Show that there is no positive integer n so that is rational.

Let us suppose there exist a positive integer for which is a rational number, where p and q are positive and q ≠ 0

Rationalizing the RHS,

⇒ (∵ (a –b) (a +b) = a² – b²)

Now

⇒ …… eq 2

Adding eq 1 and 2

⇒

Hence there is no positive integer n for which is a rational number