Primes represented by shifted quadratic forms: on primitivity and congruence classes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915265602650112 |
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| author | Fuchs, Elena Hsu, Catherine Rickards, James Schindler, Damaris Stange, Katherine E. |
| author_facet | Fuchs, Elena Hsu, Catherine Rickards, James Schindler, Damaris Stange, Katherine E. |
| contents | We prove lower bounds of the form $\gg N/(\log N)^{3/2}$ for the number of primes up to $N$ primitively represented by a shifted positive definite integral binary quadratic form, and under the additional condition that primes are from an arithmetic progression. This extends the sieve methods of Iwaniec, who showed such lower bounds without the primitivity and congruence conditions. Imposing primitivity adds some subtleties to the local criteria for representation of a shifted prime: for example, some shifted quadratic forms of discriminant $5 \pmod{8}$ do not primitively represent infinitely many primes. We also provide a careful list of the local conditions under which a genus of an integral binary quadratic form represents an integer, verified by computer, and correcting some minor errors in previous statements. The motivation for this work is as a tool for the study of prime components in Apollonian circle packings [FFH+24] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20289 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Primes represented by shifted quadratic forms: on primitivity and congruence classes Fuchs, Elena Hsu, Catherine Rickards, James Schindler, Damaris Stange, Katherine E. Number Theory 11N32, 11D09, 11E12, 11N36, 52C26 We prove lower bounds of the form $\gg N/(\log N)^{3/2}$ for the number of primes up to $N$ primitively represented by a shifted positive definite integral binary quadratic form, and under the additional condition that primes are from an arithmetic progression. This extends the sieve methods of Iwaniec, who showed such lower bounds without the primitivity and congruence conditions. Imposing primitivity adds some subtleties to the local criteria for representation of a shifted prime: for example, some shifted quadratic forms of discriminant $5 \pmod{8}$ do not primitively represent infinitely many primes. We also provide a careful list of the local conditions under which a genus of an integral binary quadratic form represents an integer, verified by computer, and correcting some minor errors in previous statements. The motivation for this work is as a tool for the study of prime components in Apollonian circle packings [FFH+24] |
| title | Primes represented by shifted quadratic forms: on primitivity and congruence classes |
| topic | Number Theory 11N32, 11D09, 11E12, 11N36, 52C26 |
| url | https://arxiv.org/abs/2504.20289 |