Re: Primitive Pythagorean Triples

On Sat, 1 Feb 2025 10:54:50 -0000 (UTC), Alan Mackenzie wrote:

There are lots.  The smallest "non-trivial" example has a hypotenuse of
65.  We have (16, 63, 65) and (33, 56, 65).  The next such has a
hypotenuse of 85: (36, 77, 85) and (13, 84, 85).
 
In general, a hypotenuse in a Pythagorean triple has prime factors of
the form (4n + 1), together with any number of factors 2, and squares of
other prime factors.  The latter two things don't really add much of
interest.
 
If the hypotenuse is a prime number (4n + 1), there is just one triple
with it.  If there are two distinct factors of the form (4n + 1), there
are two triples (as in 5 * 13 and 5 * 17 above).  The more such prime
factors there are in the hypotenuse, the more triples there are for it,
though it's not such a simple linear relationship that one might expect.

Hi Alan,

Thanks for the comprehensive reply. I see where I have gone wrong - I was
looking at hypotenuse that were prime, when I should have been looking for
co-prime with the other two sides. I'll correct that and see where it
takes me.

Best wishes,
--
David Entwistle