Balanced Primes
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Basic Info
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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.

## Examples

- 53 is a balanced prime. The previous prime number is 47, while the next prime number is 59, both of which are 6 numbers away from 53.
- 5 is a balanced prime. The pervious prime number is 3, while the next prime number is 7, both of which are 2 numbers away from 5.