Timeline for Does this surface contain all perfect cuboids?
Current License: CC BY-SA 3.0
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| Aug 28, 2011 at 6:10 | vote | accept | joro | ||
| Aug 28, 2011 at 6:08 | comment | added | joro | @John Thanks. I am familiar with Sharipov's paper. btw, rational points on his surface do not necessary produce perfect cuboids, there are additional restrictions. Check the conditions in Theorem 5.3 and "...within the open domain $D_{ab}$" Brute force search on Sharipov's surface produced many nontrivial points, but none satisfying the additional conditions. | |
| Aug 27, 2011 at 21:57 | comment | added | Noam D. Elkies | Careful: it is not true that a surface of general type always has only finitely many rational points, because it can have rational or elliptic curves that have infinitely many rational points. (Indeed this happens here with the degenerate cuboids having a "side" of length zero.) The Lang-Bombieri conjecture is that all but finitely many points are on such curves, but nobody has any idea how to prove it. Also, reducing modulo $N$ can't work, at least not by itself, because that will not distinguish trivial points (with a zero side) from nontrivial ones. | |
| Aug 27, 2011 at 20:57 | history | edited | John R Ramsden | CC BY-SA 3.0 |
added 31 characters in body; added 7 characters in body
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| Aug 27, 2011 at 20:28 | history | answered | John R Ramsden | CC BY-SA 3.0 |