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?

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    $\begingroup$ You're confusing "sum" and "mean" in your reasoning. $\endgroup$ – Jack Pfaffinger Aug 21 at 17:55
  • $\begingroup$ The mean of $7$ and $11$ is $9$. $\endgroup$ – lulu Aug 21 at 17:56
  • $\begingroup$ $7+11$ is indeed even, but the mean of $7$ and $11$ is $9$. $\endgroup$ – Brian M. Scott Aug 21 at 17:56
  • $\begingroup$ Hint: That "consecutive" bit is the point. The mean is between the two numbers. $\endgroup$ – lulu Aug 21 at 17:56
  • $\begingroup$ 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 $\frac{5+17}{2}=11$. $\endgroup$ – DreiCleaner Aug 21 at 18:00

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.

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

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