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.
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