Let's solve a fun problem relating infinite square roots,
Find the number of pairs of positive integers (m, n) such that .
First, we observe that
is equal to
since the expression is infinite, and we can ONLY DO THIS IFF IT'S INFINITE, take note. So, the problem reduces to find the number of pairs of positive integers that satisfies the equation,
We can first say that since and are positive.
Let's square both sides and simplify the expression a little bit, and at last it will become
and it's a quadratic equation in which has real solutions if the discriminant, .
which is trivial since .
Also, using the quadratic formula, we get,
since we shall take the negative sign, this leaves
. Thus, let , where is a positive integer, then .
For , , at last, there are pairs of positive integers .