Jacobi's four-square theorem

Remarks

One proof uses the series $\sum_{n \in \Z} q^{n^2}$ with $q = e^{2 \pi i \tau}$, which is $\theta(2\tau)$ for the theta constant $\theta$ of the theta function. Its fourth power is $\sum_{n \ge 0} r_4(n) q^n$. The function $\theta(2\tau)^4$ is a modular form of weight $2$ for the congruence subgroup $\Gamma_0(4)$. That space has dimension $2$ and is spanned by Eisenstein series, and comparing two coefficients gives the formula. Elementary proofs also exist. Example: $n = 4$ has divisors $1, 2$ not divisible by $4$, so $r_4(4) = 8 \cdot 3 = 24$. Check: the solutions are the $8$ quadruples with one entry $\pm 2$ and three zeros, and the $16$ quadruples with all entries $\pm 1$. Since the sum always contains $d = 1$, $r_4(n) \ge 8 > 0$, which gives Lagrange's four-square theorem.

Going deeper

For a student

The theorem is a model case of how modular forms solve counting problems. A generating function of a counting problem is shown to be modular, and the finite dimension of the space of modular forms identifies it. For two squares the same method gives $r_2(n) = 4 \sum_{d \mid n,\, d \text{ odd}} \chi(d)$, where $\chi(d) = (-1)^{(d-1)/2}$. For larger numbers of squares, cusp forms can appear, and the exact formula then has a cusp form term besides the divisor sums.