Primes and certain unit fractions

Question

Are there primes $p,q$ and a natural number $a$ such that $\frac{1}{p}+\frac{1}{q}=\frac{1}{a}$?

Answer 1

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, $p=q=2$.