IMPROVED UPPER BOUNDS ON DIOPHANTINE TUPLES WITH THE PROPERTY $D(n)$

Abstract Let n be a nonzero 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 $a,b \in S$ with $a \neq b$ . It is of special interest to estimate the quantity $M_n$ , the maximum size of a Diophantine tuple with the property $D(n)$ . We show the contribution of intermediate elements is $O(\log \log |n|)$ , improving a result by Dujella [‘Bounds for the size of sets with the property $D(n)$ ’, Glas. Mat. Ser. III 39 (59)(2) (2004), 199–205]. As a consequence, we deduce that $M_n\leq (2+o(1))\log |n|$ , improving the best known upper bound on $M_n$ by Becker and Murty [‘Diophantine m -tuples with the property $D(n)$ ’, Glas. Mat. Ser. III 54 (74)(1) (2019), 65–75].