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Hello,

I want to start by apologising for what is probably a weak attempt at a question on a site like this, but I'm having trouble understand a concept that doesn't seem to be properly explained elsewhere - perhaps someone can put it in layman's terms.

I've been trying to understand the idea on Wikipedia that discusses Pythagorean Triple - namely the section entitled Parent/child relationships. It talks about a Swedish man called Berggren who devised a set of equations that would allow you to determine the children of this parental triple. Each parent created 3, which in turn created 3 and so on.

When I started running the code, I couldn't pick up a certain triple - (200, 375, 425)

Basically, I wondered if someone could provide a little clarification. Is it either that...

  1. My code is wrong, it's definitely possible to get to that triple from a starting point of (3,4,5).
  2. I haven't understood what Berggren used these equations for, and I need to back and read it properly.

Any clarification would be superb,

Thanks

P.S - Could someone also tag this appropriately? I have no idea which subject it comes under.

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4  
Your triple $(200,375,425)$ isn't primitive: it's got by multiplying each term of the primitive triple $(8,15,17)$ by $25$. –  Robin Chapman Jul 28 '10 at 18:42
    
Right - sorry, thanks. Post that as a proper answer and let me tick it. –  Spedge Jul 28 '10 at 19:06
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Here are two other references to consider: R. C. Alperin, The Modular Tree of Pythagoras, Amer. Math. Monthly 112 (2005), 807–816 and A. Hall, Genealogy of Pythagorean Triads, The Math. Gazette 54 (1970), 377–379. –  KConrad Jul 28 '10 at 19:36
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The tree of triples has been independently rediscovered nearly a dozen times by various authors. The decidedly cute paper "Dynamics of Pythagorean triples" by Dan Romik, published in TAMS, lists nine papers, all of which I think are independent: Barning 1963, Hall 1970, Jaeger 1976, Kristensen 1976, Kanga 1990, Gollnick-Scheid-Zoellner 1992, Préau 1992, Alperin 2005... interestingly, Romik doesn't list Berggren, so perhaps there are even more! –  Ian Morris Jul 29 '10 at 0:02
    
Thanks for the further reading guys - will allow me to delve a little deeper. Appreciate it. –  Spedge Jul 29 '10 at 10:36
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2 Answers 2

up vote 14 down vote accepted

A little-known chatoyant gem of elementary number theory is that the tree of Pythagorean triples has a beautiful geometric genesis in terms of reflections. This viewpoint should clarify the points that you raise. Below is a brief sketch excerpted from some emails I sent to John Conway and R. K. Guy, after noticing that they mention this topic (too) briefly in their "Book of Numbers". Namely, on p. 172 they write: alt text alt text .

Below I explain briefly how to view this in terms of reflections and I mention some generalizations and closely related topics. I plan to discuss this at greater length in a future MO post when time permits.

Consider the quadratic space $Z$ of the form $Q(x,y,z) = x^2 + y^2 - z^2$. It has Lorentzian inner product $(Q(x+y)-Q(x)-Q(y))/2$ given by $\; v \cdot u = v_1 u_1 + v_2 u_2 - v_3 u_3$. Recall that here one defines the

$\quad$ reflection of $v$ in $u$

$\quad\quad v \mapsto v - 2 \dfrac{v \cdot u}{u \cdot u} u \quad\quad$ Reflectivity is clear: $\; u \mapsto -u$, and $\; v \mapsto v$ if $\; v\perp u, \;$ i.e. $v\cdot u = 0$.

With $\; v = (x,y,z)$ and $\; u = (1,1,1)$ of norm 1

$\quad\quad (x,y,z)\; \mapsto (x,y,z) - 2 \dfrac{(x,y,z)\cdot(1,1,1)}{(1,1,1)\cdot(1,1,1)} (1,1,1)$

$\quad\quad\quad\quad\quad\quad = (x,y,z) - 2 \; (x+y-z) \; (1,1,1)$

$\quad\quad\quad\quad\quad\quad = (-x-2y+2z, \; -2x-y+2z, \; -2x-2y+3z)$

This is the nontrivial reflection that effects the descent in the triples tree. Said simpler: if $x^2 + y^2 = z^2$ then $(x/z, y/z)$ is a rational point $P$ on the unit circle $C$. A simple calculation shows that the line through $P$ and $(1,1)$ intersects $C$ in a smaller rational point, given projectively via the above reflection, e.g.

