Near-squares in binary recurrence sequences

TL;DR

This paper investigates near-square values in binary recurrence sequences, establishing conditions under which such values occur and revealing a new factorization related to these sequences.

Contribution

It proves that at most one near-square occurs beyond a certain index in these sequences and introduces a novel Aurifeuillean-like factorization.

Findings

At most one near-square exists for n ≥ 5 in each sequence.

Near-squares are rare and follow specific modular conditions.

A new factorization technique for recurrence sequence elements is presented.

Abstract

We call an integer a \emph{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 $a \geq 3$ by $u_{0}(a)=0$, $u_{1}(a)=1$ and $u_{n+2}(a)=au_{n+1}(a)-u_{n}(a)$ for $n \geq 0$. We show that for a given $a \geq 3$, there is at most one $n \geq 5$ such that $u_{n}(a)$ is a near-square. With the exceptions of $u_{6}(3)=12^{2}$ and $u_{7}(6)=239 \cdot 13^{2}$, any such $u_{n}(a)$ can only be a near-square if $a \equiv 2 \bmod 4$, $n \equiv 3 \bmod 4$ is prime and $n \geq 19$. This is part of a more general phenomenon regarding near-squares in non-degenerate recurrence sequences defined for integers $a$ and $b=-b_{1}^{2}$ by $u_{0}(a,b)=0$, $u_{1}(a,b)=1$ and $u_{n+2}(a,b)=au_{n+1}(a,b)+bu_{n}(a,b)$ for $n \geq 0$ (see our Conjecture 1.1). It arises from a new Aurifeuillean-like factorization of elements of recurrence sequences that we have discovered (see relation (1.1)).