21 Nov
2002
21 Nov
'02
12:32 p.m.
Hi, Gene. Thanks for the answer. (I hope to see the proof sometime. I wonder if there's some use for quaternions here...) << The number, r4(n), of integer solutions (w,x,y,z) of w^2 + x^2 + y^2 + z^2 = n, for positive integer n, is 8 times the sum of the divisors of n which are not divisible by 4. Expressed another way, r4(n)/8 is multiplicative, and r4(p^n)/8 = 1 + p + p^2 + ... + p^n for p an odd prime, r4(2^n)/8 = 1 + 2 for n>0.
This formula, which I'm seeing for the first time, is amazingly interesting! --Dan