UBC Discrete Math Seminar: Kyle Yip

A set {a1,a2,…,am} of distinct positive integers is a Diophantine m-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 each n≥1 and k≥2, we call a set of positive integers a Diophantine tuple with property Dk(n) if the product of any two distinct elements is n less than a k-th power, and we denote Mk(n) be the largest size of a Diophantine tuple with property Dk(n). Using various tools from number theory, we show that there is k=k(n) such that k,n→∞ and Mk(n)=o(logn), breaking the logn barrier. A key ingredient is to study the finite field model of the same problem. Joint work with Seoyoung Kim and Semin Yoo.