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 \mathbb {Z} ^{k}\ :\ n=a_{1}^{2}+a_{2}^{2}+\cdots +a_{k}^{2}\}|

where |.| denotes the cardinality of the set. In other words, $r_{k}(n)$ is the number of ways $n$ can be written as a sum of $k$ squares.

Particular cases

The number of ways to write a natural number as sum of two squares is given by $r 2 (n)$ . It is given explicitly by

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

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

External links

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

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Sum of squares function

Definition

Particular cases

See also

External links