Let $$n= 2^a p_1^{e_1}...p_r^{e_r}q_1^{f_1}...q_s^{f_s} $$ be the prime factorization of $ n \in \mathbb{N} $, where $ p_1,\ldots,p_r $ are the primes leaving a residue of $1\bmod{4}$, $q_1,...,q_s$ are the ones leaving a residue of $3$, and $f_1,...,f_s $ are even numbers.
How can I show that $$ \# \{(a,b) \in \mathbb{Z} ^2 | a^2 + b^2=n \} = 4(e_1+1)...(e_r +1 ) ?$$
Any help is appreciated. __ _Edit
I try to get an example for this to understand it more..the answere in the linked duplikate is not very clear to me. lets take $ 65$
$65= 5* 13 $
$65=(1* 3+2*2)^2+(2*3-1*2)^2 $
I can't find a representation for the $4(e_1 +1)..(e_r+1) $ what is $e$ in this case?