Timeline for Integers $8k+3>0$ not represented by $2x^2+4y^2+4yz+9z^2$ over the integers
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 4, 2023 at 20:51 | comment | added | Will Jagy | this seems to be an expanded version of Ono Soundararajan (1997) uva.theopenscholar.com/files/ken-ono/files/025_8.pdf | |
| Nov 4, 2023 at 20:42 | comment | added | Will Jagy | the kind of thing that can be done without assumptions: the sporadic numbers are squarefree. For any specific prime, there are explicit formulas to show that any eligible multiple of $p^2$ is represented by your form. I believe this was done for $x^2 + y^2 + 10 z^2$ called Ramanujan;s form, let me look. There it is, Ono and Soundararajan, Ramanujan's Ternary Quadratic Form, Mathematische Annalen 308, 1997. Oh they display the precise GRH assumptions needed to show the known sporadic list is complete. | |
| Nov 4, 2023 at 12:15 | comment | added | Mastrem | @CommandMaster, $f(x,y,z)=(x+2z)^2+(x-2z)^2+(2y+z)^2$ as well. | |
| Nov 4, 2023 at 12:14 | comment | added | Mastrem | @GerryMyerson, I write $\overline{\mathbb{Z}}$ for the ring of all algebraic integers over $\mathbb{Z}$, and a point $\tau\in\mathcal{H}$ has discriminant $d$ if there exist $a,b,c\in\mathbb{Z}$ with $(a,b,c)=1$, $a\tau^2+b\tau+c=0$ and $d=b^2-4ac$. | |
| Nov 4, 2023 at 8:51 | comment | added | Gerry Myerson | What is meant by the discriminant of a point? What is meant by $\overline{\bf Z}$? | |
| Nov 3, 2023 at 17:28 | answer | added | Will Jagy | timeline score: 2 | |
| Nov 3, 2023 at 16:34 | comment | added | Jeremy Rouse | I think this is likely to be very hard. For comparison, there are three positive-definite ternary quadratic forms that are conjectured to represent all positive integers, but for which there is currently no proof. Proving that A306970 is complete assuming GRH is probably doable. (I did something similar in Section 6 of the paper here.) | |
| Nov 3, 2023 at 14:53 | comment | added | Daniel Weber | $f(x, y, z) = 2x^2 + (2y+z)^2 + 2(2z)^2$, there might be relevant generalizations of Legendre's three-square theorem. | |
| Nov 3, 2023 at 14:33 | history | asked | Mastrem | CC BY-SA 4.0 |