I note that Charlie has several solutions that are not of the Brian Smith sort, in that a and b are not multiples of c. So Brian's excellent method does not find all solutions.
The first of these are:
793 854 183 (gcd = 61)
737 1072 201 (gcd = 67)
730 1241 219 (gcd = 73)
They are not relatively prime. I believe I have read someplace that solutions to this equation must have a gcd > 1.
New solutions not relatively prime
