Jacobi's four-square theorem

Last updated October 05, 2023

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).

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{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.

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.

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

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 $n$ is not divisible by 4. In particular, for a prime number $p$ we have the explicit formula $r 4 (p) = 8(p + 1)$ .^[1]

Some values of $r 4 (n)$ occur infinitely often as $r 4 (n) = r 4 (2 m n)$ whenever $n$ is even. The values of $r 4 (n)$ can be arbitrarily large: indeed, $r 4 (n)$ is infinitely often larger than $8{\sqrt {\log n}}.$ ^[1]

Proof

The theorem can be proved by elementary means starting with the Jacobi triple product.^[2]

The proof shows that the 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 2 Williams 2011 , p. 119.
↑ Hirschhorn, Michael D. (2000). "Partial Fractions and Four Classical Theorems of Number Theory". The American Mathematical Monthly. 107 (3): 260–264. CiteSeerX 10.1.1.28.1615 . doi:10.2307/2589321. JSTOR 2589321.

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References

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. 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.

External links

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

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[Williams_2011-1] 1 2 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. CiteSeerX 10.1.1.28.1615 . doi:10.2307/2589321. JSTOR 2589321.

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