Every linear order isomorphic to its cube is isomorphic to its square

In 1958, Sierpiński asked whether there exists a linear order X that is isomorphic to its lexicographically ordered cube but is not isomorphic to its square. The main result of this paper is that the answer is negative. More generally, if X is isomorphic to any one of its finite powers Xn, n>1, it is isomorphic to all of them.