mathschallenge.net

Problem

If we add the primes 11 and 23 we get 34, which is twice the prime 17.

Give that $p$ and $q$ are consecutive primes, show that the equation $p + q = 2r$, where $r$ is prime, has no solutions.

Solution

If we rearrange $p + q = 2r$, we get $r = \dfrac{p + q}{2}$. That is, $r$ is the mean of $p$ and $q$. But as they are consecutive primes there can be no prime in-between them. Hence the equation has no solutions if $r$ is prime.