### Riley Becker and M. Ram Murty

Department of Mathematics, Queen's University, Kingston, Ontario, K7L 3N6, Canada
e-mail: rileydbecker@gmail.com
e-mail: murty@queensu.ca

Abstract.   Let n be a non-zero integer. A set of m positive integers { a1,a2,⋯ ,am} such that aiaj+n is a perfect square for all 1≤ i < j≤ m is called a Diophantine m-tuple with the property D(n). In a series of papers, Dujella studied the quantity Mn= sup {|𝒮|: 𝒮 has the property D(n)} and showed for |n|≥ 400 that Mn ≤ 15.476 log |n| and if |n| >10100, then Mn < 9.078 log |n|. We refine his argument to show that Cn≤ 2log |n|+ O(log |n|/(log log |n|)2), where the implied constant is effectively computable and Cn = sup {|𝒮 ∩ [1,n2]|:𝒮 has the property D(n)}. Together with earlier work of Dujella, this implies Mn≤ 2.6071 log |n|+ O(log |n|/ (log log |n|)2), where the implied constant is effectively computable.

2010 Mathematics Subject Classification.   11D25, 11N36

Key words and phrases.   Diophantine m-tuples, Gallagher's sieve, Vinogradov's inequality

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