Searcharxiv⌕ Search

arXiv subjects

Paul M. Voutier

Publications and source records attributed to Paul M. Voutier.

5 recordsLinked to original sources
arXiv2026

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

For the elliptic curves $E_{p,q}: y^{2}=x^{3}-pqx$ where $p$ and $q$ are distinct odd primes, we establish necessary and sufficient conditions under which rank$\,E_{p,q}(\mathbb{Q})$ and $\dim_{\mathbb{F}_{2}} \Sha \left( E_{p,q}/\bbQ \right)[2]$ are both $0$. We do so via a similar characterisation of when the Selmer groups associated with the degree-$2$ isogeny $ϕ$ and its dual $\widehatϕ$ are both of minimal size, along with results about a cokernel that arises from a related exact sequence.

math.NT↗
arXiv2026

Perfect powers in sequences of polygonal numbers

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 for $m>2$ and $s\in\{3,5,6,8,10,20\}$. Here we extend these results to the cases $s = 2k+4$ (where $k = 4,6$ or $5 \leq k \leq 97$ is a prime number) and $s = k+4$ (where $k = 9,15$ or $3 \leq k \leq 97$ is a prime number). The proofs of our results use the modular and hypergeometric methods, linear forms in logarithms and extensive calculations. We were unable to completely solve the above Diophantine equations, but we expect (based on GRH and the weak effective $abc$ conjecture) that there will be no additional solutions beyond those explicitly shown in Theorems~1, 2 and 3.

math.NT↗
arXiv2025

A family of cyclic quartic monogenic polynomials

We produce an explicit family of totally real cyclic quartic polynomials that are monogenic in many cases and, if the $abc$ conjecture holds, generate distinct monogenic quartic fields infinitely often. Additional families (also conjecturally generating infinitely many distinct fields) are provided in Section 4, including what appears to be an infinite collection of such families.

math.NT↗
arXiv2010

A New Approximation to the Normal Distribution Quantile Function

We present a new approximation to the normal distribution quantile function. It has a similar form to the approximation of Beasley and Springer [3], providing a maximum absolute error of less than $2.5 \cdot 10^{-5}$. This is less accurate than [3], but still sufficient for many applications. However it is faster than [3]. This is its primary benefit, which can be crucial to many applications, including in financial markets.

stat.CO↗