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).
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:
[math]\displaystyle{ \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} }[/math]
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.
[math]\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} }[/math]
Equivalently, it is eight times the sum of all its divisors which are not divisible by 4, i.e.
[math]\displaystyle{ r_4(n) = 8 \sum_{{m \mid n,} \atop {4 \nmid m}} m. }[/math]
We may also write this as
[math]\displaystyle{ r_4(n) = 8 \, \sigma(n) -32 \, \sigma(n/4) }[/math]
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(2mn) whenever n is even. The values of r4(n) can be arbitrarily large: indeed, r4(n) is infinitely often larger than [math]\displaystyle{ 8\sqrt{\log n}. }[/math][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.
See also
Notes
- ↑ 1.0 1.1 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.
References
- Hirschhorn, Michael D.; McGowan, James A. (2001). "Algebraic Consequences of Jacobi's Two— and Four—Square Theorems". Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Developments in Mathematics. 4. Springer. pp. 107–132. 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. 76. Cambridge University Press. ISBN 978-0-521-17562-3.
External links
- Weisstein, Eric W.. "Sum of Squares Function". http://mathworld.wolfram.com/SumofSquaresFunction.html.
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