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Zagier's has found a famous one sentence proof for Fermat's theorem on sums of two squares. It centers on the following involution of the set $S= \lbrace (x,y,z) \in N^3: x^2+4yz=p \rbrace $ having exactly one fixed point.

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The author of this question as well as several other sources say, that this can easily been seen. Wikipedia states that the fixed point is $(1, 1, k)$. Now I perfectly understand that $(1, 1, k)$ always falls into the second case of the involution and returns $(1, 1, k)$ thus being a fixed point, but I completely fail to recognize that there is absolutely no possibility for another fixed point to exist.

I started out trying to find another (which might be very difficult), and came up with $(x, x, 1)$ but i did not find an example for which $(x, x, 1)$ is part of the given set of prime numbers. Still I don't see why it can't be possible that there is another fixed point for some $x$ I haven't found yet.

Can anyone show that to me?

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    $\begingroup$ Maybe it is worth remarking that the crucial argument in this "one-sentence" proof is actually due to Heath-Brown $\endgroup$
    – Francesco Polizzi
    Jul 27, 2016 at 12:24

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Setting the coordinates to be equal, it's clear that the only fixed point in $\mathbb R^3$ for the top map is (0,0,0), and ditto for the bottom map. Finally, as you (almost) noted, the fixed points in $\mathbb R^3$ of the middle map are the points of the form $(x,x,z)$. But for a point of this form, the quantity $x^2+4yz$ is $x^2+4xz=x(x+4z)$. Now, if $x$ and $z$ are in $\mathbb N$, the only way for $x(x+4z)$ to be prime is for $x=1$, and then also one needs $1+4z$ to be prime.

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  • $\begingroup$ The fact that $(x,x,z)$ actually corresponds to a divisor of $n=x^2+4yz$ can be used to factor $n$ if it is not prime, using the Zagier maps. Take for example $n=p_1 p_2$, with two distinct primes $3 \mod 4$. For details see section 2.2. of my paper "A Combinatorial Approach to Sums of Two Squares and Related Problems " link.springer.com/chapter/10.1007%2F978-0-387-68361-4_8, (or paper 30 of: math.tugraz.at/~elsholtz/WWW/papers/papers.html ) Starting with $(1,1,k)$ and iterating the two Zagier maps one must come to a fixed point of the form $(x,x,z)$. Very slowly algorithm. $\endgroup$
    – Christian Elsholtz
    Jul 27, 2016 at 12:57
  • $\begingroup$ @Joe Thank you. That was prompt and very clear. Good explanation. I just noticed that i should've posted this on Math.StackExchange. My bad. Thank you anyway. $\endgroup$
    – Toastgeraet
    Jul 29, 2016 at 21:36

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