# Primes and certain unit fractions Are there primes p , q and a natural number a

Banguizb 2022-07-07 Answered
Primes and certain unit fractions
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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## Answers (1)

Janiyah Patton
Answered 2022-07-08 Author has 12 answers
Only for $p=q=2$. Indeed, if it is the case, then
$\frac{p+q}{pq}=\frac{1}{a}$
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,
$\left(p-1\right)\left(q-1\right)=pq-p-q+1=1$
Therefore, $p=q=2$

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