This question might be easy.
The hard question is this: prove that if $a,b,c\geq3$ then there are no solutions in positive integers $x,y,z$ to $x^a+y^b=z^c$ with $x,y,z$ coprime. This implies Fermat, most cases of Catalan, etc., and is an open problem.
But it's really crucial that $x,y,z$ are coprime to make this question hard. For example if I want to find any solution to $x^9+y^{10}=z^{11}$ in positive integers, I just start with a random solution to $A+B=C$, e.g. $1+1=2$, and now I multiply by an appropriate power of all the primes dividing $ABC$ to get a solution. For example, if I start with $1+1=2$ then I multiply both sides by $2^N$ and deduce $2^N+2^N=2^{N+1}$. Now it's easy to find a positive $N$ with $N=0$ mod 9, $N=0$ mod 10 and $N=-1$ mod 11, and for this value of $N$ we get a solution in positive integers to $x^9+y^{10}=z^{11}$.
But this trick relies on the fact that 9, 10, 11 are pairwise coprime. It wouldn't surprise me if an extension of the trick could give a solution in positive integers to $x^6+y^{10}=z^{15}$, where the point is that the exponents aren't pairwise coprime, but 5 minutes on the back of an envelope didn't give me the trick I needed, and I thought that here might be a great place to ask.
What's the trick I've missed?
Best Answer
I agree with Qiaochu; there can't be a similar trick in this case. You can't profit from multiplying by common factors because in this case all solutions can be reduced to coprime solutions.
To see this, consider the number of factors of an arbitrary prime $p$ in the equation. We must have
$$rp^{6k}+sp^{10l}=tp^{15m}$$
with $p \nmid r,s,t$. The lowest two powers of $p$ must coincide, since otherwise we could divide through by the lowest and the remaining equation couldn't be fulfilled mod $p$. So we can divide through by this common lowest power of $p$ and leave a power of $p$ in only one of the terms. But since the factors $6$, $10$ and $15$ are such that the least common multiple of each pair is a multiple of the third, dividing through by a multiple of that least common multiple will just substract a multiple of the third factor in the exponent of the third term, still leaving a multiple of that factor. It follows that we could have divided each of the numbers by an appropriate power of $p$ to begin with, leaving a power of $p$ in only one of the three. Doing this for all primes, we can reduce all solutions to coprime solutions.