$\quad\quad (5,12,13) \mapsto (5,12,13) - 2 \; (5+12-13) \; (1,1,1) = (-3,4,5)$

alt text
We ascend the tree by inverting this reflection, combined with trivial sign-changing reflections:

$\quad\quad (-3,+4,5) \mapsto (-3,+4,5) - 2 \; (-3+4-5) \; (1,1,1) = ( 5,12,13)$

$\quad\quad (-3,-4,5) \mapsto (-3,-4,5) - 2 \; (-3-4-5) \; (1,1,1) = (21,20,29)$

$\quad\quad (+3,-4,5) \mapsto (+3,-4,5) - 2 \; (+3-4-5) \; (1,1,1) = (15,8,17)$

Continuing in this manner one may reflectively generate the entire tree of primitive Pythagorean triples, e.g. the topmost edge of the triples tree corresponds to the ascending $C$-inscribed zigzag line $(-1,0), (3/5,4/5), (-3/5,4/5), (5/12,12/13), (-5/12,12/13), (7/25,24/25), (-7/25,24/25) \ldots$

This technique easily generalizes to the form $ x_1^2 + x_2^2 + \cdots + x_{n-1}^2 = x_n^2$ for $4 \le n \le 9$, but for $n \ge 10 $ the Pythagorean n-tuples fall into at least $[(n+6)/8]$ distinct orbits under the automorphism group of the form - see Cass & Arpaia (1990) [1]

There are also generalizations to different shape forms that were first used by L. Aubry (Sphinx-Oedipe 7 (1912), 81-84) to give elementary proofs of the 3 & 4 square theorem (see Appendix 3.2 p. 292 of Weil's: Number Theory an Approach Through History). These results show that if an integer is represented by a form rationally, then it must also be so integrally. In particular, the following class of forms is included $x^2+y^2, x^2 \pm 2y^2, x^2 \pm 3y^2, x^2+y^2+2z^2, x^2+y^2+z^2+t^2,\ldots$ More precisely, essentially the same proof as for Pythagorean triples shows

THEOREM Suppose that the $n$-ary quadratic form $F(x)$ has integral coefficients and has no nontrivial zero in ${\mathbb Z}^n$, and suppose further that for any $x \in {\mathbb Q}^n$ there exists $y \in {\mathbb Z}^n$ such that $\; |F(x-y)| < 1$. Then $F$ represents $m$ over $\mathbb Q$ $\iff$ $F$ represents $m$ over $\mathbb Z$, for all nonzero integers $m$.

The condition $|F(x-y)| < 1$ is closely connected to the Euclidean algorithm. In fact there is a function-field analog that employs the Euclidean algorithm which was independently rediscovered by Cassels in 1963. Namely, a polynomial is a sum of $n$ squares in $k(x)$ iff the same holds true in $k[x]$. Pfister immediately applied this to obtain a complete solution of the level problem for fields. Shortly thereafter he generalized Cassels result to arbitrary quadratic forms, founding the modern algebraic theory of quadratic forms ("Pfister forms").

Aubry's results are, in fact, very special cases of general results of Wall, Vinberg, Scharlau et al. on reflective lattices, i.e. arithmetic groups of isometries generated by reflections in hyperplanes. Generally reflections generate the orthogonal group of Lorentzian quadratic forms in dim < 10.

1 Daniel Cass; Pasquale J. Arpaia
Matrix Generation of Pythagorean n-Tuples.
Proc. Amer. Math. Soc. 109, 1, 1990, 1-7.
http://www.jstor.org/stable/2048355

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The whole orthogonal group of $Q$ is generated by 5 reflections (see math.uconn.edu/~kconrad/blurbs/linmultialg/descentPythag.pdf). Can you generate this group with fewer than 5 reflections? –  KConrad Jul 28 '10 at 23:19
    
Wow. That's mind-blowing. Might take me a while to get through it, but thanks for the post :) –  Spedge Jul 29 '10 at 10:38
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Chatoyant...reflections...pun intended? –  Gerry Myerson Jul 30 '10 at 0:30
    
@Gerry: Yes, pun intended - sharp eye! –  Bill Dubuque Jul 31 '10 at 2:58
    
@KConrad: The group $PO(x^2+y^2-z^2;Z)=PGL(2,Z)$. So this group is generated by three reflections. If you don't consider the projective action, then you can add in negation. Vinberg generalized this (as Bill alludes to) to higher dimensions by implementing an algorithm which finds a fundamental domain for the subgroup generated by reflections. In fact, there is some sort of generalized descent algorithm up to dimension 19. However, not all tuples will be equivalent: one must include the finitely many cusps, corresponding to n-1-dim. unimodular Euclidean lattices, which grow with dimension –  Ian Agol Jun 22 '11 at 17:29
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The Classic Tree shown above was probably first found by B. Berggren in 1933. Recently, an entirely New Tree was found. [see: H. Lee Price (2008).The Pythagorean tree: A new species. ArXiv e-prints, arXiv:0809.4324, pp 14.]

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