Romanoff (1934) showed that integers that are the sum of a prime and a power of 2 have positive lower asymptotic density in the positive integers. We adapt his method by showing more generally the existence of a positive lower asymptotic density for integers that are the sum of a prime and a term of a given nonconstant nondegenerate integral linear recurrence with separable characteristic polynomial.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa161-1-3,
author = {Christian Ballot and Florian Luca},
title = {On the sumset of the primes and a linear recurrence},
journal = {Acta Arithmetica},
volume = {161},
year = {2013},
pages = {33-46},
zbl = {1302.11079},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa161-1-3}
}
Christian Ballot; Florian Luca. On the sumset of the primes and a linear recurrence. Acta Arithmetica, Tome 161 (2013) pp. 33-46. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa161-1-3/