Let $p$ and $q$ be prime numbers. Prove by contradiction that if $p + q$ is prime, then $p = 2$ or $q = 2$.

Question

Let $p$ and $q$ be prime numbers. Prove by contradiction that if $p + q$ is prime, then $p = 2$ or $q = 2$.

I'm having trouble with this proof, if anyone could help out I would appreciate it.

Answer 1

Assume as hypothesis:

• $p+q$ is prime with $p,q\neq 2$

hence $p$ and $q$ are odds (since $2$ is the unique even prime) and then $p+q$ is even which is in contradiction with the hypothesis.