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148Pythagorean Triples

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  • harrogate22
    Jun 14, 2003
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      I was looking recently at the frequency of Pythagorean Triples. In
      particular I was interested in the set with opposite and adjacent as
      n, (n+1), - 3,4,5 being the first in this series. The next one is
      20,21,29, and this is followed by 119, 120, 169.

      I set my PC generating these triples with a simple program, and I
      found that the distribution – i.e., the ratio of the gap between
      successive triples approximates close and closer to 3 + root(8) –
      i.e., 5.828……..

      Does anyone have any idea why root-eight should be linked to the
      distribution of this particular set of Pythagorean Triples?

      Incidentally, my interest was in looking at a shape which – as the
      numbers got larger – approximated closer and closer to a square, the
      hypotenuse of the triangle being the diagonal. In all of the triples
      the hypotenuse is an integer, but of course for a true square the
      diagonal/side ratio would be non-rational.

      Any thoughts?