Near-squares in binary recurrence sequences

Abstract

We call an integer a near-square if its absolute value is a square or a prime times a square. We investigate such near-squares in the binary recurrence sequences defined for integers [Formula: see text] by [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text]. We show that for a given [Formula: see text], there is at most one [Formula: see text] such that [Formula: see text] is a near-square. With the exceptions of [Formula: see text] and [Formula: see text], any such [Formula: see text] can be a near-square only if [Formula: see text], [Formula: see text] is prime and [Formula: see text]. This is a part of a more general phenomenon regarding near-squares in nondegenerate recurrence sequences defined for the integers [Formula: see text] and [Formula: see text] by [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text]. This arises from a novel Aurifeuillean-type factorization of [Formula: see text] we have found.