10

A Pythagorean triple is a solution to the equation $x^2+y^2=z^2$ over the positive integers. We write a Pythagorean triple as a tuple $(a, b, c)$, where $a\leq b\leq c$. I am interested in the existence of distinct Pythagorean triples of the following form: $$(a, b_1, c_1), (a, b_2, c_2), (b_1, b_2, c_3).$$

I am aware that there are pairs of Pythagorean triples of the form $(a, b_1, c_1), (a, b_2, c_2)$, for instance $(57,176,185)$ and $(57,1624,1625)$. I am now asking for the existence of this third triple. I am especially interested in this when $a$ is somewhat small.

J. W. Tanner
  • 63,683
  • 4
  • 43
  • 88
Mathieu Rundström
  • 3,647
  • I don't think that there will exist $c_3$ according to your criteria. Because $(176)^{2}+(1624)^{2}=2668352$ which implies that $2668352$ is not the square of any positive integer. – Dev Nov 21 '24 at 21:55
  • 1
    This just shows that that pair does not extend to a triple. I am asking does there exist a triple? – Mathieu Rundström Nov 21 '24 at 22:01
  • Oops sorry. I mainly focused on your that triplets. Sorry @Mathieu Rundström – Dev Nov 21 '24 at 23:00
  • 1
    A small stylistic note: The naming of variables here is a bit unhelpful; it hides the fact that the desired configuration is symmetric between $a$, $b_1$, and $b_2$, and so makes it harder to notice the geometric interpretation pointed out in the accepted answer. A nicer naming choice might be to call $a,b_1,b_2$ something like $a_1,a_2,a_3$ instead. – Peter LeFanu Lumsdaine Nov 23 '24 at 13:09

2 Answers2

24

The triples $(a, b_1, b_2)$ you describe are the side lengths of Euler bricks; that is, cuboids with integer side lengths and integer face diagonals (namely $c_1, c_2, c_3$). The smallest example is given by $$ (a, b_1, b_2, c_1, c_2, c_3) = (44, 117, 240, 125, 244, 267); $$ you can find more examples at the link.

Brady Gilg
  • 1,246
  • 8
  • 15
Ravi Fernando
  • 7,948
  • 5
    Note that one of the triples is not primitive. All primitive Pythagorean triples have an even leg and an odd leg, which forces any set of primitive triples to have an even number of distinct legs. – Oscar Lanzi Nov 22 '24 at 00:26
9

A Euler brick is defined by a triple of positive integers $(a,b,c)$ such that

$(a^2+b^2, a^2+c^2, b^2+c^2)$ are all squares.

This solution is due to Tito Piezas

If $u^2+v^2=5w^2$ then

$(a,b,c) = [(u^2-w^2)(v^2-w^2), 4uvw^2, 2vw(u^2-w^2)]$

For more related identities about Euler Brick visit Piezas Website, Link :

https://web.archive.org/web/20230326021937/https://sites.google.com/site/tpiezas/0021

Guruprasad
  • 803