# Jacobi's four-square theorem

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:

{\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.

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

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

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

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)/n can be arbitrarily large: indeed, r4(n)/n is infinitely often larger than 8log 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.

## Notes

1. ^ a b Williams 2011, p. 119.
2. ^ Hirschhorn, Michael D. (2000). "Partial Fractions and Four Classical Theorems of Number Theory". The American Mathematical Monthly. 107 (3): 260–264. doi:10.2307/2589321.