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Balanced Primes are primes which are the mean of the next primes number and the previous prime number. This can be expressed as $ p_n = {{p_{n - 1} + p_{n + 1}} \over 2}. $, where $ {{p_n}} $ is the nth prime number and $ {{p_{n - 1}}} $ and $ {{p_{n + 1}}} $ are the previous prime number and the next prime number, respectively. 5 is the first balanced prime, as 3+7/2 =5. It is thought, although not proven, that there are infinitely many balanced primes. In order to be a balanced prime, the difference between the nth prime number and the n+1th prime number can be any even number, as long as it is the same as the gap between the nth and the n-1th. The first few balanced primes are 5, 51, 173, 257.

The First Few Balanced Primes

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