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:

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

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.

See also

• Lagrange's four-square theorem
• Lambert series
• Sum of squares function

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.
• Williams, Kenneth S. (2011). Number theory in the spirit of Liouville. London Mathematical Society Student Texts 76. Cambridge: Cambridge University Press. ISBN 978-0-521-17562-3. Zbl 1227.11002.

External links

• Weisstein, Eric W., "Sum of Squares Function", MathWorld.