Prime-power Diophantine tuples
Abstract
A positive D(n)-m-tuple is a set A={a1,…,am} of distinct positive integers such that ai aj+n is a square for every i≠ j. In 2005, Dujella and Luca obtained an absolute bound for the cardinality of a D(p)- or D(−p)-tuple of positive integers, uniformly in the prime p. We extend the underlying gap argument to prime powers. An explicit toric elimination certificate, proved by elementary linear algebra, replaces the unsaturated homogeneous elimination step and is valid modulo every prime power. Explicit degree and height bounds for this eliminant yield a uniform gap principle. Combined with the general bound for bounded |n|, this shows that every positive reduced D(± pr)-tuple (i.e. tuple with elements not divisible by p) has less than 2121 elements, independently of p and r. Thus, every positive D(± p)-tuple has at most 2121 elements, positive D(± p2)-tuples have less than 2122 elements, and the maximal cardinality of a positive D(± pr)-tuple is O(r) uniformly in p.
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BibTeX
@misc{dujella2026primepower,
title = {{Prime-power Diophantine tuples}},
author = {Andrej Dujella},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03448},
eprint = {2609.03448},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03448},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Prime-power Diophantine tuples
AU - Andrej Dujella
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03448
UR - https://arxiv.org/abs/2609.03448
AB - A positive D(n)-m-tuple is a set A={a_1,…,a_m} of distinct positive integers such that a_i a_j+n is a square for every i≠ j. In 2005, Dujella and Luca obtained an absolute bound for the cardinality of a D(p)- or D(−p)-tuple of positive integers, uniformly in the prime p. We extend the underlying gap argument to prime powers. An explicit toric elimination certificate, proved by elementary linear algebra, replaces the unsaturated homogeneous elimination step and is valid modulo every prime power. Explicit degree and height bounds for this eliminant yield a uniform gap principle. Combined with the general bound for bounded |n|, this shows that every positive reduced D(± p^r)-tuple (i.e. tuple with elements not divisible by p) has less than 2^121 elements, independently of p and r. Thus, every positive D(± p)-tuple has at most 2^121 elements, positive D(± p^2)-tuples have less than 2^122 elements, and the maximal cardinality of a positive D(± p^r)-tuple is O(r) uniformly in p.
ER -