Given integers n,b what (all) the integer solutions for $a_1^2+a_2^2+....+a_n^2 = b.C^2$ ??

Question

Given integers b,n, what are integer C,$a_i$ who solves $a_1^2+a_2^2+....+a_n^2 = b.C^2$ ??

Example for n=4 , b=7-> $a_1^2+a_2^2+a_3^2+a_4^2 = 7C^2$

or

for n=3, b=1 -> $a_1^2+a_2^2+a_3^2 = C^2$

Please if any reference, book, author I will appreciate thanks

Answer 1

Every positive integer is the sum of four squares, in particular there exist $a_1,a_2,a_3,a_4$ with $a_1^2+\cdots +a_4^2=7C^2$ for every $C$. For two squares and three squares there are well-known theorems as well:

Sum of one, two, and three squares

Show that an integer of the form $8k + 7$ cannot be written as the sum of three squares.