Hi

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?

Ken