Let n be a non-zero integer. A set of m positive integers {a 1, a 2, ... , a(m)} such that a(i)a(j) + n is a perfect square for all 1 <=3D i < j <=3D m is called a Diophantine m-tuple with the property D(n). In a series of papers, Dujella studied the quantity M-n =3D sup {vertical bar S vertical bar : S has the property D(n)} and showed for vertical bar n vertical bar >=3D 400 that M-n <=3D 15.476 log vertical bar n vertical bar and if vertical bar n vertical bar > 10(100), then M-n < 9.078 log vertical bar n vertical bar. We refine his argument to show that C-n <=3D 2 log vertical bar n vertical bar + O (log vertical bar n vertical bar/(log log vertical bar n vertical bar)(2)), where the implied constant is effectively computable and C-n =3D sup {|S boolean AND [1, n(2)]vertical bar : S has the property D(n)}. Together with earlier work of Dujella, this implies M-n <=3D 2.6071 log vertical bar n vertical bar + O (log vertical bar n vertical bar/(log log vertical bar n vertical bar(2)), where the implied constant is effectively computable.
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