Jacobi's four-square theorem

For other theorems of Jacobi see Jacobi's theorem (disambiguation).

In 1834, Carl Gustav Jakob Jacobi found an exact formula for the total number of ways a given positive integer n can be represented as the sum of four squares. Two representations are considered different if their terms are in different order or if the integer being squared (not just the square) is different; to illustrate, these are three of the eight different ways to represent 1:

${\begin{aligned}&1^{2}+0^{2}+0^{2}+0^{2}\\&0^{2}+1^{2}+0^{2}+0^{2}\\&(-1)^{2}+0^{2}+0^{2}+0^{2}.\end{aligned}}$

The number of ways to represent n as the sum of four squares is eight times the sum of the divisors of n if n is odd and 24 times the sum of the odd divisors of n if n is even (see divisor function), i.e.

$r_{4}(n)={\begin{cases}8\sum \limits _{m|n}m&{\text{if }}n{\text{ is odd}}\\[12pt]24\sum \limits _{\begin{smallmatrix}m|n\\m{\text{ odd}}\end{smallmatrix}}m&{\text{if }}n{\text{ is even}}.\end{cases}}$

Equivalently, it is eight times the sum of all its divisors which are not divisible by 4, i.e.

$r_{4}(n)=8\sum _{m\,:\,4\nmid m|n}m.$

In particular, for a prime number p we have the explicit formula r₄(p) = 8(p + 1).

The proof shows that the Theta series for the lattice Z⁴ is a modular form of a certain level, and hence equals a linear combination of Eisenstein series.

See also

• Lagrange's four-square theorem
• Lambert series

References

• Hirschhorn, Michael D.; James A. Mcgowan. "Algebraic consequences of Jacobi's two– and four–square theorems". Ismail (eds), Developments in Mathematics: 107–132.
• Hirschhorn, Michael D. (1987). "A simple proof of Jacobi's four-square theorem". Proc. Amer. Math. Soc.