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 Template:Mvar 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:

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

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

$$r_4(n)=\{\begin{array}{ll}8\sum _{m|n}m& \text{if }n\text{ is odd},\\ 24\sum _{\genfrac{}{}{0ex}{}{m|n}{m\text{ odd}}}m& \text{if }n\text{ is even}.\end{array}$$

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

$$r_4(n)=8\sum _{\genfrac{}{}{0ex}{}{m\mid n,}{4\nmid m}}m.$$

An immediate consequence is $r_4(2n)=r_4(8n)$; for odd $n$, $r_4(2\cdot 4^{a}n)=r_4(2n)$.

We may also write this as

$$r_4(n)=8\sigma (n)-32\sigma (n/4)$$

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

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

Proof

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

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
• Sum of squares function