arXiv subjects

Paul M Voutier

Publications and source records attributed to Paul M Voutier.

6 recordsLinked to original sources

arXiv2026

Sharp bounds on the number of squares in recurrence sequences and solutions of $X^{2}-\left( a^{2}+b \right) Y^{4}=-b$

We obtain best possible results for the number of coprime positive integer solutions of the equation in the title when $a$ is a positive integer, $b=p^{m}$, $2p^{m}$ or $4p^{m}$, where $m$ is a non-negative integer, $p$ is prime, $\gcd \left( a^{2}, b \right)$ is squarefree and $X^{2}- \left( a^{2}+b \right) Y^{2}=-4$ has a solution in positive integers. We prove our results by establishing best possible bounds for the number of distinct squares in certain binary recurrence sequences, including those associated with such equations.

math.NT↗

arXiv2026

On the Guy-Kelly Conjecture for the No-Three-In-Line Problem

We provide details of the error Gabor Ellmann found in 2004 in a heuristic argument of Guy and Kelly on this problem. This led to a correction of their conjectured upper bound for the no-three-in-line problem. However, details of the issue and its correction, including the actual location of the issue, while simple, do not seem to have appeared in the literature previously. That said, very recent work of Prellberg [5] does contain a derivation of the corrected conjectured upper bound.

math.CO↗

arXiv2025

Bounds on the number of squares in recurrence sequences

We investigate the number of squares in a very broad family of binary recurrence sequences with $u_{0}=1$. We show that there are at most two distinct squares in such sequences (the best possible result), except under such very special conditions where we prove there are at most three such squares.

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arXiv2025

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

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.

math.NT↗

arXiv2025

Bounds on the number of squares in recurrence sequences: arbitrary $b$, III

We generalise our earlier work on the number of squares in binary recurrence sequences, $\left\{ y_{k} \right\}_{k \geq -\infty}$. In the notation of our previous papers, here we consider the case when $N_α$ is any negative integer and $y_{0}=b^{2}$ for any positive integer, $b$. We show that there are at most $4$ distinct squares with $y_{k}$ sufficiently large. This allows us to also show that there are at most $9$ distinct squares in such sequences when $b=1,2$ or $3$, or once $d$ is sufficiently large.

math.NT↗

arXiv2012

Primitive divisors of Lucas and Lehmer sequences

Stewart reduced the problem of determining all Lucas and Lehmer sequences whose $n$-th element does not have a primitive divisor to solving certain Thue equations. Using the method of Tzanakis and de Weger for solving Thue equations, we determine such sequences for $n \leq 30$. Further computations lead us to conjecture that, for $n > 30$, the $n$-th element of such sequences always has a primitive divisor.

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