Constellations in prime elements of number fields

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.

Cryptanalysis of the quantum public-key cryptosystem OTU under heuristics from combinatorial statements

“…For this direction, a large generalization of Green-Tao theorem is appeared in preprint [18], recently. In [18], they focus on the two types of generalizations. One is a generalization of arithmetic progressions.…”

“…As a "relative" Szemerédi's theorem, Green-Tao theorem [14] is well known, This theorem states that there exists an arbitrarily long arithmetic progression in the set of prime numbers. For this direction, a large generalization of Green-Tao theorem is appeared in preprint [18], recently. In [18], they focus on the two types of generalizations.…”