Bounds on the number of squares in recurrence sequences: y0 = b2 (I)

Abstract

We continue and generalise our earlier investigations of the number of squares in binary recurrence sequences. Here we consider sequences, ${\left(y_k\right)}_{k=-\infty }^{\infty }$, arising from the solutions of generalised negative Pell equations, $X^2-d{Y}^2=c$, where −c and $y_0$ are any positive squares. We show that there are at most 2 distinct squares larger than an explicit lower bound in such sequences. From this result, we also show that there are at most 5 distinct squares when $y_0=b^2$ for infinitely many values of b, including all $1\le b\le 24$, as well as once d exceeds an explicit lower bound, without any conditions on the size of such squares.