Improved upper bounds on Diophantine tuples with the property D (n)

Let n be a non-zero integer. A set S of positive integers is a Diophantine tuple with the property D (n) if ab+n is a perfect square for each a, b S with a b. It is of special interest to estimate the quantity Mₙ, the maximum size of a Diophantine tuple with the property D (n). In this notes, we show the contribution of intermediate elements is O (|n|), improving a result by Dujella. As a consequence, we deduce that Mₙ (2+o (1) ) |n|, improving the best-known upper bound on Mₙ by Becker and Murty.