This is an extra question from an old examination paper:
VI. Let $n>1$ in $\mathbf{Z}$ and let $r(n) = \#\{(a,b)\in \mathbf{Z}^{2};n=a^{2}+b^{2}\}$ Let also n=n'n''n''' where n',n'',n''' \in \mathbf{N} and
p|n' \Longrightarrow p\equiv 1\mod 4 ; p|n'' \Longrightarrow p=2 ; p|n''' \Longrightarrow p\equiv 3 \mod 4
for $p\in \mathbf{N}$ prime.
i) Show that if n''' is not a square, then $r(n)=0$.
ii) Show that if n''' is a square and n' = 1, then $r(n)=4$
iii) Show that if n''' is a square and 1
I am completely dumbstruck and can't see how to begin (and neither did an older student who took this exam and whom I asked for hints). Help is greatly appreciated.