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arXiv · 1902.09048

Diophantine Tuples over $\mathbb{Z}_p$

Nitya Mani·Simon Rubinstein-Salzedo
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Abstract

For an element $r$ of a ring $R$, a Diophantine $D(r)$ $m$-tuple is an $m$-tuple $(a_1,a_2,\ldots,a_m)$ 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 its residue field $\mathbb{F}_p$.

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Nitya Mani, Simon Rubinstein-Salzedo. 2019-02-25. Diophantine Tuples over $\mathbb{Z}_p$. https://arxiv.org/abs/1902.09048

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