Speaker: Kyle Pratt (Brigham Young University)

Title: Diophantine problems with perfect powers

Abstract: Many Diophantine equations feature perfect powers in an essential way. For instance, the famous Catalan conjecture (proved by Mih$\breve{\text{a}}$ilescu) asserts that the only solution to $x^m-y^n=1$ in positive integers $x,y,m,n$ with $m,n \geq 2$ is $(x,y,m,n) = (3,2,2,3)$. In this talk, I discuss some recent work on other Diophantine problems involving perfect powers. In particular, I will focus on Lebesgue--Nagell equations, and a conjecture of Erd\H{o}s concerning when products of terms in an arithmetic progression can form a perfect power.