Primes of the form $p^2 + nq^2$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912071236452352 |
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| author | Green, Ben Sawhney, Mehtaab |
| author_facet | Green, Ben Sawhney, Mehtaab |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04189 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Primes of the form $p^2 + nq^2$ Green, Ben Sawhney, Mehtaab Number Theory Combinatorics 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. |
| title | Primes of the form $p^2 + nq^2$ |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2410.04189 |