P. Voutier

activity

20082026

most citedAn effective lower bound for the height of algebraic numbers

120 citations · 320 across the 31 of their papers we have counts for

collaborators

32 papers

math.NT2026

Smyth's Conjecture on the Mahler Measure of non-reciprocal trinomials of height $1$

Paul M Voutier

We prove a conjecture of Smyth on the Mahler measure of non-reciprocal trinomials of height $1$ , determining exactly when their Mahler measures are greater than or less than $M(x+y…

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

For the elliptic curves $E_{p, q} : y^{2} = x^{3} - pq x$ where $p$ and $q$ are distinct odd primes, we establish necessary and sufficient conditions under which rank$\,E_{p,q}(\mathbb{Q}…

math.NT2026

Perfect powers in sequences of polygonal numbers

Andrzej Dąbrowski, Salah Eddine Rihane, Gökhan 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.CO2026

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

Paul M Voutier

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…

math.NT2025

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

Paul M Voutier

We generalise our earlier work on the number of squares in binary recurrence sequences, ${y_{k}}_{k \geq - \infty}$ . In the notation of our previous papers, here we c…

math.NT2025★ 1 cited

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.NT2024

$n$ -defective Lehmer pairs for small $n$ : Corrections and Clarifications

Paul M. Voutier

We provide some corrections and clarifications of statements in \cite{Ab} and \cite{BHV} regarding elements in Lehmer sequences that have no primitive divisors for small $n$ .

math.NT2024

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…

math.NT2024★ 2 cited

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.NT2022

A kit for linear forms in three logarithms

Maurice Mignotte, Paul Voutier

We provide a technique to obtain explicit bounds for problems that can be reduced to linear forms in three complex logarithms of algebraic numbers. This technique can produce bound…

math.NT2021★ 1 cited

Improved Constants for Effective Irrationality Measures from Hypergeometric Functions

Paul Voutier

In this paper, we simplify and improve the constant, $c$ , that appears in effective irrationality measures, $∣ (a / b)^{m / n} - p / q ∣ > c ∣ q ∣^{- (κ + 1)}$ , obtained from the hypergeometric meth…

math.NT2021

Properties of Trinomials of Height at least 2

V. Flammang, P. Voutier

This paper is concerned with trinomials of the form $z^{n} + a z^{m} + b$ , where $0 < m < n$ are relatively prime integers and $a$ and $b$ are non-zero complex numbers. Typically (but not e…

math.NT2021

Near-squares in binary recurrence sequences

Nikos Tzanakis, Paul Voutier

We call an integer a \emph{near-square} if its absolute value is a square or a prime times a square. We investigate such near-squares in the binary recurrence sequences defined for…

math.NT2020★ 2 cited

The greatest common valuation of $ϕ_{n}$ and $ψ_{n}^{2}$ at points on elliptic curves

Paul Voutier, Minoru Yabuta

Given a minimal model of an elliptic curve, $E / K$ , over a finite extension, $K$ , of $Q_{p}$ for any rational prime, $p$ , and any point $P \in E (K)$ of infinite order, we…

math.NT2018★ 9 cited

Indecomposable integers in real quadratic fields

Magdaléna Tinková, Paul Voutier

In 2016, Jang and Kim stated a conjecture about the norms of indecomposable integers in real quadratic number fields $Q (D)$ where $D > 1$ is a squarefre…

math.NT2018

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.NT2015★ 5 cited

Families of periodic Jacobi-Perron algorithms for all period lengths

Paul Voutier

For all integers $m \geq n \geq 2$ , we exhibit infinite families of purely periodic Jacobi-Perron Algorithm (JPA) expansions of dimension $n$ with period length equal to $m$ along…

math.NT2014★ 1 cited

Modules with many non-associates and norm form equations with many families of solutions

Paul Voutier

For every number field $K$ , with $[K : Q] \geq 3$ , we show that the number of non-associates of the same norm in a full module in $K$ does not dep…

math.NT2014★ 54 cited

Complete solution of the diophantine equation $X^{2} + 1 = d Y^{4}$ and a related family of quartic Thue equations

Chen Jian Hua, Paul Voutier

In this paper, we use the method of Thue and Siegel, based on explicit Pade approximations to algebraic functions, to completely solve a family of quartic Thue equations. From this…

math.NT2013

Lang's conjecture and sharp height estimates for the elliptic curves $y^{2} = x^{3} + b$

Paul Voutier, Minoru Yabuta

For $E_{b} : y^{2} = x^{3} + b$ , we establish Lang's conjecture on a lower bound for the canonical height of non-torsion points along with upper and lower bounds for the difference betw…

math.NT2012★ 120 cited

An effective lower bound for the height of algebraic numbers

Paul Voutier

We prove that if $α$ is a non-zero algebraic number of degree $d \geq 2$ which is not a root of unity, then $dh(α)>(1/4) (\log(\log (d))/\log(d))^3.

math.NT2012★ 17 cited

Primitive divisors of Lucas and Lehmer sequences, III

Paul Voutier

In this paper we prove that if $n > 30, 030$ then the $n$ -th element of any Lucas or Lehmer sequence has a primitive divisor.

math.NT2012★ 11 cited

Primitive divisors of Lucas and Lehmer sequences, II

Paul Voutier

Let $\al$ and $\be$ be conjugate complex algebraic integers which generate Lucas or Lehmer sequences. We present an algorithm to search for elements of such sequences which have no…

math.NT2012★ 66 cited

Primitive divisors of Lucas and Lehmer sequences

Paul M Voutier

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…

Smyth's Conjecture on the Mahler Measure of non-reciprocal trinomials of height 1

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

Perfect powers in sequences of polygonal numbers

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

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

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

n-defective Lehmer pairs for small n: Corrections and Clarifications

A family of cyclic quartic monogenic polynomials

Bounds on the number of squares in recurrence sequences

A kit for linear forms in three logarithms

Improved Constants for Effective Irrationality Measures from Hypergeometric Functions

Properties of Trinomials of Height at least 2

Near-squares in binary recurrence sequences

The greatest common valuation of ϕn​ and ψn2​ at points on elliptic curves

Indecomposable integers in real quadratic fields

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

Families of periodic Jacobi-Perron algorithms for all period lengths

Modules with many non-associates and norm form equations with many families of solutions

Complete solution of the diophantine equation X2+1=dY4 and a related family of quartic Thue equations

Lang's conjecture and sharp height estimates for the elliptic curves y2=x3+b

An effective lower bound for the height of algebraic numbers

Primitive divisors of Lucas and Lehmer sequences, III

Primitive divisors of Lucas and Lehmer sequences, II

Primitive divisors of Lucas and Lehmer sequences

Smyth's Conjecture on the Mahler Measure of non-reciprocal trinomials of height $1$

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

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

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

$n$ -defective Lehmer pairs for small $n$ : Corrections and Clarifications

The greatest common valuation of $ϕ_{n}$ and $ψ_{n}^{2}$ at points on elliptic curves

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

Complete solution of the diophantine equation $X^{2} + 1 = d Y^{4}$ and a related family of quartic Thue equations

Lang's conjecture and sharp height estimates for the elliptic curves $y^{2} = x^{3} + b$