Timeline for Irrationality of $ \pi e, \pi^{\pi}$ and $e^{\pi^2}$
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 13, 2015 at 18:30 | comment | added | Fan Zheng | as well as in theory, which I of course believe. To put it more concretely, you need to compute at least $10^{17}$ digits of $\pi$ in order to tell if that number is an integer, and the best approximation of $\pi$ that I'm aware of gives $10^{12}$ digits. There's still someway to go. | |
| Nov 13, 2015 at 18:04 | comment | added | Fan Zheng | @VladimirReshetnikov OK, after putting it in wolframalpha, I realized like LTS above that it is much larger than I had thought... so your scenario may well be true, at least in practice. | |
| Nov 13, 2015 at 17:02 | comment | added | Vladimir Reshetnikov | @FanZheng It's still possible that none of the programs terminates, because of incompleteness (there are true statements without proofs). | |
| Nov 13, 2015 at 15:33 | comment | added | Fan Zheng | @VladimirReshetnikov In that case you can run two programs in parallel: one obtaining higher and higher precision of the said number, and the other enumerate all valid proofs and see if one is a proof that the said number is an integer. I bet the first program terminate first. | |
| Jan 31, 2015 at 21:07 | comment | added | Daniel McLaury | $\pi^{\pi^{\pi^{\pi}}}$ has over a hundred quadrillion digits. It would take more than two exabytes of storage just to write down the integer part of that number. | |
| Apr 21, 2014 at 20:49 | comment | added | user85798 |
@VladimirReshetnikov Oh, if it actually is an integer then of course this wouldn't work. I'm assuming it's not an integer. (I see no reason why we would get x.00000000000...)
|
|
| Apr 21, 2014 at 20:11 | comment | added | Vladimir Reshetnikov |
@Oliver Even if we had such a computer, and got a result like 435643…85685.00000000000…, how do we know if it is the exact integer, or we just need higher precision to discover non-zero fractional part?
|
|
| Apr 21, 2014 at 19:45 | comment | added | user85798 | ok, pi^pi^pi^pi is a hell of a lot bigger than I thought it was. Should still work though if you have a good computer and enough time. | |
| Apr 21, 2014 at 19:14 | comment | added | user85798 | @VladimirReshetnikov Then we would fail, and be unable to bound it between two consecutive integers. But that (probably) won't happen.. just get a computer to find a good enough upper and lower bound, which would count as a proof. | |
| Apr 21, 2014 at 18:09 | comment | added | Vladimir Reshetnikov | @Oliver But what if it is actually an integer? | |
| Apr 21, 2014 at 14:36 | comment | added | user85798 | Surely it can be proven that pi^pi^pi^pi is not an integer? Just bound it between two consecutive integers... It would be inelegant as factorial but it would work ? | |
| May 4, 2013 at 14:53 | vote | accept | Vladimir Reshetnikov | ||
| May 3, 2013 at 22:02 | comment | added | Oksana Gimmel | It is mentioned on the Russian Wikipedia page Open mathematical problems. A very similar question was discussed at math.stackexchange.com/questions/13050/eee79-and-ultrafinitism | |
| May 3, 2013 at 21:42 | comment | added | Peter LeFanu Lumsdaine | @Oksana Gimmel: very interesting! Can you suggest any references for reading on that last bit? (It’s rather difficult to search about!) | |
| May 3, 2013 at 20:54 | comment | added | Stefan Kohl♦ | You raise a nice question! (Though of course an answer 'yes' would be a lot nicer than 'no'!) | |
| May 3, 2013 at 19:58 | history | answered | Oksana Gimmel | CC BY-SA 3.0 |