Timeline for Are the number of solutions to $ax^2+bxy+cy^2\equiv u\pmod{p}$, $(x,y)\in\{0,\dotsc,p-1\}$, the same for all units $u$?
Current License: CC BY-SA 4.0
9 events
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| Jun 17, 2022 at 21:47 | comment | added | KConrad | @MarkLewko the intended question is "why the same for all units in $F$", not that the count should be independent of $F$. That is consistent with your comment (as $F$ varies). | |
| Jun 17, 2022 at 20:37 | vote | accept | user47804 | ||
| Jun 17, 2022 at 6:05 | comment | added | Mark Lewko | Clearly, this isn't true. Consider x^2+y^2 =0. If -1 is not a square in F then this has one solution (0,0). If -1 is a square it has at least order |F| solutions of the form (a,ia), with i^2=-1. The number of solutions is well understood but depends on more than just if u is a unit. There is a rich classical theory to binary quadratic forms in finite fields. | |
| Jun 17, 2022 at 0:51 | answer | added | KConrad | timeline score: 5 | |
| Jun 16, 2022 at 22:06 | answer | added | Joe Silverman | timeline score: 8 | |
| Jun 16, 2022 at 17:54 | comment | added | aorq | math.stackexchange.com/questions/398200/… | |
| Jun 16, 2022 at 17:52 | answer | added | LSpice | timeline score: 5 | |
| Jun 16, 2022 at 17:30 | history | edited | LSpice | CC BY-SA 4.0 |
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| Jun 16, 2022 at 17:20 | history | asked | user47804 | CC BY-SA 4.0 |