Primitive Pythagorean Triples

[David Entwistle]

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?

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...

Thanks,

[Alan Mackenzie]

Post by David Entwistle
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.

Post by David Entwistle
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.

Post by David Entwistle
Thanks,
--
David Entwistle

[David Entwistle]

Post by Alan Mackenzie
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,

[sobriquet]

Post by David Entwistle
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?
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...
Thanks,

On the topic of Pythagorean triples, or perhaps slightly tangential,
I was just watching this interesting youtube video regarding points
in the plane that are not all co-linear at integer distances from one
another: