Balanced Primes

Balanced Primes
Basic Info
Discovered by	N/A
Number of	Conjectured infinite
Description	A prime which is the mean of the previous prime and the following prime
First Few	5, 53, 157, 173, 211, 257, 263

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 theory, 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. Most of said even numbers are multiples of three, as a result of the pigeonhole principle. If the difference is not a multiple of three, one of the three numbers n-x, n or n+x will be a multiple of three.

The First Few Balanced Primes