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.


Bruce Yoshiwara said...

Sigh. The colleague decided he now wants just the face diagonals to be integers. So he wants an Euler brick, for example a box with sides 44, 117 and 240.

Bruce Yoshiwara said...

On Thanksgiving Day, November 25, 2010, an exhaustive computer search for a perfect cuboid completed the range of 1 to 10^10 (actually 10,000,676,880) for the smallest edge of the cuboid with no such perfect cuboid found. This search was delayed for 721 days due to unforseen circumstances, so it was gratifying to finally complete the goal started 3 years ago. Over the range, 22,806 body cuboids were found, 13,123 edge cuboids and 21,922 face cuboids, wherein the cuboid is named for one irrational value of the 7 (3 edge, 3 face diagonals, 1 body diagonal) which occurs, making a total of 57,851 unique cuboids over this range. There were 35 pairs of cuboids sharing two common edges, of these, there were 28 body cuboid pairs, only 1 body & edge cuboid pair, 3 body & face cuboid pairs, no edge & face cuboid pairs, and 3 face cuboid pairs. Unfortunately no example of 4 edges such that any pair squared sum to a square was yet found. Interestingly, the distribution of the cuboids over the range is approximated very well by a polynomial equation in log(x), log(y) where x is the occurance of the cuboid and y is the smallest edge size. A fitted quartic polynomial in log(x) is: (1) log(y) = a + b*log(x) + c*log^2(x) + d*log^3(x) + e*log^4(x) for the coefficients: a = 3.780968876038006 b = 0.1610159794568313 c = 0.2344105833655249 d = -0.01445284862761818 e = 0.0005571886117749887 All 57,851 cuboids were checked, no perfect cuboid was found for the smallest edge in the range 1 to 1e10. - Randall Rathbun