# Jacobi's four-square theorem

In number theory, 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 (of integers).

## 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:

{\displaystyle {\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.

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

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

${\displaystyle r_{4}(n)=8\sum _{{m\mid n,} \atop {4\nmid m}}m.}$

We may also write this as

${\displaystyle r_{4}(n)=8\,\sigma (n)-32\,\sigma (n/4)}$

where the second term is to be taken as zero if n is not divisible by 4. In particular, for a prime number p we have the explicit formula r4(p) = 8(p + 1).[1]

Some values of r4(n) occur infinitely often as r4(n) = r4(2mn) whenever n is even. The values of r4(n) can be arbitrarily large: indeed, r4(n) is infinitely often larger than ${\displaystyle 8{\sqrt {\log n}}.}$[1]

## Proof

The theorem can be proved by elementary means starting with the Jacobi triple product.[2]

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.