet review: 4109570
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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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Additional Information

Gianna M. Del Corso
Affiliation: Dipartimento di Informatica, Università di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, PI, Italy
Email: gianna.delcorso@unipi.it

Ilaria Del Corso
Affiliation: Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56123 Pisa, PI, Italy
Email: ilaria.delcorso@unipi.it

Roberto Dvornicich
Affiliation: Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo, 5, 56123 Pisa, PI, Italy
Email: roberto.dvornicich@unipi.it

Francesco Romani
Affiliation: Dipartimento di Informatica, Università di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Pl, Italy
Email: francesco.romani@unipi.it

DOI: https://doi.org/10.1090/mcom/3537
Keywords: Sums of primes and powers, odd integers
Received by editor(s): April 2, 2019
Received by editor(s) in revised form: January 17, 2020
Published electronically: April 28, 2020
Additional Notes: The research of the first and last authors was partially supported by the INdAM-GNCS project “Analisi di matrici sparse e data-sparse: metodi numerici ed applicazioni”.
The second author was partially supported by MIUR (Italy) through PRIN 2017 “Geometric, algebraic and analytic methods in arithmetic”, and by Università di Pisa through PRA 2018-19 “ Spazi di moduli, rappresentazioni e strutture combinatorie”.
The third author has been partially supported by MIUR (Italy) through PRIN 2017 “Geometric, algebraic and analytic methods in arithmetic”, and by Università di Pisa through PRA 2018-19 “Geometira e topologia delle varietà”.
Article copyright: © Copyright 2020 American Mathematical Society