Sums of primes and quadratic linear recurrence sequences

Abstract

Let U be a sequence of positive integers which grows essentially as a geometric progression. We give a criterion on U in terms of its distribution modulo d, d = 1, 2, …, under which the set of positive integers expressible by the sum of a prime number and an element of U has a positive lower density. This criterion is then checked for some second order linear recurrence sequences. It follows, for instance, that the set of positive integers of the form \(p + \left\lfloor {(2 + \sqrt 3 )^n } \right\rfloor \), where p is a prime number and n is a positive integer, has a positive lower density. This generalizes a recent result of Enoch Lee. In passing, we show that the periods of linear recurrence sequences of order m modulo a prime number p cannot be “too small” for most prime numbers p.