Assistance with understanding parent/child relationships in Pythagorean Triples - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T13:38:28Z http://mathoverflow.net/feeds/question/33697 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/33697/assistance-with-understanding-parent-child-relationships-in-pythagorean-triples Assistance with understanding parent/child relationships in Pythagorean Triples Spedge 2010-07-28T18:37:48Z 2011-07-14T04:10:40Z <p>Hello,</p> <p>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.</p> <p>I've been trying to understand the idea on Wikipedia that discusses <a href="http://en.wikipedia.org/wiki/Pythagorean_triple" rel="nofollow">Pythagorean Triple</a> - 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.</p> <p>When I started running the code, I couldn't pick up a certain triple - (200, 375, 425) </p> <p>Basically, I wondered if someone could provide a little clarification. Is it either that...</p> <ol> <li>My code is wrong, it's definitely possible to get to that triple from a starting point of (3,4,5). </li> <li>I haven't understood what Berggren used these equations for, and I need to back and read it properly.</li> </ol> <p>Any clarification would be superb,</p> <p>Thanks</p> <p>P.S - Could someone also tag this appropriately? I have no idea which subject it comes under.</p> http://mathoverflow.net/questions/33697/assistance-with-understanding-parent-child-relationships-in-pythagorean-triples/33726#33726 Answer by Bill Dubuque for Assistance with understanding parent/child relationships in Pythagorean Triples Bill Dubuque 2010-07-28T22:35:09Z 2010-07-31T02:53:43Z <p>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: <img src="http://i30.tinypic.com/156fm2g.jpg" alt="alt text"> <img src="http://i31.tinypic.com/25ptso3.jpg" alt="alt text"> . </p> <p>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.</p> <p>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 </p> <p>$\quad$ <strong>reflection</strong> of $v$ in $u$</p> <p>$\quad\quad v \mapsto v - 2 \dfrac{v \cdot u}{u \cdot u} u \quad\quad$ Reflectivity is clear: $\; u \mapsto -u$, and $\; v \mapsto v$ if $\; v\perp u, \;$ i.e. $v\cdot u = 0$.</p> <p>With $\; v = (x,y,z)$ and $\; u = (1,1,1)$ of norm 1</p> <p>$\quad\quad (x,y,z)\; \mapsto (x,y,z) - 2 \dfrac{(x,y,z)\cdot(1,1,1)}{(1,1,1)\cdot(1,1,1)} (1,1,1)$</p> <p>$\quad\quad\quad\quad\quad\quad = (x,y,z) - 2 \; (x+y-z) \; (1,1,1)$</p> <p>$\quad\quad\quad\quad\quad\quad = (-x-2y+2z, \; -2x-y+2z, \; -2x-2y+3z)$</p> <p>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 <em>smaller</em> rational point, given projectively via the above reflection, e.g. </p> <p>$\quad\quad (5,12,13) \mapsto (5,12,13) - 2 \; (5+12-13) \; (1,1,1) = (-3,4,5)$ </p> <p><img src="http://i27.tinypic.com/ohimfs.jpg" alt="alt text"><br> We ascend the tree by inverting this reflection, combined with trivial sign-changing reflections:</p> <p>$\quad\quad (-3,+4,5) \mapsto (-3,+4,5) - 2 \; (-3+4-5) \; (1,1,1) = ( 5,12,13)$</p> <p>$\quad\quad (-3,-4,5) \mapsto (-3,-4,5) - 2 \; (-3-4-5) \; (1,1,1) = (21,20,29)$</p> <p>$\quad\quad (+3,-4,5) \mapsto (+3,-4,5) - 2 \; (+3-4-5) \; (1,1,1) = (15,8,17)$</p> <p>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 $(-1,0), (3/5,4/5), (-3/5,4/5), (5/12,12/13), (-5/12,12/13), (7/25,24/25), (-7/25,24/25) \ldots$</p> <p>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 <a href="http://www.jstor.org/stable/2048355" rel="nofollow">Cass &amp; Arpaia (1990) [1]</a> </p> <p>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 &amp; 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</p> <p>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 $\; |F(x-y)| &lt; 1$. Then $F$ represents $m$ over $\mathbb Q$ $\iff$ $F$ represents $m$ over $\mathbb Z$, for all nonzero integers $m$.</p> <p>The condition $|F(x-y)| &lt; 1$ 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").</p> <p>Aubry's results are, in fact, very special cases of general results of Wall, Vinberg, Scharlau et al. on <strong>reflective lattices</strong>, i.e. arithmetic groups of isometries generated by reflections in hyperplanes. Generally reflections generate the orthogonal group of Lorentzian quadratic forms in dim &lt; 10.</p> <p><a href="http://www.jstor.org/stable/2048355" rel="nofollow">1</a> Daniel Cass; Pasquale J. Arpaia<br> Matrix Generation of Pythagorean n-Tuples.<br> Proc. Amer. Math. Soc. 109, 1, 1990, 1-7.<br> <a href="http://www.jstor.org/stable/2048355" rel="nofollow">http://www.jstor.org/stable/2048355</a></p> http://mathoverflow.net/questions/33697/assistance-with-understanding-parent-child-relationships-in-pythagorean-triples/68300#68300 Answer by kyzyl for Assistance with understanding parent/child relationships in Pythagorean Triples kyzyl 2011-06-20T16:40:49Z 2011-06-20T16:40:49Z <p>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.]</p>