Linear combinations of prime powers in binary recurrence sequences

Abstract

We give finiteness results concerning terms of linear recurrence sequences having a representation as a linear combination, with fixed coefficients, of powers of fixed primes. On one hand, under certain conditions, we give effective bounds for the terms of binary recurrence sequences with such a representation. On the other hand, in the case of some special binary recurrence sequences, all terms having a representation as sums of powers of [Formula: see text] and [Formula: see text] are explicitly determined.