===
k = 2 === The number of ways to write a
natural number as sum of two squares is given by . It is given explicitly by :r_2(n) = 4(d_1(n)-d_3(n)) where is the number of
divisors of which are
congruent to 1
modulo 4 and is the number of divisors of which are congruent to 3 modulo 4. Using sums, the expression can be written as: :r_2(n) = 4\sum_{d \mid n \atop d\,\equiv\,1,3 \pmod 4}(-1)^{(d-1)/2} The prime
factorization n = 2^g p_1^{f_1}p_2^{f_2}\cdots q_1^{h_1}q_2^{h_2}\cdots , where p_i are the
prime factors of the form p_i \equiv 1\pmod 4, and q_i are the prime factors of the form q_i \equiv 3\pmod 4 gives another formula :r_2(n) = 4 (f_1 +1)(f_2+1)\cdots , if
all exponents h_1, h_2, \cdots are
even. If one or more h_i are
odd, then r_2(n) = 0. ===
k = 3 === Gauss proved that for a
squarefree number , :r_3(n) = \begin{cases} 24 h(-n), & \text{if } n\equiv 3\pmod{8}, \\ 0 & \text{if } n\equiv 7\pmod{8}, \\ 12 h(-4n) & \text{otherwise}, \end{cases} where denotes the
class number of an integer . There exist extensions of Gauss' formula to arbitrary integer . ===
k = 4 === The number of ways to represent 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. Representing , where
m is an odd integer, one can express r_4(n) in terms of the
divisor function as follows: :r_4(n) = 8\sigma(2^{\min\{k,1\}}m). ===
k = 6 === The number of ways to represent as the sum of six squares is given by :r_6(n) = 4\sum_{d\mid n} d^2\big( 4\left(\tfrac{-4}{n/d}\right) - \left(\tfrac{-4}{d}\right)\big), where \left(\tfrac{\cdot}{\cdot}\right) is the
Kronecker symbol. ===
k = 8 === Jacobi also found an
explicit formula for the case : :r_8(n) = 16\sum_{d\,\mid\,n}(-1)^{n+d}d^3. == Generating function ==