Jacobi's theorem is: the number of ways of representing $N$ as a sum of two squares is $4(d_1(N)-d_3(N))$ where $d_i(N)$ is the number of divisors of $N$ that are of the form $4k+i$. I was wondering if there is a combinatorial proof of this (involving, of course, the divisors of $N$). I am working on such a proof but don't want to repeat others' work.

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    $\begingroup$ Papers by Christian Elsholtz could be a good place to start. Also, once you have rewritten the theorem in terms of q-series, you could check the q-series literature, which has a lot of combinatorial proofs. $\endgroup$ Aug 17, 2017 at 16:15
  • $\begingroup$ Ah -- it's a "Corollaire (Jacobi)" (one of many) in Bodo Lass, Démonstration de la conjecture de Dumont, math.univ-lyon1.fr/~lass/articles/pub13dumontj.pdf . But I'm not sure if you count this proof as combinatorial. $\endgroup$ Aug 17, 2017 at 16:30
  • $\begingroup$ @darijgrinberg At least, Bodo really works in combinatorics. I think it is a combinatorial proof - well, "Théorie des nombres et Combinatoire". $\endgroup$ Aug 17, 2017 at 20:14
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    $\begingroup$ @darijgrinberg: I hadn’t seen the Lass paper before — wonderfully general. It uses, like a short proof by Hirschhorn, theta functions and the like. I’m curious about instead developing the argument of Liouville, Heath-Brown, Zagier proving Fermat’s 2-square theorem (covered in a nice survey by Elsholtz) by, e.g., defining a set and an involution on it and then cleverly using it to show Jacobi’s result. $\endgroup$
    – user47804
    Aug 18, 2017 at 1:21
  • $\begingroup$ Is the Elsholtz survey mentioned above math.tugraz.at/~elsholtz/WWW/papers/…? Interesting to hear about Lucas's Satin squares and the movement at the time to write about mathematics for general audiences. $\endgroup$ Aug 18, 2017 at 13:36


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