Arithmetic progressions consisting of unlike powers

In this paper we present some new results about unlike powers in arithmetic progression. We prove among other things that for given $k\ge 4$ and $L\ge 3$ there are only finitely many arithmetic progressions of the form $(x_0^{l_0},x_1^{l_1},\mathrm{.}\mathrm{.}\mathrm{.},x_{k-1}^{l_{k-1}})$ with $x_i\in \mathbb{Z},$ gcd$(x_0,x_1)=1$ and $2\le l_i\le L$ for $i=0,1,\mathrm{.}\mathrm{.}\mathrm{.},k-1.$ Furthermore, we show that, for L=3, the progression $(1,1,\mathrm{.}\mathrm{.}\mathrm{.},1)$ is the only such progression up to sign.