Sunday, August 05, 2007
A Mathematical Puzzle
The French mathematician Fermat claimed to have discovered a proof that the equation: A^n + B^n = C^n,
[where "^n" means "raised to the power of n"]
has integer solutions only for n=2. It is easy to show that the numerical solutions for A, B, C are (3,4,5), (5,12,13), (7,24,25), and so on. More than 200 years after Fermat wrote the tantalizing note in the margin of a book some mathematician was able to find a proof.
I have been looking at the equation A^n + B^n + C^n = D^n where n=3 and A, B, C, and D are integers. I haven't been able to develop an expression for generating the solutions. I have found a few solutions by trial and error for A, B, C, and D. As in the case of the Fermat equation, I have looked only at cases D = C+1. Here are a few solutions:
(3,4,5,6), (1,6,8,9), (3,10,18,19), (2,17,40,41), (14,23,70,71), (12,31,102,103)
I expect that some mathematical wizard who happens to read this blog will COMMENT that this problem has already been examined extensively and that I'm just wasting my time to repeat work that has been done already. I will welcome the comment. Otherwise, I may continue plodding on and finding more solutions to the problem without hitting on a general formula for generating solutions.
In the case of the Fermat equation, for any odd number A, B = (A^2-1)/2.