Suppose with and . Then . The exponent of the prime in the prime factorisation of a square is even, while in it is odd, contradicting uniqueness of prime factorisation. The same argument shows that if satisfies for a natural number , then must be a perfect square.

Suppose with and . Every number of the form with integers equals for some , so if it is positive it is at least . But , so for large we have ; replacing even powers of by powers of , each has the form , a contradiction.

The gap this exposes motivates the real numbers: the set of positive rationals with square less than has no largest element, since lies in the set and exceeds for every member , and no rational least upper bound exists.