Timeline for Representation of two related integers by the same binary quadratic form
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 21, 2019 at 18:47 | vote | accept | Stanley Yao Xiao | ||
| Jun 19, 2019 at 17:08 | comment | added | Will Jagy | @AllenHatcher thank you. I knew about $34$ and $-1,$ it did not occur to me to just find values of small absolute value and compare $\pm$ lists. Similar for $205$ and $221,$ but never for $pq$ with primes $p \equiv q \equiv 1 \pmod 4$ with $(p|q) = -1,$ then there is always an $x^2 - pq y^2 = -1.$ | |
| Jun 19, 2019 at 15:34 | comment | added | Allen Hatcher | @Will Jagy: It's a good thing you don't recall the proof since there are counterexamples! The principal form $x^2-34y^2$ takes the values $15$ at $(x,y)=(7,1)$ and $-15$ at $(11,2)$ but it does not take the value $-1$, as one can see from its topograph. | |
| Jun 18, 2019 at 20:30 | answer | added | Allen Hatcher | timeline score: 8 | |
| Jun 17, 2019 at 23:03 | comment | added | Will Jagy | any particular examples in mind? The only thing that comes to mind is that when a form represents both $1$ and $-1,$ all forms inherit the $\pm$ and the discriminant is greatly restricted. I may have once proved that when any primitive form represents both some $w$ and $-w,$ then the principal form represents both $1,-1.$ If so, i don't recall how it went. | |
| Jun 17, 2019 at 13:51 | history | asked | Stanley Yao Xiao | CC BY-SA 4.0 |