Improved upper bounds on Diophantine tuples with the property $D(n)$
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916733529358336 |
|---|---|
| author | Yip, Chi Hoi |
| author_facet | Yip, Chi Hoi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00840 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improved upper bounds on Diophantine tuples with the property $D(n)$ Yip, Chi Hoi Number Theory 11D09, 11D45 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. |
| title | Improved upper bounds on Diophantine tuples with the property $D(n)$ |
| topic | Number Theory 11D09, 11D45 |
| url | https://arxiv.org/abs/2406.00840 |