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(log log |n|), improving a result by Dujella. As a consequence, we deduce that Mₙ≤ (2+o(1))log |n|, improving the best-known upper bound on Mₙ by Becker and Murty.