A set of natural numbers $\{a_1, a_2, \cdots, a_m\}$ is said to be a Diophantine $m$-tuple with property $D(n)$ if $a_ia_j +n$ is a perfect square for $i \neq j$. One may ask, what is the largest $m$ for which such a tuple exists. This problem has a long history, attracting the attention of many, including Fermat, Baker, Davenport etc, with significant progress made in recent times due to Dujella and others. In this talk, we consider a similar question by replacing the condition $a_ia_j+n$ from being a square to $k$-th powers. This is joint work with Ram Murty and Seoyoung Kim.

This talk will be accessible to graduate students!

The following link gives a quick introduction to the topic. It should help bring students up to speed on the history and recent developments on the problem:

https://web.math.pmf.unizg.hr/~duje/dtuples.html

Additional introductory reading material:

http://www.fields.utoronto.ca/sites/default/files/uploads/Paley%20graph_...

http://www.fields.utoronto.ca/sites/default/files/uploads/Diophantine%20...

http://www.fields.utoronto.ca/sites/default/files/uploads/Diophantine%20...