1 results
IMPROVED UPPER BOUNDS ON DIOPHANTINE TUPLES WITH THE PROPERTY
$D(n)$
- Part of
-
- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 08 October 2024, pp. 1-5
-
- Article
-
- You have access
- HTML
- Export citation
-
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].
