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arXiv · 2012.15669

Constellations in prime elements of number fields

Wataru Kai·Masato Mimura·Akihiro Munemasa·Shin-ichiro Seki·Kiyoto Yoshino
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Abstract

Given any number field, we prove that there exist arbitrarily shaped constellations consisting of pairwise non-associate prime elements of the ring of integers. This result extends the celebrated Green-Tao theorem on arithmetic progressions of rational primes and Tao's theorem on constellations of Gaussian primes. Furthermore, we prove a constellation theorem on prime representations of binary quadratic forms with integer coefficients. More precisely, for a non-degenerate primitive binary quadratic form $F$ which is not negative definite, there exist arbitrarily shaped constellations consisting of pairs of integers $(x,y)$ for which $F(x,y)$ is a rational prime. The latter theorem is obtained by extending the framework from the ring of integers to the pair of an order and its invertible fractional ideal.

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Wataru Kai, Masato Mimura, Akihiro Munemasa, Shin-ichiro Seki, Kiyoto Yoshino. 2022-04-04. Constellations in prime elements of number fields. https://arxiv.org/abs/2012.15669

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