Timeline for A quadratic Diophantine equation
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 17, 2015 at 4:38 | vote | accept | M. Farrokhi D. G. | ||
| Dec 17, 2015 at 4:38 | vote | accept | M. Farrokhi D. G. | ||
| Dec 17, 2015 at 4:38 | |||||
| Dec 17, 2015 at 4:37 | vote | accept | M. Farrokhi D. G. | ||
| Dec 17, 2015 at 4:38 | |||||
| Dec 17, 2015 at 0:42 | vote | accept | M. Farrokhi D. G. | ||
| Dec 17, 2015 at 4:37 | |||||
| Dec 17, 2015 at 0:42 | vote | accept | M. Farrokhi D. G. | ||
| Dec 17, 2015 at 0:42 | |||||
| Dec 16, 2015 at 17:06 | answer | added | GH from MO | timeline score: 7 | |
| Dec 16, 2015 at 12:06 | answer | added | few_reps | timeline score: 4 | |
| Dec 16, 2015 at 10:39 | comment | added | Felipe Voloch | This curve has $p \pm 1$ points depending on what happens at infinity, as long as it's smooth (i.e. $p \ne 2,5$). There are many simple proofs of this or one can appeal to the Weil bounds. To find a solution, pick $x$ at random and solve for $y$, there is a 50% chance that a given $x$ succeeds. To solve for $y$ you can use the quadratic formula. To take square roots modulo $p$ using one of the known algorithms (google will help). | |
| Dec 16, 2015 at 10:35 | comment | added | joro | There is parametrization of all solutions if p>1 and -5 is square residue modulo p, are interested in this? | |
| Dec 16, 2015 at 10:19 | comment | added | M. Farrokhi D. G. | Thanks for your answer. Where can I find the terminology and the universality of the non-degenerate quadratic planes? | |
| Dec 16, 2015 at 9:30 | comment | added | few_reps | Call $q$ your quadratic form and $M$ the quadratic module $(\mathbf Z^2, q)$. Then $M\otimes \mathbf F_p$ is non-degenerate for $p>5$ (and $p=3$), and non-degenerate quadratic planes are universal, so solutions exist. Finding them doesn't seem to be as easy. | |
| Dec 16, 2015 at 8:54 | history | asked | M. Farrokhi D. G. | CC BY-SA 3.0 |