arXiv subjects

Philippe Michaud-Rodgers

Publications and source records attributed to Philippe Michaud-Rodgers.

2 recordsLinked to original sources

arXiv2021

A Unique Perfect Power Decagonal Number

Let $\mathcal{P}_s(n)$ denote the $n$th $s$-gonal number. We consider the equation \[\mathcal{P}_s(n) = y^m, \] for integers $n,s,y,$ and $m$. All solutions to this equation are known for $m>2$ and $s \in \{3,5,6,8,20 \}$. We consider the case $s=10$, that of decagonal numbers. Using a descent argument and the modular method, we prove that the only decagonal number $>1$ expressible as a perfect $m$th power with $m>1$ is $\mathcal{P}_{10}(3) = 3^3$.

math.NT↗

arXiv2020

Quadratic Points on Non-Split Cartan Modular Curves

In this paper we study quadratic points on the non-split Cartan modular curves $X_{ns}(p)$, for $p = 7, 11,$ and $13$. Recently, Siksek proved that all quadratic points on $X_{ns}(7)$ arise as pullbacks of rational points on $X_{ns}^+(7)$. Using similar techniques for $p=11$, and employing a version of Chabauty for symmetric powers of curves for $p=13$, we show that the same holds for $X_{ns}(11)$ and $X_{ns}(13)$. As a consequence, we prove that certain classes of elliptic curves over quadratic fields are modular.

math.NT↗