Timeline for Is there an $n\ge1$ such that every prime $p\equiv1\pmod{9}$ is representable in the form $x^2+ny^2$? [closed]
Current License: CC BY-SA 3.0
20 events
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| Dec 11, 2017 at 20:24 | history | closed |
Lucia Pace Nielsen Henry.L Alex B. Greg Martin |
Not suitable for this site | |
| Dec 11, 2017 at 20:15 | comment | added | square-free | @nfdc23, It is not a homework problem or take-home exam question. | |
| Dec 11, 2017 at 20:13 | vote | accept | square-free | ||
| Dec 11, 2017 at 16:25 | history | edited | GH from MO |
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| Dec 11, 2017 at 16:11 | answer | added | GH from MO | timeline score: 8 | |
| Dec 11, 2017 at 15:38 | review | Close votes | |||
| Dec 11, 2017 at 20:27 | |||||
| Dec 11, 2017 at 13:02 | comment | added | nfdc23 | Is this a homework problem or take-home exam question? What is the motivation for asking about it here? There is such an $n$ that works, and this is a standard exercise in the initial parts of algebraic number theory. As a hint, there is an $n$ which even works for all $p\equiv 1 \bmod 3$. | |
| Dec 11, 2017 at 12:20 | comment | added | square-free | @TobiasKildetoft, Yes I would like, that $n$ be different from $-1$ | |
| Dec 11, 2017 at 12:19 | history | edited | Joe Silverman | CC BY-SA 3.0 |
Fixed grammar
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| Dec 11, 2017 at 12:19 | comment | added | square-free | Prof @JoeSilverman, I don't have access to this book. | |
| Dec 11, 2017 at 12:17 | comment | added | Joe Silverman | Have you read David Cox's book *Primes of the form $X^2+nY^2$"? | |
| Dec 11, 2017 at 12:15 | comment | added | Tobias Kildetoft | So presumably you also want to specify that $n$ should be positive, as otherwise $n=-1$ obviously works. Once that is done my guess would be that there is no such $n$. | |
| Dec 11, 2017 at 12:11 | history | edited | square-free | CC BY-SA 3.0 |
added 29 characters in body
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| Dec 11, 2017 at 12:08 | comment | added | Gerry Myerson | Then please edit your question to say so. | |
| Dec 11, 2017 at 12:07 | comment | added | square-free | @GerryMyerson, Exactly, I would like to find an integer $n$ which doesn't depend on $p$. | |
| Dec 11, 2017 at 12:07 | comment | added | Gerry Myerson | But $x=y=1$ satisfies $(x,y)=1$. You'll have to do better than that! | |
| Dec 11, 2017 at 12:06 | comment | added | square-free | @JasonStarr, Thanks for your comment, I am trying to find a representation of $p$ in the form $x^2+ny^2$ with $(x,y)=1$. | |
| Dec 11, 2017 at 12:06 | comment | added | Gerry Myerson | I imagine, Jason, that OP doesn't want $n$ to depend on $p$. Perhaps OP could edit the question, to clarify this point? | |
| Dec 11, 2017 at 12:04 | comment | added | Jason Starr | Set $n$ equal to $p-1$ and set $x$ and $y$ equal to $1$. Is that what you are asking? | |
| Dec 11, 2017 at 12:01 | history | asked | square-free | CC BY-SA 3.0 |