Near-squares in binary recurrence sequences
Nikos Tzanakis, Paul Voutier
TL;DR
This paper analyzes when terms of binary recurrence sequences $u_n(a,b)$ can be near-squares, i.e., lie in $c\mathrm{S}$ with $|c|=1$ or a prime, focusing on $b=-b_1^2$. It proves that for fixed $a\ge3$ there is at most one $n\ge5$ with $u_n(a,-b_1^2)$ a near-square, aside from two explicit small instances; near-squares occur only for $a\equiv2\pmod4$ and prime subscripts $n\ge19$ with $n\equiv3\pmod4$, plus a special $u_7(6)=239\cdot13^2$ case and $u_6(3)=12^2$. A central tool is an Aurifeullian-like factorisation of $u_{2n+1}(a,-b_1^2)$, writing $u_{2n+1}=t_n w_n$ with gcd$(t_n,w_n)\le2$, which reduces the problem to checking whether $t_n$ or $w_n$ is a square. The authors combine this with Lucas-sequence and Diophantine-analytic results to obtain sharp, uniform constraints (in particular modulo $4$ and for the primality of indices), and they propose conjectures linking these phenomena to broader non-degenerate recurrence structures. The results substantially constrain the occurrence of near-squares in binary recurrences and illuminate a novel Aurifeullian-type structure in such sequences.
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)).