Are there any other twin primes with this property?

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The discussion centers on the property of twin primes, specifically the pair 5 and 7, where half their sum equals a perfect number. It is established that no other twin primes exhibit this property due to the rarity of perfect numbers. The reasoning provided indicates that any even perfect number, apart from 6, cannot be divisible by three, leading to the conclusion that one of the twin primes must be non-prime. Thus, no additional twin primes meet the criteria.

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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
 
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nikolany said:
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.
 
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.
 

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