Timeline for Number of solutions to $mx^2+ny^2 \equiv k\pmod{p}$
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 3, 2012 at 23:55 | answer | added | Gerry Myerson | timeline score: 3 | |
| Jul 3, 2012 at 23:10 | answer | added | GH from MO | timeline score: 1 | |
| Jul 3, 2012 at 22:04 | comment | added | Will Sawin | @Igor: When $-n/m$ is a quadratic residue, the same argument of course works. When $-n/m$ is not a quadratic residue, several arguments show that the answer is $p+1$. (kernel of the norm map $\mathbb F_{p^2}^{\times}\to \mathbb F_p^{\times}$, comparing the number of solutions for a single $x$ to a corresponding equation of the first type, using the isomorphism of the projective completion and $\mathbb P^1$ and counting points at $\infty$, probably etc.) Unfortunately I think Mike Decaro knows the answer but wants a reference for it. | |
| Jul 3, 2012 at 21:49 | comment | added | Igor Rivin | Well, when $m$ and $-n$ are both quadratic residues, (and $p\neq 2$) the answer seems to be $p-1,$ by factoring the left hand side. | |
| Jul 3, 2012 at 21:03 | history | asked | Mike Decaro | CC BY-SA 3.0 |