On prime factors of the sum of two k-Fibonacci numbers

Abstract

We consider for integers [Formula: see text] the [Formula: see text]-generalized Fibonacci sequences [Formula: see text], whose first [Formula: see text] terms are [Formula: see text] and each term afterwards is the sum of the preceding [Formula: see text] terms. We give a lower bound for the largest prime factor of the sum of two terms in [Formula: see text]. As a consequence of our main result, for every fixed finite set of primes [Formula: see text], there are only finitely many positive integers [Formula: see text] and [Formula: see text]-integers which are a non-trivial sum of two [Formula: see text]-Fibonacci numbers, and all these are effectively computable.