**Theorem.**

Let be the least common multiple of and with for all , for some and for some . Then for infinitely many .

*Proof.*

It is clear that it is necessary that the numerator of does not equal infinitely often. So let’s start with that. Assume that the numerator of *does* equal for some large , where is not a prime power. Then: . And since , we have , which is never a unit fraction. Let be an integer for which and be a prime larger than such that divides . Note that if such a prime doesn’t exist for any , we would have that all the prime divisors of (which, as we saw, exist for infinitely many ), are smaller than or equal to . But a prime divisor of which is smaller than or equal to also divides . And this would immediately imply our theorem. So we may assume the existence of a prime dividing . Now, let be any (positive) multiple of lcm and set ;