Near-squares in binary recurrence sequences

Nikos Tzanakis, Paul Voutier

Published 2021-06-08Version 1

We call an integer a \emph{near-square} if it is a prime times a square. In 1993, Mignotte and Peth\H{o} proved that, for all integers $a>3$, there are no elements that are squares, two times squares or three times squares in the sequence defined by $u_{0}=0$, $u_{1}=1$ and $u_{n+2}=au_{n+1}-u_{n}$ for $n \geq 0$, once $n>6$. Our investigations suggest that something stronger is happening in this sequence. For $n>7$, it appears that there are no elements in this sequence that are near squares (i.e., it appears that $2$ and $3$ in Mignotte and Peth\H{o}'s paper can be replaced with any prime whatsoever at the expense of increasing the bound on $n$ only slightly). Furthermore, if we generalize the recurrence relation to $u_{n+2}=au_{n+1}-b^{2}u_{n}$ with $a>b^{2}$, then the same appears to hold once $n>13$. We explain why this appears to be the case. There is something special about such sequences. While we have not been able to prove this in the generality that we state here, we give the partial results we have obtained in this direction. More specifically, for $b=-1$ and prime $p>3$ (unspecified) we show that a relation of the form $u_n = p\Box$ is impossible for $n\geq 5$ if $a$ is of the form $a=c^2+1$ with $c$ an integer $\geq 2$. Various intermediate results concerning terms of second-order binary sequences being a near square are included. Some of these results are valid for general $a\geq 3$.