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The fact that the set of all primitive Pythagorean triples naturally has the structure of a ternary rooted tree may have first been published in 1970:

http://www.jstor.org/stable/3613860

I learned of it from Joe Roberts' book Elementary number theory: A problem oriented approach, published in 1977 by MIT Press. In 2005 it was published again by Robert C. Alperin:

http://www.math.sjsu.edu/~alperin/pt.pdf

He seems to say in a footnote on the first page that he didn't know that someone had discovered this before him until a referee pointed it out. That surprised me. Maybe because of Roberts' book, I had thought this was known to all who care about such things.

If we identify a Pythagorean triple with a rational point (or should I say a set of eight points?) on the circle of unit radius centered at $0$ in the complex plane, then we can say the set of all nodes in that tree is permuted by any linear fractional transformation that leaves the circle invariant. The function $f(z) = (az+b)/(bz+a)$, where $a$ and $b$ are real, leaves the circle invariant and fixes $1$ and $-1$. There are also LFTs that leave the circle invariant and don't fix those two points.

Are there any interesting relationships between the geometry of the LFT's action on the circle and that of the permutation of the nodes in the tree?

(Would it be too far-fetched if this reminds me of the recent discovery by Ruedi Suter of rotational symmetries in some well-behaved subsets of Young's lattice?)

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As beautifully written (literally) as the book by Joe Roberts is, the fact that something is mentioned there doesn't mean all who care about number theory are supposed to know it. It's not the universal bible of number theory. –  KCd Oct 9 '11 at 17:54
    
That the primitive Pythagorean triples are generated from (3,4,5) by the action of 3 integral matrices was found before 1970. It is in the paper "Pytagoreiska triangular" by B. Berggren, which appeared in Tidskrift för elementär matematik, fysik och kemi 17 (1934), 129--139. But I haven't been able to see a copy of this article directly. –  KCd Oct 9 '11 at 18:07
    
@KCd: Of course, no one would think that because it's in that book, it's universally known. Nonetheless, having read it in that book (without remembering where one read it) might be the cause of an impression that it's universally known. (The cause of an impression, as opposed to information from which the impression is inferred.) –  Michael Hardy Oct 9 '11 at 21:57
    
It appears that maybe the fact that each Pythagorean triple shows up eight times on the circle might mean the answer is not as pleasant as I had hoped. However, I've found that by applying inverse matrices, you can find each Pythagorean triple occurring many times in the tree as well. –  Michael Hardy Oct 10 '11 at 19:35
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I have created a new Wikipedia article titled Tree of primitive Pythagorean triples. –  Michael Hardy Oct 12 '11 at 15:28
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