Are there any other twin primes with this property?

[nikolany]

The twin primes 5 and 7 are such that one half their sum is a perfect number. Are there any other twin primes with this property?

It works for p=5. I think it should be of the form 1/2*(p+P+2). Is this true? How can I prove it?

Thx

 

[CRGreathouse]

Are there any other twin primes with this property?

I didn't find any. It's not too surprising, given how rare perfect numbers are.

 

[MrJB]

There aren't any more. An even perfect number (other than 6) is not divisible by three since it is the product of a power of 2 and a mersenne prime. Therefore one of P-1 or P+1 must be divisible by 3, and thus not prime.