Sum of squares function

The sum of squares function is an arithmetic function that gives the number of representations for a given positive integer n as the sum of k squares, where representations that differ only in the order of the summands or in the signs of the square roots are counted as different, and is denoted by r_k(n).

Definition

The function is defined as

$r_{k}(n)=|\{a_{1},a_{2},\dots ,a_{k}\in \mathbf {Z} \ :\ n=a_{1}^{2}+a_{2}^{2}+\cdots +a_{k}^{2}\}|$

where |.| denotes the cardinality of the set.

Particular cases

The number of ways to write a natural number as sum of two squares is given by r₂(n). It is given explicitly by

$r_{2}(n)=4(d_{1}(n)-d_{3}(n))$

where d₁(n) is the number of divisors of n which are congruent with 1 modulo 4 and d₃(n) is the number of divisors of n which are congruent with 3 modulo 4.

The number of ways to represent n as the sum of four squares was due to Carl Gustav Jakob Jacobi and it is eight times the sum of all its divisors which are not divisible by 4, i.e.

$r_{4}(n)=8\sum _{d\mid n;4\nmid d}d.$

See also

• Jacobi's four-square theorem

External links

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