A multi-Frey approach to Fermat equations of signature $(r,r,p)$
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2017
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| _version_ | 1866913419468210176 |
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| author | Billerey, Nicolas Chen, Imin Dieulefait, Luis Freitas, Nuno |
| author_facet | Billerey, Nicolas Chen, Imin Dieulefait, Luis Freitas, Nuno |
| contents | In this paper, we give a resolution of the generalized Fermat equations $$x^5 + y^5 = 3 z^n \text{ and } x^{13} + y^{13} = 3 z^n,$$ for all integers $n \ge 2$, and all integers $n \ge 2$ which are not a multiple of $7$, respectively, using the modular method with Frey elliptic curves over totally real fields. The results require a refined application of the multi-Frey technique, which we show to be effective in new ways to reduce the bounds on the exponents $n$.
We also give a number of results for the equations $x^5 + y^5 = d z^n$, where $d = 1, 2$, under additional local conditions on the solutions. This includes a result which is reminiscent of the second case of Fermat's Last Theorem, and which uses a new application of level raising at $p$ modulo $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1703_06530 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | A multi-Frey approach to Fermat equations of signature $(r,r,p)$ Billerey, Nicolas Chen, Imin Dieulefait, Luis Freitas, Nuno Number Theory 11D41, 11F80, 11G05 In this paper, we give a resolution of the generalized Fermat equations $$x^5 + y^5 = 3 z^n \text{ and } x^{13} + y^{13} = 3 z^n,$$ for all integers $n \ge 2$, and all integers $n \ge 2$ which are not a multiple of $7$, respectively, using the modular method with Frey elliptic curves over totally real fields. The results require a refined application of the multi-Frey technique, which we show to be effective in new ways to reduce the bounds on the exponents $n$. We also give a number of results for the equations $x^5 + y^5 = d z^n$, where $d = 1, 2$, under additional local conditions on the solutions. This includes a result which is reminiscent of the second case of Fermat's Last Theorem, and which uses a new application of level raising at $p$ modulo $p$. |
| title | A multi-Frey approach to Fermat equations of signature $(r,r,p)$ |
| topic | Number Theory 11D41, 11F80, 11G05 |
| url | https://arxiv.org/abs/1703.06530 |