Timeline for Is there a simple way to compute the number of ways to write a positive integer as the sum of three squares?
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14 events
| when toggle format | what | by | license | comment | |
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| Jul 27, 2021 at 15:08 | comment | added | Mason | Is there anything in the literature that explores $\sum_{n=1}^x R(n)$( or $\sum_{n=1}^x R_0(n)$)? It might be interesting to see how this class number relates to $\pi$ (or $\pi/\zeta(3)$) in a series. | |
| Jan 7, 2017 at 2:39 | comment | added | John Voight | Just to finish for now, Shemanske projecteuclid.org/euclid.pjm/1102702131 gives several other references, and generalizes to other ternary quadratic forms. | |
| Jan 7, 2017 at 2:26 | comment | added | Will Jagy | @John I vaguely remember deciding to list all the cases mod 8 instead of combining in any way. I think Grosswald writes $3 \pmod 8$ in a different way, you need to know how $h(-n)$ and $h(-4n)$ relate. I get that sort of thing from Buell, Binary Quadratic Forms. | |
| Jan 7, 2017 at 2:19 | comment | added | John Voight | I'm looking at Grosswald (and Gauss) now. | |
| Jan 7, 2017 at 2:18 | comment | added | John Voight | Also, there is duplication in the cases--is there a reason for writing it this way? | |
| Jan 7, 2017 at 2:18 | comment | added | Will Jagy | @John Yes, primitive. I believe I kept copies of the pages in Grosswald's book, if I can find them and only one or two pages are involved, I'll scan those and email. | |
| Jan 7, 2017 at 2:15 | comment | added | John Voight | Ah! OK, so did you mean primitive also in the class number of binary quadratic forms? I think now you mean "yes". | |
| Jan 7, 2017 at 2:12 | comment | added | Will Jagy | @JohnVoight sum of three squares, $(\pm 1, \pm 1, \pm 1)$ so just $8$ There is an interesting question on representations by quaternaries you might like, mathoverflow.net/questions/256576/… Not one hundred percent finished | |
| Jan 7, 2017 at 2:02 | comment | added | John Voight | Just checking: you meant $h(d)$ to count the class number of not-necessarily-primitive binary quadratic forms of discriminant $d<0$? So $h(d)$ is not the class number of the quadratic order of discriminant $d$? Something must be up because $R_0(3) = 4\cdot 8 = 32$ by counting orbits of $(0,\pm 1, \pm 1, \pm 1)$, but the only reduced primitive quadratic form of discriminant $-12$ is $x^2+3y^2$. | |
| Jun 27, 2012 at 19:22 | comment | added | Will Jagy | The stuff relating to class number is due to Gauss, is in the Disquisitiones, and is described in Grosswald's 1985 book Representations of Integers as Sums of Squares. Chapter 4 is called Representations as Sums of Three Squares. Section 8, Gauss's Theorem is pages 51-53. There is a difference in presentation, you do need to know that if $n >3, \; n \equiv 3 \pmod 8,$ then $h(-4n) = 3 h(-n).$ | |
| Apr 27, 2010 at 1:45 | comment | added | JSE | (also copied from the duplicate thread) We (me, Michel, and Venkatesh) write something about this question in the preprint "Linnik's Ergodic method and the distribution of integral points on spheres." In particular, we explain how the set of (SO_3(Z) classes of) representations of n is naturally a torsor for a class group, thus recovering the above formulas for R_0(n) in the squarefree case. (None of this is really original to us except maybe the use of the word "torsor.") | |
| Apr 27, 2010 at 0:07 | comment | added | David E Speyer | This is a very nice answer! You organized the data much more clearly than any of the articles I found. | |
| Apr 27, 2010 at 0:00 | comment | added | Gerry Myerson | The paper is Michael D. Hirschhorn and James A. Sellers, On representations of a number as a sum of three squares, Discrete Math. 199 (1999), no. 1-3, 85-101, MR1675913 (2000a:11141). | |
| Apr 26, 2010 at 22:36 | history | answered | Will Jagy | CC BY-SA 2.5 |