Bounds on the number of squares in recurrence sequences: $y_{0}=b^{2}$ (I)

TL;DR

This paper investigates the maximum number of perfect squares in sequences derived from solutions to generalized negative Pell equations, establishing explicit bounds and extending previous results in recurrence sequence analysis.

Contribution

It generalizes earlier work by providing explicit bounds on the number of squares in sequences from negative Pell equations, including cases with initial squares and large parameters.

Findings

At most 2 squares larger than a certain bound in such sequences.

At most 5 squares for infinitely many initial squares, including all from 1 to 24.

Bounded the number of squares once the parameter d exceeds a certain threshold.

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, $X^{2}-dY^{2}=c$, 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 $1 \leq b \leq 24$, as well as once $d$ exceeds an explicit lower bound, without any conditions on the size of such squares.