2020/12/31 by Wataru Kai, Kai, Wataru, Masato Mimura +7 · 1 citation
Computer Science · Mathematics · #05C55 (Secondary) #11B30 (Primary) 11B25 #11H55 #11N05 #11R04 #Analytic Number Theory Research #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2012.15669
openalex publication_date 2020/12/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.