What is the probability of two random positive integers being both nonsquare? Suppose we pick two integers m and n at randomly and independently from the set {1,2,3.......N}. What is the probability that both m and  n are nonsquares?By a nonsquare I mean a positive integer which is not a perfect square.What happens to this probability as N goes to infinity?

Well given any set of $N$ integers, there are exactly $\lfloor\sqrt{N}\rfloor$ integers that are perfect squares. Meaning that there are $N - \lfloor\sqrt{N}\rfloor$ integers that aren't perfect squares.
Therefore the probability of choosing 2 non-squares is $$\left(\frac {N - \lfloor\sqrt{N}\rfloor}{N}\right)^2 = \left(1 - \frac {\lfloor\sqrt{N}\rfloor}{N}\right)^2$$
As $\lim_{N\to \infty}$, the expression approaches 1.
