Complete solutions of a Lebesgue-Ramanujan-Nagell type equation
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866916136183922688 |
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| author | Baruah, Priyanka Das, Anup Hoque, Azizul |
| author_facet | Baruah, Priyanka Das, Anup Hoque, Azizul |
| contents | We consider the Lebesgue-Ramanujan-Nagell type equation $x^2+5^a13^b17^c=2^m y^n$, where $a,b,c, m\geq 0, n \geq 3$ and $x, y\geq 1$ are unknown integers with $\gcd(x,y)=1$. We determine all integer solutions to the above equation. The proof depends on the classical results of Bilu, Hanrot and Voutier on primitive divisors in Lehmer sequences, and finding all $S$-integral points on a class of elliptic curves. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2104_12680 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Complete solutions of a Lebesgue-Ramanujan-Nagell type equation Baruah, Priyanka Das, Anup Hoque, Azizul Number Theory 11D61, 11D41, 11Y50 We consider the Lebesgue-Ramanujan-Nagell type equation $x^2+5^a13^b17^c=2^m y^n$, where $a,b,c, m\geq 0, n \geq 3$ and $x, y\geq 1$ are unknown integers with $\gcd(x,y)=1$. We determine all integer solutions to the above equation. The proof depends on the classical results of Bilu, Hanrot and Voutier on primitive divisors in Lehmer sequences, and finding all $S$-integral points on a class of elliptic curves. |
| title | Complete solutions of a Lebesgue-Ramanujan-Nagell type equation |
| topic | Number Theory 11D61, 11D41, 11Y50 |
| url | https://arxiv.org/abs/2104.12680 |