One should verify that x1​,x2​ and x3​ are not integers. Here is a sketch of one way to do this. First, prove by induction that the sequence {xn​} is increasing. Next, since 1<x1​<2 and x4​=3, it suffices to show that neither x2​ nor x3​ is 2. As for x2​, note that x1​<3/2 (since x1​3=3<27/8 ). Thus
x2​=x1x1​​<(3/2)3/2=27/8​<2
To show x3​>2, show that x1​>2​ and do a similar manipulation.
Note. x2​ and x3​ are integers raised to radicals, e.g.,
x2​=3(31/9​)
There is a general theorem which says (as a special case) that such numbers cannot be integers (except when the base is 0 or 1), but the proof is very deep.