Diophantine tuples over $\mathbb {Z}_p$

Nitya Mani; Simon Rubinstein-Salzedo

doi:10.4064/aa190331-15-3

Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

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Diophantine tuples over $\mathbb {Z}_p$

Volume 197 / 2021

Nitya Mani, Simon Rubinstein-Salzedo 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 $\mathbb F _p$ .

Authors

Nitya ManiDepartment of Mathematics
Stanford University
531 Lasuen Mall
Stanford, CA 94305, U.S.A.
e-mail
Simon Rubinstein-SalzedoEuler Circle
Palo Alto, CA 94306, U.S.A.
e-mail

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Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

Acta Arithmetica

Diophantine tuples over \mathbb {Z}_p

Volume 197 / 2021

Abstract

Authors

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Diophantine tuples over $\mathbb {Z}_p$