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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2401.10930 |
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| _version_ | 1866914657042694144 |
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| author | Gica, Alexandru |
| author_facet | Gica, Alexandru |
| contents | The main purpose of this paper is to find all the prime numbers p for which whenever we add to p an odd square less than p we obtain a number which has at most two different prime factors. We solve completely the cases $p\equiv 1,3,5 \pmod 8$. The idea of the proof in these cases is to find the class number for the quadratic imaginary field $\mathbb{Q}(i\sqrt p)$. Since we know all these quadratic imaginary fields with class number 1, 2 or 4, we are able to solve these cases. The most interesting case is $p\equiv 7 \pmod 8$. We prove in this case that the Ono invariant of the field equals the class number. S. Louboutin succeeded to find all these fields, with one possible exception. Assuming a Restricted Riemann Hypothesis, the list of Louboutin is complete. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10930 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Class numbers, Ono invariants and some interesting primes Gica, Alexandru Number Theory The main purpose of this paper is to find all the prime numbers p for which whenever we add to p an odd square less than p we obtain a number which has at most two different prime factors. We solve completely the cases $p\equiv 1,3,5 \pmod 8$. The idea of the proof in these cases is to find the class number for the quadratic imaginary field $\mathbb{Q}(i\sqrt p)$. Since we know all these quadratic imaginary fields with class number 1, 2 or 4, we are able to solve these cases. The most interesting case is $p\equiv 7 \pmod 8$. We prove in this case that the Ono invariant of the field equals the class number. S. Louboutin succeeded to find all these fields, with one possible exception. Assuming a Restricted Riemann Hypothesis, the list of Louboutin is complete. |
| title | Class numbers, Ono invariants and some interesting primes |
| topic | Number Theory |
| url | https://arxiv.org/abs/2401.10930 |