Are there infinitely many natural numbers whose square root is rational?

Question

True or false? There are infinitely many natural numbers $n$ for which $\sqrt{n}$ is rational.

Is this statement true/false? And how can I construct a proof of its statement or negation?

Answer 1

Try showing that there are infinitely-many natural numbers whose square root is a natural number. (It turns out that that's the only way for the square root of a natural number to be rational, but that's a side note.)

Answer 2

Hint: can you see a pattern here? $$ 0,1,4,9,\dotsc $$

Answer 3

Supose you have an infinite number of with of rationals q with the form q = a/b where a and b are integers and coprimes. So, q is rational.

But q^2 = a^2/b^2 is rational and sqrt (q^2) = a/b, so is true.