Exercise 9. Find all pairs ( $p, q$ ) of prime numbers for which the numbers $2 p+q, p+2 q$ and $p+q-18$ are all three prime numbers.

We recall that a prime number is an integer greater than or equal to 2, which is divisible only by 1 and itself.
Solution to Exercise 9 Let $\mathrm{p}$ and $\mathrm{q}$ be two prime numbers, if they exist, that satisfy the given conditions.

Since $p \geqslant 2$ and $q \geqslant 2$, we have $2 p+q \geqslant 6$ and $p+2 q \geqslant 6$. As these are prime numbers, it follows that $2 p+q$ and $p+2 q$ are odd, and thus $p$ and $q$ are odd. However, the number $\mathrm{p}+\mathrm{q}-18$ is an even prime number, hence $\mathrm{p}+\mathrm{q}-18=2$.

We are thus led to find the prime numbers $\mathrm{p}$ and $\mathrm{q}$ such that $\mathrm{p}+\mathrm{q}=20$. A direct and "by hand" exploration shows that the pair $(p, q)$ is one of the pairs $(3,17),(7,13),(13,7),(17,3)$.

If we now return to the condition that $2 p+q$ and $p+2 q$ are prime, we observe that only the cases $p=3, q=17$ and $p=17, q=3$ actually correspond to solutions of the problem posed.