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

TL;DR

This paper improves the upper bounds on the size of Diophantine tuples with property D(n), showing the contribution of intermediate elements is logarithmic in the logarithm of |n|, thus tightening previous bounds.

Contribution

It establishes a tighter upper bound on the maximum size of D(n) Diophantine tuples, improving previous results by analyzing the contribution of intermediate elements.

Findings

Intermediate elements contribute O(log log |n|) to the size.

Upper bound on M_n is improved to (2+o(1)) log |n|.

Enhanced bounds refine understanding of Diophantine tuples with 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 $a \neq b$. It is of special interest to estimate the quantity $M_n$, the maximum size of a Diophantine tuple with the property $D(n)$. 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 $M_n\leq (2+o(1))\log |n|$, improving the best-known upper bound on $M_n$ by Becker and Murty.