Timeline for Does $2^x-3p^y=5$ (with $p$ an odd prime) have only finitely many positive integer solutions?
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| Sep 4, 2018 at 15:21 | comment | added | Mike Bennett | One should be able to handle the more general equation $2^x-3z^y=5$ by appealing to lower bounds for linear forms in 2-adic logs to bound $x$ and $y$ and then using Frey curves ( a la Kraus) to knock off the remaining values of $y$ (likely one would have to do the cases where $2$ or $3$ divide $y$ separately via integral points on elliptic curve arguments or something similar). The most obvious couple of Frey curves give you congruences to forms of level $30$. The details shouldn't be too bad, but I'm not sure my claim follows explicitly from anything currently in the literature. | |
| Sep 4, 2018 at 3:18 | comment | added | GH from MO | @MikeBennett: Very interesting. Can you give some details or references for your claim? | |
| Sep 3, 2018 at 22:34 | comment | added | Mike Bennett | One can show that there are only two solutions with $y > 1$ (corresponding to $x=5$ and $x=9$), for arbitrary $p$. Presumably, there are infinitely many $p$ of the shape $(2^x-5)/3$. | |
| Sep 2, 2018 at 7:47 | history | edited | GH from MO | CC BY-SA 4.0 |
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| Sep 2, 2018 at 5:55 | comment | added | KConrad | The book you mention did not have Lang as a co-author. | |
| Sep 2, 2018 at 5:48 | vote | accept | qrilove | ||
| Sep 2, 2018 at 2:46 | history | edited | GH from MO | CC BY-SA 4.0 |
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| Sep 2, 2018 at 2:26 | history | edited | GH from MO | CC BY-SA 4.0 |
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| Sep 2, 2018 at 2:20 | history | edited | GH from MO | CC BY-SA 4.0 |
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| Sep 2, 2018 at 2:05 | history | edited | GH from MO | CC BY-SA 4.0 |
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| Sep 2, 2018 at 1:57 | history | answered | GH from MO | CC BY-SA 4.0 |