USAMO

Since $q(p-1)$ is a perfect square and $q$ is prime, we should have $p - 1 = qb^2$ for some positive integer $b$ . Let $a^2 = p - q$ . Therefore, $q = p - a^2$ , and substituting that into the $p - 1 = qb^2$ and solving for $p$ gives $$p = \frac{a^2b^2 - 1}{b^2 - 1} = \frac{(ab - 1)(ab + 1)}{b^2 - 1}.$$ Notice that we also have $$p = \frac{a^2b^2 - 1}{b^2 - 1} = a^2 + \frac{a^2 - 1}{b^2 - 1}$$ and so $b^2 - 1 | a^2 - 1$ . We run through the cases
: Then so , which works. : This means , so , a contradiction. : This means that . Since can be split up into two factors such that and , we get
$$p = \frac{ab - 1}{F_1} \cdot \frac{ab + 1}{F_2}$$ and each factor is greater than $1$ , contradicting the primality of $p$ .
Thus, the only solution is $(p, q) = (3, 2)$ .