Timeline for A Diophantine equation with prime powers
Current License: CC BY-SA 3.0
10 events
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| Aug 30, 2015 at 2:32 | history | edited | knsam | CC BY-SA 3.0 |
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| May 20, 2015 at 10:34 | vote | accept | darya | ||
| May 19, 2015 at 16:55 | comment | added | Joe Silverman | The probability that $x$ is prime is $O(1/\log x)=O(1/n)$ and since $y\sim x$, the probability that $y$ is prime is the same. So the probability that $x$ and $y$ are both primes is $O(1/n^2)$. (Of course, I'm making some randomness and independence assumptions that can't be justified, but just to get an idea...) Summing, this suggests that there are only finitely many such solutions. | |
| May 19, 2015 at 16:54 | comment | added | GH from MO | @LinusHamilton: Your remark is very appropriate. Let me add that the sequence $2^n-1$ is very similar, e.g. it satisfies a similar recursion, and it contains infinitely many primes if and only there are infinitely many even perfect numbers. This is one of the oldest unsolved problems in mathematics. | |
| May 19, 2015 at 16:10 | comment | added | Linus Hamilton | This sequence is similar to Fibonacci, and we don't know whether there are infinitely many Fibonacci primes. And $n=8$ is another prime solution $x=2288805793$, $y=1321442641$, so any proof would need that as a special case. I don't know a strategy for solving this. | |
| May 19, 2015 at 16:09 | comment | added | knsam | I still don't see how to do this with "prime" condition. | |
| May 19, 2015 at 15:36 | history | edited | knsam | CC BY-SA 3.0 |
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| May 19, 2015 at 15:31 | comment | added | Geoff Robinson | In the Pell like case, how do you decide whether $x$ is prime or not, and if so, whether $y$ is also prime? | |
| May 19, 2015 at 15:06 | history | edited | knsam | CC BY-SA 3.0 |
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| May 19, 2015 at 14:16 | history | answered | knsam | CC BY-SA 3.0 |