P. Voutier

works on

2-descent 1 elliptic curves 1 isogeny 1 rank zero 1 selmer groups 1 tate-shafarevich group 1

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math.NT2026

Exact classification of elliptic curves $y^{2} = x^{3} - pq x$ with rank $0$ and trivial $\Sha[2]$

Arkabrata Ghosh, Paul M. Voutier

The paper determines exactly when the elliptic curves $y^{2} = x^{3} - pq x$ (with distinct odd prime parameters) have Mordell‑Weil rank zero and a trivial 2‑torsion part of the Tate–Sh…

math.NT2026

Perfect powers in sequences of polygonal numbers

Andrzej DÄ browski, Salah Eddine Rihane, GÃ¶khan Soydan +1

Let $P_{s} (n)$ denote the $n$ -th $s$ -gonal number. Consider the Diophantine equation $P_{s} (n) = t^{m}$ for integers $n, s, t$ and $m > 2$ . All solutions to this equation are known f…

math.NT2026

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

Paul M Voutier

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, $b = p^{m}$ , $2 p^{m}$ or $4 p^{m}$ , w…

math.NT2025

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

Paul M Voutier

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…

math.NT2025

Bounds on the number of squares in recurrence sequences

Paul M Voutier

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 (th…

math.NT2025

A family of cyclic quartic monogenic polynomials

Paul M. Voutier

We produce an explicit family of totally real cyclic quartic polynomials that are monogenic in many cases and, if the $ab c$ conjecture holds, generate distinct monogenic quartic fi…

Exact classification of elliptic curves y2=x3−pqx with rank 0 and trivial $\Sha[2]$

Perfect powers in sequences of polygonal numbers

Sharp bounds on the number of squares in recurrence sequences and solutions of X2−(a2+b)Y4=−b

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

Bounds on the number of squares in recurrence sequences

A family of cyclic quartic monogenic polynomials

Exact classification of elliptic curves $y^{2} = x^{3} - pq x$ with rank $0$ and trivial $\Sha[2]$

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

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