On the complete solutions of a generalized Lebesgue-Ramanujan-Nagell equation
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866918170137198592 |
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| author | Chakraborty, Kalyan Hoque, Azizul |
| author_facet | Chakraborty, Kalyan Hoque, Azizul |
| contents | We consider the generalized Lebesgue-Ramanujan-Nagell equation $x^2+17^k41^\ell 59^m=2^δy^n$ in the unknown integers $x\geq 1, y>1,n\geq 3$ and $k, \ell, m\geq 0$ satisfying $\gcd(x,y)=1$. We first find all the integer solutions of the above equation, and then use this result to determine all the integer solutions of some other Lebesgue-Ramanujan-Nagell type equations. Our method uses the classical results of Bilu, Hanrot and Voutier on existence of primitive divisors of Lehmer sequences in combination with number theoretic arguments and computer search. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_02832 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the complete solutions of a generalized Lebesgue-Ramanujan-Nagell equation Chakraborty, Kalyan Hoque, Azizul Number Theory 11D61, 11D41, 11B39, 11Y50 We consider the generalized Lebesgue-Ramanujan-Nagell equation $x^2+17^k41^\ell 59^m=2^δy^n$ in the unknown integers $x\geq 1, y>1,n\geq 3$ and $k, \ell, m\geq 0$ satisfying $\gcd(x,y)=1$. We first find all the integer solutions of the above equation, and then use this result to determine all the integer solutions of some other Lebesgue-Ramanujan-Nagell type equations. Our method uses the classical results of Bilu, Hanrot and Voutier on existence of primitive divisors of Lehmer sequences in combination with number theoretic arguments and computer search. |
| title | On the complete solutions of a generalized Lebesgue-Ramanujan-Nagell equation |
| topic | Number Theory 11D61, 11D41, 11B39, 11Y50 |
| url | https://arxiv.org/abs/2105.02832 |