On the size of Diophantine $m$-tuples in imaginary quadratic number rings

Nikola Adžaga

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arXiv · 1807.01896

On the size of Diophantine $m$-tuples in imaginary quadratic number rings

Nikola Adžaga

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Abstract

A Diophantine $m$-tuple is a set of $m$ distinct integers such that the product of any two distinct elements plus one is a perfect square. It was recently proven that there is no Diophantine quintuple in positive integers. We study the same problem in the rings of integers of imaginary quadratic fields. By using a gap principle proven by Diophantine approximations, we show that $m\leqslant 42$. Our proof is relatively simple compared to the proofs of the similar results in positive integers.

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Nikola Adžaga. 2018-07-05. On the size of Diophantine $m$-tuples in imaginary quadratic number rings. https://arxiv.org/abs/1807.01896

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