Wednesday, October 27, 2010

Rope-stretching a right corner

A colleague asked today if one could find positive integers a, b, c, and d so that a2 + b2 + c2 = d2, or if that was an open problem.  He added that he'd heard that Egyptians stretched ropes to create 3-4-5 triangles in order to form right angles, and was wondering about the possibility of a three-dimensional analog.

I mentioned the Google Group investigating the harder problem of finding a rectangular box with integer sides, integer diagonals, and integer main diagonal.  (See

But only while driving home did it occur to me that it's straightforward to produce lots of examples of my colleague's easier problem.

Start with your favorite primitive Pythagorean triple (a, b, c).  (See my earlier post about Pythagorean triples:

Then c is odd, so c = 2n + 1, and

a2 + b2 + [2n(n + 1)]2 = [2n(n + 1) + 1]2

For example, 32 + 42 + 122 = 132.