6. How many pairs of prime (not necessarily distinct) numbers ( $p ; q$) exist such that $p^{q}-p q$ is also prime? Answers:

A) at least one (+) B) at least two (+) C) at least three (-) D) none (-) E) infinitely many (-)
# Solution.

pq-pq is divisible by p, so for this expression to be a prime number, it is necessary that pq-pq=p, from which $\mathrm{p}^{\mathrm{q}-1}-\mathrm{q}=1$. Therefore, $\mathrm{p}$ and $\mathrm{q}$ must be of different parity, so one of them must be 2. If $\mathrm{q}=2$, then $\mathrm{p}^{1}-2=1, \mathrm{p}=3$. If $p=2$, then $2^{q-1}=q+1$. The inequality $2^{q-1}>q+1$ holds for $q>3$. It remains to check $q=3$.

Answer: $p=2, q=3 ; p=3, q=2$.