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B Pythagorean triples that sum to 60

  1. Jul 18, 2016 #1
    I'm trying to find pythagorean triples that sum to 60. Just from memory, I kow that 3-4-5 and 5-12-13, scaled to some factor, will give triples that sum to 60. These seem to be the only ones that sum to 60, but how can I be sure that there aren't more triples that sum to 60?
     
  2. jcsd
  3. Jul 18, 2016 #2

    fresh_42

    Staff: Mentor

    What do you mean by "sum up"? I don't see the ##60##.
    You can find all by ##x = u^2 - v^2 \; , \; y = 2uv \; , \; z = u^2 + v^2##.
     
  4. Jul 18, 2016 #3
    Let [itex]x,y,z[/itex] be the side lengths of a triangle you describe, so that [itex]x^2+y^2=z^2[/itex] and [itex]x+y+z=60[/itex]. You can solve for [itex]z[/itex] in the last equation and plug it into the first, getting [itex]x^2+y^2=(60-(x+y))^2=3600-120(x+y)+x^2+2xy+y^2[/itex]. This gives the equation [itex]60x+60y-xy=1800[/itex], which you can rearrange into [itex](60-x)(60-y)=1800[/itex]. Now you just need to look for positive factor pairs of [itex]1800[/itex] with both factors smaller than [itex]60[/itex] (since [itex]0<x,y<60[/itex]).
     
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