1. Forum
  2. Usenet
  3. SCI.MATH

  • Primitive Pythagorean Triples

    From David Entwistle@21:1/5 to All on Sat Feb 1 10:02:24 2025
    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,
    --
    David Entwistle

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Alan Mackenzie@21:1/5 to David Entwistle on Sat Feb 1 10:54:50 2025
    David Entwistle <[email protected]> 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,

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From sobriquet@21:1/5 to All on Sat Feb 1 15:03:10 2025
    Op 01/02/2025 om 11:02 schreef 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:

    https://www.youtube.com/watch?v=fpIc-FE4c5U

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From David Entwistle@21:1/5 to Alan Mackenzie on Sun Feb 2 09:45:16 2025
    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

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)