# Jacobi's four-square theorem

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

Jacobi's four-square theorem gives a formula for the number of ways that a given positive integer n can be represented as the sum of four squares.

## History

The theorem was proved in 1834 by Carl Gustav Jakob Jacobi.

## Theorem

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{align} & 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{align}

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).

## Proof

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.