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Find all pairs $(p, q)$ of prime numbers, the difference of whose fifth powers is also a prime number.

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We can assume that $p>q$. Since $p^{5}-q^{5}$ is divisible by $p-q$, and the quotient is certainly greater than 1, then $p-q=1$.

Therefore, $p=3, q=2$.

Indeed, the number $3^{5}-2^{5}=211-$ is a prime number.

## Answer

$\{3,2\}$.