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.
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:
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.
Equivalently, it is eight times the sum of all its divisors which are not divisible by 4, i.e.
We may also write this as
Some values of r4(n) occur infinitely often as r4(n) = r4(2mn) whenever n is even. The values of r4(n)/n can be arbitrarily large: indeed, r4(n)/n is infinitely often larger than 8√.
- 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. doi:10.2307/2589321.
- 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. doi:10.1090/s0002-9939-1987-0908644-9.
- 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.