We consider a real $\alpha$ such that for every prime number $p$, $p^\alpha$ is a natural number. We apply relation
$$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} f(x+j) = f^{(n)}(x + y_n) \quad \text{(IV.1)}$$
to the function $f_\alpha(x) = x^\alpha$ and to the integer $n = \lfloor \alpha \rfloor + 1$.
Conclude that $\alpha$ is a natural number.