Given a right triangle with lengths that are reciprocals of integers, what is the smallest possible sum of these the integers?

In other words, given a right triangle with lengths 1/a, 1/b, and 1/c, where a, b, and c are all integers, what is the lowest value of a+b+c? Also, prove it.

Taken from CAML, which did not ask for a proof.

If the sides of a right triangle are 1/a, 1/b, and 1/c, then the Pythagorean equation becomes 1/a² + 1/b² = 1/c².

In the recent problem, Reciprocal Equations #2, we found all the sets of positive integers a, b, and c for which 1/a + 1/b = 1/c. For this problem, we need to find such a case where a, b, and c are all perfect squares.

In the other problem, the given solution was that for a given value of a and some number x, b=a+x, and c=a²/x + a. I don't know how to solve this directly, but it looks like a good place to start..

Posted by DJ on 2003-08-05 22:21:24