Timeline for Variation of Fermat's Last Theorem for n = 5
Current License: CC BY-SA 3.0
6 events
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| Aug 9, 2011 at 18:15 | comment | added | Noam D. Elkies | @Martin: you're welcome. I'm afraid the short answer to your next question is that it's exceedingly unlikely that there's a nontrivial rational solution (unlike the fourth-power case, where it was actually likely a priori that there would be infinitely many primitive solutions), but we have no techniques to prove such a result. I might post an answer later detailing why, but have some things to tend to first before dinner... | |
| Aug 9, 2011 at 17:58 | comment | added | Martin | Thank you for your answer Noam. Could you imagine there is some astronomical solution like the one you gave for the fourth powers or now it is more reasonable to look for a non-existence proof? | |
| Aug 9, 2011 at 17:18 | comment | added | Noam D. Elkies | It's also been known for some time that there are number fields $K$, such as ${\bf Q}(i)$, over which there are infinitely many nontrivial solutions. The nice way to see this is to intersect the quintic projective surface $w^5+x^5+y^5+z^5=0$ with the plane $w+x+y+z=0$: the resulting quintic curve contains the three lines $w+x=y+z=0$, $w+y=x+z=0$, $w+z=x+y=0$, and thus a residual conic which contains nontrivial rational points. Alas this conic has no real points, let alone rational ones, so a necessary condition on $K$ for this construction to work over $K$ is that $K$ be totally imaginary. | |
| Aug 9, 2011 at 12:51 | comment | added | Gerry Myerson | Many solutions exist, but all the known ones are trivial, e.g., $(x,y,z,t)=(17,42,-17,42)$. | |
| Aug 9, 2011 at 9:28 | answer | added | Koen S | timeline score: 4 | |
| Aug 9, 2011 at 8:44 | history | asked | Martin | CC BY-SA 3.0 |