Detailed explanation of Zagier's one sentence proof

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A detailed explanation of Zagier's one-sentence proof

0:00 start
3:00 construct the involution from S to S and basic lemma
14:00 compute the fixed point
16:00 final prove that prime is 4k+1 is sum over two squares

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Автор

13:15
How do you know that |S| must be odd?
Given the condition you wrote down, it is possible that |A|=|C| and |B| is even, right?
I think the logic you present here has the wrong order.
We should do the following:
1. Study the fixed point condition for the set B as you did 15:15
2. Using the property the p is the prime and p=4k+1, there is only one solution of fixed point x=y=1 and z=k.
3. Conclude that |S| is odd and has exactly one fixed point.
4. Consider the other involution (x, y, z) -> (x, z, y), which must have some fixed point since we know that |S| is odd
5. This fixed point is equivalent to p = x^2 + (2y)^2, which shows that p is the sum of two integers.

YosiChen