Timeline for General solution of the quartic $a^4+b^4=c^4+d^4$?
Current License: CC BY-SA 4.0
10 events
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| Nov 3, 2023 at 15:52 | comment | added | David | Richard guy in his book 'Unsolved Problems' on pages 212 & 213 has this to say. Quote, [ "Euler knew that (133^4 + 134^4 = 59^4 + 158^4). A method is known for generating parametric solutions of (a^4 + b^4 = c^4 + d^4) which will generate all published solutions from the trivial one (x, 1,x, 1); it will only produce solutions of degree (6n + 1). " ] End of quote. Guy's statement can be interpreted as, that since a complete solution is possible for (6n+1) degree, than maybe down the road there could be other solutions for degree's other than degree (6n+1). | |
| Oct 24, 2023 at 8:49 | comment | added | Daniel Loughran | To add to Silverman's remark: In the case $x_0^d + x_1^d = x_2^d + x_3^d$ with $d \geq 5$, depending on one's viewpoint a "formal for a general solution" should indeed exist. This is because the Bombieri-Lang conjecture predicts that all but finitely many rational points lie on finitely many rational and elliptic curves. So it just suffices to list these curves with their rational points and the rational points not on the curves. However there is no hope of proving anything like this at all with current tools! | |
| Oct 22, 2023 at 3:10 | comment | added | Noam D. Elkies | @JoeSilverman Your comment does answer the question, because the question concerns the special case $(n,d) = (4,3)$ (once we work over a field in which $-1$ is a 4th power). | |
| Oct 20, 2023 at 23:16 | comment | added | paul garrett | @JoeSilverman Ah, well... | |
| Oct 20, 2023 at 22:19 | comment | added | Joe Silverman | @paulgarrett I'm usually hesitant about posting an MO "answer" that doesn't actually answer the question. So I tend to think that generalizations of this sort belong in the comments section. | |
| Oct 20, 2023 at 21:49 | comment | added | paul garrett | @JoeSilverman, perhaps you could convert your interesting comment into an answer, so that it doesn't get lost ... ? :) | |
| Oct 20, 2023 at 21:25 | comment | added | paul garrett | @JoeSilverman, whoa! Very good! :) | |
| Oct 20, 2023 at 20:00 | comment | added | Joe Silverman | @paulgarrett Right. And moving in the opposite direction (dimension-wise), an analogous statement holds for $$x_1^n+x_2^n+\cdots+x_d^n=1$$ whenever $n\ge d+1$. This is because the canonical divisor of this smooth hypersurface in $\mathbb P^d$ is $\mathcal O(n-d-1)$, so its anti-canonical divisor $\mathcal O(d+1-n)$ is not ample if $n\ge d+1$; but the anti-canonical divisor of a rational variety is ample. As you say, hooray for general theory! | |
| Oct 20, 2023 at 15:56 | comment | added | paul garrett | (Corrected following @Joe Silverman's observtions...) Exactly. This is a fancier version of a point that Joe Lipman made to me many years ago, that there are no (non-constant) rational-function solutions to Fermat equations $x^n+y^n=1$ for $n>2$, because those curves are not genus zero. It was dumbfounding to me at the time! :) That is, in the face of extensive computations I'd done (oof), "it is obvious". :) That is, maybe concepts are worth something!?! :) | |
| Oct 19, 2023 at 20:37 | history | answered | Daniel Loughran | CC BY-SA 4.0 |