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From: Alan Mackenzie <acm@muc.de>
Newsgroups: sci.math
Subject: Re: Primitive Pythagorean Triples
Date: Sat, 1 Feb 2025 10:54:50 -0000 (UTC)
Organization: muc.de e.V.
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David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> wrote:
> Hello,

> Are there any primitive Pythagorean triples where the one hypotenuse
> has
> more than the two values for the other two sides? So, in the case of
> the
> 3, 4, 5 right triangle, there's the two possible arrangement of sides
> 3,
> 4, 5 and 4, 3, 5. Are there any triangles with more than two
> arrangements
> for the one single size of hypotenuse?

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.

> I haven't found any, looking at hypotenuse up to 10,000, but don't
> immediately see why there couldn't be solutions of: a, b, h; b, a, h;
> c,
> d, h and d, c, h.

> Apologies if this is inappropriate here. My maths is okay, but just
> high-
> school level, nothing more...

No apologies needed.  It's much more appropriate than most posts on this
group.

> Thanks,
> --
> David Entwistle

-- 
Alan Mackenzie (Nuremberg, Germany).> Hello,