On the sum of fifth powers in arithmetic progression
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914741157363712 |
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| author | Torcomian, Lucas Villagra |
| author_facet | Torcomian, Lucas Villagra |
| contents | In this paper we study equation $$(x-dr)^5+\cdots+x^5+\cdots+(x+dr)^5=y^p$$ under the condition $\gcd(x,r)=1$. We present a recipe for proving the non-existence of non-trivial integer solutions of the above equation, and as an application we obtain explicit results for the cases $d=2,3$ (the case $d=1$ was already solved). We also prove an asymptotic result for $d\equiv 1, 7\pmod9$. Our main tools include the modular method, employing Frey curves and their associated modular forms, as well as the symplectic argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_03457 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the sum of fifth powers in arithmetic progression Torcomian, Lucas Villagra Number Theory In this paper we study equation $$(x-dr)^5+\cdots+x^5+\cdots+(x+dr)^5=y^p$$ under the condition $\gcd(x,r)=1$. We present a recipe for proving the non-existence of non-trivial integer solutions of the above equation, and as an application we obtain explicit results for the cases $d=2,3$ (the case $d=1$ was already solved). We also prove an asymptotic result for $d\equiv 1, 7\pmod9$. Our main tools include the modular method, employing Frey curves and their associated modular forms, as well as the symplectic argument. |
| title | On the sum of fifth powers in arithmetic progression |
| topic | Number Theory |
| url | https://arxiv.org/abs/2404.03457 |