Timeline for Find all solution $a,b,c$ with $(1-a^2)(1-b^2)(1-c^2)=8abc$
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Jun 11, 2015 at 15:48 | comment | added | Allan MacLeod | I can assure you that no offence was meant, if I guess correctly what "snarky" means - never heard the word before! I was just trying to point out that, to experts such as yourself and Noam Elkies, K3 Surfaces are simple objects, but not perhaps to others. | |
| Jun 11, 2015 at 14:05 | comment | added | Joe Silverman | ... method to find lots of solutions involves the group law on elliptic curves, while the solution I described, albeit without working out the explicit equations, involves solving quadratic polynomials. So despite my admiration for the theory of elliptic curves, I think it would actually be easier to explain the quadratic equation method to, say, a high school student. OTOH, the rational curves you've found are completely explicit, which is even better. | |
| Jun 11, 2015 at 13:59 | comment | added | Joe Silverman | "I am sure that Noam Elkies and Joe Silverman feel their answers are extremely simple. The following discussion is, in my humble opinion, simpler." Hi Allan, maybe that's a bit snarky? In any case, your solution and my solution are complementary. You've explained how to transform the equation into an explicit elliptic surface, which is great. I explained a very general procedure for using one solution to find lots of others. It is also very simple, since it comes down to the fact that if you know one solution of a quadratic equation, it's easy to write down the other one! So using your ... | |
| Jun 11, 2015 at 10:18 | comment | added | math110 | It's nice solution!+1 | |
| Jun 11, 2015 at 6:58 | history | answered | Allan MacLeod | CC BY-SA 3.0 |