Updated 2012-11-06 07:25:02 by arjen

GS (20121104) A Pythagorean triple is a triple of positive integers (a,b,c) such that a right triangle exists with legs a,b and hypotenuse c satisfying : a**2 + b**2 = c**2

For instance the most obvious solution is (3,4,5).

A scatter plot of the legs (a,b) and their symetries show parabolic patterns.

 # pytriplet.tcl
 # Author:      Gerard Sookahet
 # Date:        04 Nov 2012
 # Description: Plot pythagorean triplet

 package require Tk
 bind all <Escape> {exit}

 proc Pytriplet {H N} {
  .c delete all
  set pix [image create photo]
  .c create image 0 0 -anchor nw -image $pix
  set HN [expr {double($H)/(2*$N)}]
  set mid [expr {$H/2}]
  for {set a 1} {$a <= $N} {incr a} {
     for {set b $a} {$b <= $N} {incr b} {
        set c [expr {sqrt($a*$a + $b*$b)}]
        if {$c == round($c)} then {
          if [Prime $a $b] {
            set ahn [expr {$a*$HN}]
            set bhn [expr {$b*$HN}]
            set mpahn [expr {int($mid + $ahn)}]
            set mpbhn [expr {int($mid + $bhn)}]
            set x(1) $mpahn
            set y(1) $mpbhn
            set x(2) [expr {int($mid - $ahn)}]
            set y(2) [expr {int($mid - $bhn)}]
            set x(3) $mpahn
            set y(3) [expr {int($mid - $bhn)}]
            set x(4) [expr {int($mid - $ahn)}]
            set y(4) $mpbhn
            foreach i {1 2 3 4} {$pix put #00FF00 -to $x($i) $y($i)}
            foreach i {1 2 3 4} {$pix put #00FF00 -to $y($i) $x($i)}
            update idletask
 proc Prime {a b} {
  for {set i 2} {$i <= $a} {incr i} {
     set ai [expr {double($a)/$i}]
     set bi [expr {double($b)/$i}]
     if {$ai == round($ai) && $bi == round($bi)} {return 0}
  return 1
 proc Main {H N} {
   wm title . "Pythagorean"
   pack [canvas .c -width $H -height $H -bg black]
   set f1 [frame .f1 -relief sunken -borderwidth 2]
   pack $f1 -fill x
   button $f1.bu -text Run -width 12 -bg blue -fg white \
         -command "Pytriplet $H $N"
   button $f1.bq -text Quit -width 5 -bg blue -fg white -command exit
   eval pack [winfo children $f1] -side left
 Main 600 6000

For more mathematical details, see : Manuel Benito and Juan L. Varona, Pythagorean triangles with legs less than n, Journal of Computational and Applied Mathematics 143 (2002) pp117–126 [1]

arjen - 2012-11-06 07:25:02

You know of course that there is an analytical (parametric) solution for them:
    a = 2uv
    b = u**2 - v**2
    c = u**2 + v**2

(u and v positive integers), u > v. For the well-known (a,b,c) = (3,4,5) solution, (u,v) = (2,1)).

Having such a solution does not lessen the magic of these triples, though.