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