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

Abstract

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 $a,b\in S$ with $a\ne b$. It is of special interest to estimate the quantity $M_n$, 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 $M_n\le (2+o(1))\log |n|$, improving the best-known upper bound on $M_n$ by Becker and Murty.

4 pages, revised based on referee comments

References in corpus (1)

• There is no Diophantine $D(-1)$--quadruple

Cited by in corpus (2)

• Multiplicative structure of shifted multiplicative subgroups and its applications to Diophantine tuples
• Bipartite Diophantine tuples and their applications