Near-squares in binary recurrence 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 $a\ge 3$ by $u_0(a)=0$, $u_1(a)=1$ and $u_{n+2}(a)=a{u}_{n+1}(a)-u_n(a)$ for $n\ge 0$. We show that for a given $a\ge 3$, there is at most one $n\ge 5$ 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 $a\equiv 2\mathrm{mod}4$, $n\equiv 3\mathrm{mod}4$ is prime and $n\ge 19$. 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 $u_{n+2}(a,b)=a{u}_{n+1}(a,b)+b{u}_n(a,b)$ for $n\ge 0$ (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)).

27 pages. This version corrects a few not serious misprints of the previous (2 Nov. 2023) one. To appear in Int. J. Number Theory