number of ways to write $p^e$ as sum of two positive integer squares

Question

I am reading the proof of the sum of two square function, and I got stuck at this little segment

let $p^e$ be a prime factor of n, $p \equiv 1 \pmod4$, there are $e+1$ positive integer solutions (x,y) tuple of the equation $p^e = x^2+y^2$

I know that Fermat proved that $p$ can be expressed uniquely as $x^2+y^2$ and $y^2+x^2$, and I also know that $(x^2+y^2)(a^2+b^2) = (ax+by)^2+(ay-bx)^2$

the website I am reading suggests that given $p = x^2+y^2$ all the solution is of the form$$p^e = (x^2+y^2)^i(y^2+x^2)^{e-i}, i\in\{0,1,,,e\}$$

but I have no idea how to prove that first these solutions do not overlap and second they cover every single possible solutions

my intuition is that since Fermat provides me with the base case(e=1), should I use induction?...but I don't even know how to show it cover ever solution for e=2

thanks for the help

Answer 1

If $p = a^2 + b^2$ and $p^e = c^2 + d^2$, then $p^{e+1} = (a^2 + b^2)(c^2 + d^2)$. Do you know how to write $(a^2 + b^2)(c^2 + d^2)$ as the sum of two squares?