Primes of the form $p^2 + nq^2$ (2410.04189v2)

Abstract: Suppose that $n$ is $0$ or $4$ modulo $6$. We show that there are infinitely many primes of the form $p^2 + nq^2$ with both $p$ and $q$ prime, and obtain an asymptotic for their number. In particular, when $n = 4$ we verify the `Gaussian primes conjecture' of Friedlander and Iwaniec. We study the problem using the method of Type I/II sums in the number field $\mathbf{Q}(\sqrt{-n})$. The main innovation is in the treatment of the Type II sums, where we make heavy use of two recent developments in the theory of Gowers norms in additive combinatorics: quantitative versions of so-called concatenation theorems, due to Kuca and to Kuca--Kravitz-Leng, and the quasipolynomial inverse theorem of Leng, Sah and the second author.

• The paper establishes an asymptotic formula for counting primes of the form p²+nq², confirming a variant of the Gaussian primes conjecture.
• It applies advanced analytic number theory and Gowers norms to investigate the distribution of primes in quadratic forms.
• The study opens new pathways for combining combinatorial techniques with classical methods in addressing prime distribution in number fields.

Primes of the form $p^2 + nq^2$

The paper by Ben Green and Mehtaab Sawhney addresses a classical question in number theory concerning the distribution of primes. Specifically, it investigates primes of the form $p^2 + nq^2$ where both $p$ and $q$ are prime numbers. Their results affirmatively resolve a conjecture involving Gaussian primes, a foundational topic linked to quadratic forms and number fields.

Main Results

The paper establishes an asymptotic formula for counting pairs of primes $x$ and $y$ such that $x^2 + ny^2$ is prime. The primary theorem indicates that for $n\equiv 0$ or $n\equiv 4 \mod 6$, there are infinitely many primes of the form $x^2 + ny^2$. In particular, when $n = 4$, this confirms a variant of the Gaussian primes conjecture posited by Friedlander and Iwaniec.

Methodological Framework

The authors employ advanced techniques from analytic number theory, leveraging tools such as the method of Type I/II sums over quadratic number fields. A key innovation lies in their treatment of Type II sums, which involve substantial use of recent advances in additive combinatorics, specifically Gowers norms. These norms, originally developed to paper uniformity and distribution in arithmetic progressions, are deftly applied here to parse the distribution of Gaussian primes.

The utilization of developments in Gowers norm theory is notable. The authors employ quantitative versions of concatenation theorems and exploit the quasipolynomial inverse theorem, which further extend the reach of these analytical methods into new domains.

Implications and Further Directions

The implications of these results are twofold. Practically, they advance the understanding of prime distributions in quadratic forms, a topic with deep historical roots. Theoretically, they open pathways to applying analytic number theory in concert with combinatorial techniques, hinting at potential further breakthroughs in both understanding and application of primes in algebraic number fields.

Moreover, this work may influence future research into related areas, such as the problem of finding other polynomial classes generating primes or enhancing sieve methods applicable in higher-dimensional cases or other rings of integers.

Conclusion

Green and Sawhney's exploration into primes defined by specific quadratic forms represents a significant stride in number theory. This marriage of classical analytic techniques with modern combinatorial insights not only resolves longstanding conjectures but also enriches the methodological toolkit available to mathematicians investigating prime distributions in various algebraic structures. These techniques promise to be applicable to a broad class of problems involving polynomial-defined integers in number fields.