Constellations in prime elements of number fields (2012.15669v2)

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.

• The paper proves that constellations of arbitrary shape, composed of prime elements, exist within the ring of integers of any number field, extending the Green-Tao theorem.
• The authors address challenges like non-unique prime factorization by formulating the constellation theorem in terms of ideals rather than elements in the number field.
• This research deepens the connection between additive number theory and algebraic number theory, with potential implications for cryptography and computational number theory.

Analysis of "Constellations in Prime Elements of Number Fields"

This essay provides an analysis of the paper authored by Wataru Kai, Masato Mimura, Akihiro Munemasa, Shin-ichiro Seki, and Kiyoto Yoshino, focused on the paper of constellations within prime elements of number fields. Given the mathematical depth and complexity, the paper makes significant strides in extending previous results in the field of additive number theory, particularly those surrounding prime constellations.

Theorem and Its Foundations

The central result of the paper is the demonstration that, for any number field $K$, there exist constellations of arbitrary shape in the ring of integers $O_K$, comprising prime elements of $O_K$. This theorem broadens the scope of the seminal Green--Tao theorem, which assures the existence of arbitrary-length arithmetic progressions of primes. It also extends Tao's work on Gaussian primes.

The authors detail the concept of an $S$-constellation—a generalization involving shapes derived from subsets of a $\mathbb{Z}$-module—and prove the existence of such constellations within prime elements of number fields beyond the standard integers. This approach involves constructing these configurations in the ring of integers $O_K$, which itself presents unique challenges due to the properties of different number fields.

Novel Contributions and Methods

One of the methodological novelties presented in this paper is the formulation of the constellation theorem in terms of ideals rather than elements, which facilitates handling difficulties such as non-uniqueness of prime factorization in number fields with class numbers greater than one. The paper systematically navigates these challenges by employing the framework of Dedekind domains and ideal theory.

The work also introduces a significant extension to quadratic forms. It is shown that for any primitive, indefinite binary quadratic form $F(x, y) = ax^2 + bxy + cy^2$, constellations can be formed such that $F(x, y)$ takes distinct prime values at each point of the constellation. This is a non-trivial application of extending the constellation theorem to binary quadratic forms, conveying both the diversity and applicability of the method.

Implications and Future Directions

This research outputs rich implications for both theoretical and practical applications. Theoretically, it deepens the connection between additive number theory and algebraic structures intrinsic to number fields. It combines ideas from algebra, combinatorics, and number theory to explore new frontiers in finding prime constellations across varied numerical settings.

Practically, the results offer potential insights into cryptography and computational number theory, where understanding the distribution of prime numbers in different algebraic structures plays a fundamental role.

Conclusion

Overall, the paper by Kai, Mimura, Munemasa, Seki, and Yoshino offers a detailed and rigorous enhancement of the understanding of prime constellations within number fields, successfully extending foundational theorems. Their work paves the way for further exploration of prime patterns across mathematical structures, suggesting intriguing directions for future research in computational and theoretical avenues related to primes in algebraic settings.