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).

The theorem was proved in 1834 by Carl Gustav Jakob Jacobi.

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.

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

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

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

An immediate consequence is ${\displaystyle r_{4}(2n)=r_{4}(8n)}$; for odd ${\displaystyle n}$, ${\displaystyle r_{4}(2\cdot 4^{a}n)=r_{4}(2n)}$.^[1]

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).^[2]

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 ${\displaystyle 8{\sqrt {\log n}}.}$^[2]

The theorem can be proved by elementary means starting with the Jacobi triple product.^[3]

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.

The first few values of the formula are as follows:

${\displaystyle n}$	0	1	2	3	4	5	6	7	8	9	10
${\displaystyle r_{4}(n)}$	1	8	24	32	24	48	96	64	24	104	144

Additional values may be seen at sequence A000118 in the Online Encyclopedia of Integer Sequences (OEIS).

The number of representations of n as the sum of k squares is known as the sum of squares function. Jacobi's four-square theorem gives an analytic formula for the case k = 4.

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

• Hirschhorn, Michael D.; McGowan, James A. (2001). "Algebraic Consequences of Jacobi's Two— and Four—Square Theorems". In Garvan, F. G.; Ismail, M. E. H. (eds.). Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Developments in Mathematics. Vol. 4. Springer. pp. 107–132. CiteSeerX 10.1.1.26.9028. doi:10.1007/978-1-4613-0257-5_7. ISBN 978-1-4020-0101-7.
• Hirschhorn, Michael D. (1987). "A simple proof of Jacobi's four-square theorem". Proceedings of the American Mathematical Society. 101 (3): 436–438. doi:10.1090/s0002-9939-1987-0908644-9.
• Williams, Kenneth S. (2011). Number theory in the spirit of Liouville. London Mathematical Society Student Texts. Vol. 76. Cambridge University Press. ISBN 978-0-521-17562-3. Zbl 1227.11002.

• Weisstein, Eric W. "Sum of Squares Function". MathWorld.
• OEIS Foundation Inc. (2025). "A000118: Number of ways of writing n as a sum of 4 squares". The On-Line Encyclopedia of Integer Sequences.