I'm wondering if there is a way to solve for x given the following equation:
$$A^x + B^x = C^x$$
where A, B, and C are known constants. For pythagorean triples, $x = 2$. I've seen a lot of stuff for sums of exponents (often using Taylor expansion) but I'm wondering if fixing it to 2 or 3 terms provides any shortcuts?
Another way to represent the problem could be finding the zeroes of the function
$$f(x) = A^x + B^x + C^x$$
WolframAlpha is somehow able to solve it but I'm not sure exactly what they are doing.
Alpha is doing a numeric solution to the equation. You can tell because the answer comes out as a decimal, not as some expression in roots. Techniques for this are discussed in any numerical analysis book. I like chapter 9 of Numerical Recipes-old versions are free online, but any other text will have it as well.