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.