here's a solution:
Suppose is an integer. Then we have
Set and where gcd(a, b) is the greatest common divisor of a and b. Note that both p and q are integers but that they don't have any common prime factor.
Every prime factor in k also has to be in and , but since , and share no prime factors. So
But there are no integers in the interval (q, q+1).
So cannot be an integer.