From "Representations of Integers as Sums of Squares" by Emil Grosswald, Springer, 1985:

n is representable with 3 squares iff n is not of the form 4^a (8k+7). R3(n) is # of _primitive_ solutions; r3(n) is # of all solutions. r3(n) = sum(d^2|n, R3(n/d^2)) R3(n) was determined by Gauss, although not in the following form. Let h = number of classes of primitive binary quadratic forms, corresponding to the discriminant D=-n if n=3 (mod 8), D=-4n if n=1,2,5,6 (mod 8), and let d_1=1/2, d_3=1/3, d_n=1 otherwise. Then R3(n) = 12 h d_n, if n=1,2,5,6 (mod 8), 24 h d_n, if n=3 (mod 8). Also, if n is squarefree and (r/n) is the Jacobi symbol, R3(n) = 24 sum(r=1,[n/4],(r/n)) if n=1 (mod 4), 8 sum(r=1,[n/2],(r/n)) if n=3 (mod 8). Don't ask me if I can follow the proof -- I can barely understand the statement. I assume that there are no simpler formulations, else Grosswald would have shown them. ----- At 12:21 PM 11/20/02 -0800, Eugene Salamin wrote:

There is a nice way to calculate the number r2(n), n>0, of integer solutions (x,y) of

x^2 + y^2 = n.

Factor n as

n = p1^a1 p2^a2 ... q1^b1 q2^b2 ... 2^c,

where the p's are primes == 1 mod 4, and the q's are primes == 3 mod 4. Then

r2(n) = if any b is odd, then 0 else, 4 (1 + a1) (1 + a2) ... .

Is there a similar formula for the number r3(n) of integer solutions (x,y,z) of

x^2 + y^2 + z^2 = n ?