TITLE: Primes and certain unit fractions
QUESTION [9 upvotes]: Are there primes $p,q$ and a natural number $a$ such that $\frac{1}{p}+\frac{1}{q}=\frac{1}{a}$?

REPLY [26 votes]: Only for $p=q=2$. Indeed, if it is the case, then
$$\frac{p+q}{pq}=\frac1a$$
and $p+q$ divides $pq$. But only $1$, $p$, $q$ and $pq$ divide $pq$. Certainly $p+q$ is not any of the three first numbers. The other possibility is
$$p+q=pq$$
But in this case,
$$(p-1)(q-1)=pq-p-q+1=1$$
Therefore, $p=q=2$.