MATH Seminar

Title: Diophantine tuples over integers and finite fields
Seminar: Combinatorics
Speaker: Kyle Yip, PhD of Georgia Institute of Technology
Contact: Dr. Cosmin Pohoata, apohoat@emory.edu
Date: 2024-09-06 at 10:00AM
Venue: MSC N306
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Abstract: 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 $D_{k}(n)$. In this talk, I will present improved upper bounds on $M_k(n)$. I will also discuss the analogue of Diophantine tuples over finite fields, which is of independent interest. Joint work with Seoyoung Kim and Semin Yoo.