Can the mean of 2 consecutive prime numbers be prime?

Question

This is apparently a "hard" question, and I don't know if I'm missing something, but it seems trivial to me. Aside from 2, all other prime numbers are odd. So the mean of any consecutive prime number is of course even and hence divisible by 2. So the answer is obviously no.

Am I missing something here or is this problem really this simple?

Hint: That "consecutive" bit is the point. The mean is between the two numbers. — lulu, Aug 21 '20 at 17:56
Without the "consecutive" requirement, there are plenty of examples of the mean of two primes being prime. For examples, $\frac{3+7}{2}=5$ and $\frac{5+17}{2}=11$. — DreiCleaner, Aug 21 '20 at 18:00
If the mean is prime as well, the primes cannot be consecutive. — Peter, Aug 21 '20 at 18:11

score 3 · Answer 1 · answered Aug 21 '20 at 17:56

3

No, the mean of two numbers is necessarily between them. If there were a prime between your two primes, they would no longer be consecutive.

answered Aug 21 '20 at 17:56

Telastyn

131

1

looks like I botched this problem with my first thought, but nonetheless, it seems this problem is simple based on your approach. – David Aug 21 '20 at 18:01

Can the mean of 2 consecutive prime numbers be prime?

1 Answers1