Diophantine m-tuples with the property D(n)

Sa？etak 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 {|????|: ???? 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 {|???? ∩ [1,n2]|:???? 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