Timeline for what are all possible pairs (k,m) such that n=2k^2+ m^2
Current License: CC BY-SA 4.0
11 events
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| Sep 3, 2021 at 3:46 | comment | added | David E Speyer | I missed that the original question asked for a generating function. As a generating function, I don't think you'll do better than LeechLattice's comment (you can give a different formula, but I don't think a better one), so I was probably unfair to LeechLattice. | |
| Sep 3, 2021 at 3:37 | comment | added | David E Speyer | The number of ideals of index $n$ is a multiplicative function of $n$, so we just have to do the prime power case. For $p \equiv 1$ or $3 \bmod 8$, there are $k+1$ ideals of index $p^k$; for $p \equiv 5$ or $7 \bmod 8$, there are no ideals of index $p^k$ for $k$ odd and one for $k$ even. Finally, there is one ideal of index $2^k$ for any $k$. Put that all together in a multiplicative function, and then deal with the details of signs and the factor of $2$ from the unit group, and you'll get a formula analogous to mathworld.wolfram.com/SumofSquaresFunction.html | |
| Sep 3, 2021 at 3:35 | comment | added | David E Speyer | For $n = 2 k^2 + m^2$, there is a similar formula. Ignoring the condition that $k$ and $m$ be positive, we are counting elements in $\mathbb{Z}[\sqrt{-2}]$ of norm $n$. Since $\mathbb{Z}[\sqrt{-2}]$ is a PID and has unit group $\pm 1$, the number of elements of number $n$ is twice the number of ideals of index $n$. | |
| Sep 3, 2021 at 3:33 | comment | added | David E Speyer | For $n = 4 k^2 + m^2$, look at the formula for the number of ways to write $n$ as $\ell^2+m^2$ mathworld.wolfram.com/SumofSquaresFunction.html . If $n \equiv 1 \bmod 4$, then half of these will have $\ell$ even and half will have $\ell$ odd; if $n \equiv 0 \bmod 4$, then all solutions have $\ell$ even; if $n \equiv 2 \bmod 4$ or $3 \bmod 4$, there are no solutions with $\ell$ even. | |
| Sep 3, 2021 at 3:30 | comment | added | David E Speyer | Uh, guys, this is a perfectly reasonable question. It could be assigned as homeowrk in an algebraic number theory course, but that doesn't mean that every mathematician knows how to do it, and it has much better answers than the one LeechLattice gives. | |
| Sep 3, 2021 at 1:17 | review | Close votes | |||
| Sep 14, 2021 at 3:02 | |||||
| Sep 3, 2021 at 1:01 | comment | added | user44191 | This is not research-level math; further, it looks like homework. As such, it does not belong on mathoverflow, which is for research-level problems. It may be better suited for math.stackexchange; however, if you do choose to post there, you should give more context - especially by showing what you have tried to do and where you are stuck. See math.stackexchange.com/help/how-to-ask for more information. | |
| Sep 2, 2021 at 22:56 | comment | added | LeechLattice | Is this a reasonable answer: $\sum_{n=1}^{\infty} s(n)x^n=\sum_{p=1}^{\infty} x^{p^2} \sum_{q=1}^{\infty} x^{2q^2}$? | |
| Sep 2, 2021 at 22:54 | history | edited | Ali | CC BY-SA 4.0 |
added 27 characters in body
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| S Sep 2, 2021 at 22:49 | review | First questions | |||
| Sep 3, 2021 at 7:30 | |||||
| S Sep 2, 2021 at 22:49 | history | asked | Ali | CC BY-SA 4.0 |