What determines the number of options for decomposing a number into the sum of two squares of integers?

Question

Mathematics

I return the question (the main thing is that the numbers are integers), and the whole essence of the question is described in the title. an example of expansion 5^2=4^2+3^2,5^2=5^2+0^2, 5^2=0^2+5^2, and does not expand when one of its prime divisors is factorized has the form 4k + 3, and stands in an odd power (for example, 1st ) , or if it is impossible to say what the number of options depends on, then how to understand which number has more decomposition options (not head-on)
(if you voice the task after the fact: then the radius (integer) of the circle with the center at the origin is given, find the number of points lying on a circle and having integer coordinates, it is simply impossible to enumerate)

Answer 1

Gauss circle problem