## Number Theory Seminar

### Thursday, October 10th, 2019

**Time:** 4:30-5:30 p.m. **Place:** Jeffery Hall 422

**Speaker:** M. Ram Murty (Queen's University)

**Title:** THE PALEY GRAPH CONJECTURE AND DIOPHANTINE TUPLES.

**Abstract:** Let $n$ be a fixed natural number. An $m$-tuple $(a(1), ..., a(m))$ is said to be a Diophantine $m$-tuple with property $D(n)$ if $a(i)a(j)+n$ is a perfect square for $i, j$ distinct and less than or equal to $m$. We will show that the Paley graph conjecture in graph theory implies that the number of such tuples is $O((log n)^c)$ for any $c>0$. This is joint work with Ahmet Guloglu.