Jacobi's four-square theorem
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In number theory, Jacobi's four-square theorem gives a formula for the number of ways that a given positive integer Template:Mvar 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:
The number of ways to represent Template:Mvar as the sum of four squares is eight times the sum of the divisors of Template:Mvar if Template:Mvar is odd and 24 times the sum of the odd divisors of Template:Mvar if Template:Mvar 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.
An immediate consequence is ; for odd , .[1]
We may also write this as
where the second term is to be taken as zero if Template:Mvar is not divisible by 4. In particular, for a prime number Template:Mvar we have the explicit formula Template:Math.[2]
Some values of Template:Math occur infinitely often as Template:Math whenever Template:Mvar is even. The values of Template:Math can be arbitrarily large: indeed, Template:Math is infinitely often larger than [2]
Proof
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.