Explanation of Zagiers Proof for primes of the form $4k+1$ What is the content of Zagiers proof? 
What is the actual proof and why does it work? I am not sure I understand why,


*

*there is only one fixed point, and

*why that implies that the involution $(x,y,z) \to (x,z,y)$ proves the theorem.
 A: *

*It's unimportant that there's exactly one fixed point, what's important is that there's an odd number of them. Unfortunately, to see that it's an involution with a fixed point you need to do the calculation. It's not hard but it is a grind, which is why I feel it's a bit disingenuous to call this a "one sentence proof". 

*Involutions either preserve points or swap a pair of them (as they are self-inverse), so if an odd number of points are preserved then the total order of the set is odd (if it is finite) because all other elements come in distinct pairs (those swapped by the involution). Hence the order of $S$ is odd. therefore all involutions of $S$ have a fixed point as if there were no fixed points then $|S|$ would be even.

*$(x,y,z) \mapsto (x,z,y)$ is an invoulution of $S$ so it has a fixed point, which implies that there's an $(x,y,z) \in S$ with $y = z$, hence $p = x^{2} + 4yz = x^{2} + 4y^{2} = x^{2} + (2y)^{2}$.
A: You have $x^2 + 4yz=p$.  If the mapping $(x,y,z)\mapsto(x,z,y)$ has a fixed point, then that is a point where $(x,y,z)=(x,z,y)$.  That implies $y=z$, so you've got $x^2 + 4y^2=p$.  That implies $x^2+(2y)^2=p$, so $p$ is a sum of two squares.
(This is not a complete answer, but it answers at least one of the questions in the posting.)
