The Paley graph conjecture and Diophantine m-tuples

Abstract

A Diophantine m-tuple with property D(n), where n is a non-zero integer, is a set of m positive integers {a1,...,am} such that aiaj+n is a perfect square for all 1⩽i<j⩽m. It is known that Mn=sup⁡{|S|:S is a D(n) m-tuple} exists and is O(log⁡|n|). In this paper, we show that the Paley graph conjecture implies that the upper bound can be improved to ≪(log⁡|n|)ϵ, for any ϵ>0.