On the generalized Fermat equation of signature $(5,p,3)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912929176092672 |
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| author | Pacetti, Ariel Torcomian, Lucas Villagra |
| author_facet | Pacetti, Ariel Torcomian, Lucas Villagra |
| contents | In this article we study solutions to the generalized Fermat equation $x^q+y^p+z^r=0 $ using hypergeometric motives within the framework of the modular method. In doing so, we give an explicit description of the ramification behavior at primes dividing $2qr$ and analyze the contribution of trivial solutions. We identify a general obstruction to the modular method that accounts for its failure in many instances. As an application, assuming a standard large image conjecture, we prove that the previous equation admits no nontrivial primitive solutions $(a,b,c)$ with $3 \nmid c$, when $q=5,$ $r=3$ and $p$ is a prime sufficiently large. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_17845 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the generalized Fermat equation of signature $(5,p,3)$ Pacetti, Ariel Torcomian, Lucas Villagra Number Theory In this article we study solutions to the generalized Fermat equation $x^q+y^p+z^r=0 $ using hypergeometric motives within the framework of the modular method. In doing so, we give an explicit description of the ramification behavior at primes dividing $2qr$ and analyze the contribution of trivial solutions. We identify a general obstruction to the modular method that accounts for its failure in many instances. As an application, assuming a standard large image conjecture, we prove that the previous equation admits no nontrivial primitive solutions $(a,b,c)$ with $3 \nmid c$, when $q=5,$ $r=3$ and $p$ is a prime sufficiently large. |
| title | On the generalized Fermat equation of signature $(5,p,3)$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.17845 |