SFU Number Theory and Algebraic Geometry Seminar: Kyle Yip

A set of positive integers is called a Diophantine tuple if the product of any two distinct elements in the set is one less than a square. There is a long history and extensive literature on the study of Diophantine tuples and their generalizations in various settings. In this talk, we focus on the following generalization: for integers $n \neq 0$ and $k \ge 3$, we call a set of positive integers a Diophantine tuple with property $D_{k}(n)$ if the product of any two distinct elements is $n$ less than a $k$-th power, and we denote $M_k(n)$ be the largest size of a Diophantine tuple with property $D_{k}(n)$. I will present an improved upper bound on $M_k(n)$ and discuss its bipartite analogue (where we have a pair of sets instead of a single set). Joint work with Seoyoung Kim and Semin Yoo.