Timeline for A cubic equation, and integers of the form $a^2+32b^2$
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 17 at 18:49 | answer | added | Bogdan Grechuk | timeline score: 0 | |
| Dec 16, 2023 at 21:29 | vote | accept | Bogdan Grechuk | ||
| Dec 13, 2023 at 21:17 | answer | added | Denis Shatrov | timeline score: 4 | |
| Dec 12, 2023 at 20:35 | answer | added | Piquito | timeline score: -1 | |
| Dec 7, 2023 at 4:24 | comment | added | Fedor Petrov | @Will $p=8k+1$ is of form $x^2+32y^2$ iff $i\sqrt{2}$ is a square modulo $p$, maybe the corresponding Jacobi symbol extends to composites with such prime divisors? | |
| Dec 7, 2023 at 2:41 | comment | added | Will Jagy | @Fedor, the only easy result is divisors of some primitively represented number are all represented by forms of the same discriminant. This is in Dickson's Introduction to the Theory of Numbers, page 95. I suspect there is no alternative to class field theory for this problem. So far, I can't prove that the product of an odd number of primes $q=4u^2 + 4uv + 9 v^2 $ fails to be represented by $x^2 + 32 y^2 .$ Maybe something will come up. | |
| Dec 7, 2023 at 1:54 | comment | added | Fedor Petrov | Bogdan, $P(t)$ is not if the form $4x^2+4xy+9y^2$ at least for $t=0$. | |
| Dec 7, 2023 at 1:44 | comment | added | Fedor Petrov | Is not it true that if the product of two numbers congruent to 1 modulo 8 is represented by $4x^2+4xy+9y^2$ , then so is each of them? And the same for $x^2+32y^2$ ? | |
| Dec 6, 2023 at 20:13 | answer | added | Will Jagy | timeline score: 3 | |
| Dec 6, 2023 at 17:31 | comment | added | Will Jagy | And, with no prime factors $ 5,7 \pmod 8$ you get both ways primitively; If there are such "bad" prime factors but each one is squared, you get imprimitive representations....that is, each such exponent must be even | |
| Dec 6, 2023 at 17:28 | comment | added | Will Jagy | Right. The other two forms (classes) are $3 x^2 \pm 2xy + 11 y^2,$ these forming the non-principal genus. These are more predictable, both represent all primes $3 \pmod 8.$ The square of one of these by Gauss composition is $\langle 4,4,9 \rangle, $ but they are "opposites" and composition takes that pair to the identity $\langle 1,0,32 \rangle, $ Actually, you get both ways for any numberr $1 \pmod 8$ once there are no prime factors $5,7 \pmod 8$ | |
| Dec 6, 2023 at 17:16 | comment | added | Bogdan Grechuk | Interesting. If this statement (either representation $s^2+32z^2$ or $4x^2+4xy+9y^2$ is true but not both) remained correct for all integers equal to $1$ mod $4$, then it would be left to represent $P(t)$ as $4x^2+4xy+9y^2$ for all $t$. But this is not true for composites: integer $33$ is representable in both ways. | |
| Dec 6, 2023 at 17:13 | comment | added | Will Jagy | sorry, your prime must have roots of $u^4 - 2 u^2 + 2,$ or $(u^2 - 1)^2 \equiv -1 \pmod p.$ This is from Liu and Williams (1994) Tamkang J., link above | |
| Dec 6, 2023 at 17:06 | comment | added | Will Jagy | the rest of the primes $1 \pmod 8$ are represented by $4x^2 + 4xy + 9 y^2 $ | |
| Dec 6, 2023 at 17:05 | comment | added | Will Jagy | journals.math.tku.edu.tw/index.php/TKJM/article/view/4461/1510 | |
| Dec 6, 2023 at 16:18 | history | asked | Bogdan Grechuk | CC BY-SA 4.0 |