SFU Number Theory and Algebraic Geometry Seminar: Chi Hoi Yip

A set $\{a_{1}, a_{2},\ldots, a_{m}\}$ 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 $n \ge 1$ and $k \ge 2$, 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 $D_{k}(n)$. Using various tools from number theory, we show that there is $k=k(n)$ such that $k,n \to \infty$ and $M_k(n)=o(\log n)$, breaking the $\log n$ barrier. A key ingredient is to study the finite field model of the same problem. Joint work with Seoyoung Kim and Semin Yoo.