How can I find a non-prime number whose square root is irrational?

Question

I already know that when $n$ is prime, that $\sqrt n$ is irrational (this is true in every case), but I know that this isn't only true for primes, $\sqrt 8$ is irrational, but it's not a prime number.

So how could I find numbers like these, where it's square root is an irrational number, but yet it's not prime?

Answer 1

Hint: if $n \in \mathbb{N}$ is not a perfect square, then $\sqrt{n}$ is irrational.

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[ EDIT ]    Examples of such non-prime $n$ whose square root is irrational:

• any non-prime integer whose prime factorization includes a prime at an odd power;

• $m!\;$ for any $\;m \gt 2$;

• $m^2 - 1\;$ for any $\;m \gt 2$.