Description
Online Number Theory Seminar
Abstract: For $m\geq 3,$ we define the $m$th order pyramidal number by $\mbox{Pyr}_m(x)=(1/6)x(x+1)((m-2)x+5-m).$ In a previous paper written by Dujella, Győry and the speaker all solutions to the equation $\mbox{Pyr}_m(x)=y^2$ are found in positive integers $x$ and $y,$ for $3\leq m\leq 100$ with $m\neq 5.$ In this talk we consider the question of higher powers, and find all solutions to the equation $\mbox{Pyr}_m(x)=y^n$ in positive integers $x,y$ and $n,$ with $n\geq 3, 3\leq m\leq 50.$ We reduce the problem to a study of systems of binomial Thue equations, and use a combination of local arguments, the modular method via Frey curves, and bounds arising from linear forms in logarithms. Joint work with Andrej Dujella, Kálmán Győry and Philippe Michaud-Jacobs.
For access please contact the organizers (ntrg[at]science.unideb.hu).