Diophantine tuples over $\mathbb {Z}_p$
Volume 197 / 2021
Acta Arithmetica 197 (2021), 331-351
MSC: Primary 11D88; Secondary 11D45, 11D72.
DOI: 10.4064/aa190331-15-3
Published online: 30 December 2020
Abstract
For an element $r$ of a ring $R$, a $D(r)$ $m$-tuple is an $m$-tuple $(a_1,\ldots ,a_m)$ of elements of $R$ such that for all $i,j$ with $i\neq j$, $a_ia_j+r$ is a perfect square in $R$. In this article, we compute and estimate the measures of the sets of $D(r)$ $m$-tuples in the ring $\mathbb {Z}_p$ of $p$-adic integers, as well as in its residue field $\mathbb F _p$.
