Jacobi's four-square theorem

In

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History

The theorem was proved in 1834 by

Carl Gustav Jakob Jacobi

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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(2^mn) 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

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

Notes