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


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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, $p=q=2$.

  • 3
    $\begingroup$ Possible simplification: Assume both p and q are odd. Then pq is odd and p + q is even. Assume one is odd and the other is even (i.e. equal to 2). Then p + q is odd and pq is even. Both cases have an odd number equal to an even number, which doesn't work. Thus, p and q are both even, and 2 is the only even prime. $\endgroup$ – Kevin Jul 20 '15 at 19:41

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