Any positive integer solutions to $x^6+y^{10}=z^{15}$?

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.

I claim that a trick of this kind doesn't exist in this case. As evidence, observe that the equation has no solutions for which $31 \nmid xy$. To see this, note that

$$x^6 \equiv 0, 1, 2, 4, 8, 16 \bmod 31$$ $$y^{10} \equiv 0, 1, 5, 25 \bmod 31$$ $$z^{15} \equiv 0, 1, 30 \bmod 31.$$

But any reasonable perturbation of this trick that I can think of would allow you to write down a solution in which $x, y, z$ are all coprime to $31$.