In number theory, a balanced prime is a prime number with equal-sized prime gaps above and below it, so that it is equal to the arithmetic mean of the nearest primes above and below. Or to put it algebraically, given a prime number $p_{n)$ , where n is its index in the ordered set of prime numbers,

$p_{n}=((p_{n-1}+p_{n+1)) \over 2}.$ For example, 53 is the sixteenth prime; the fifteenth and seventeenth primes, 47 and 59, add up to 106, and half of that is 53; thus 53 is a balanced prime.

## Examples

The first few balanced primes are

5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903 (sequence A006562 in the OEIS).

## Infinitude

Unsolved problem in mathematics:

Are there infinitely many balanced primes?

It is conjectured that there are infinitely many balanced primes.

Three consecutive primes in arithmetic progression is sometimes called a CPAP-3. A balanced prime is by definition the second prime in a CPAP-3. As of 2023 the largest known CPAP-3 has 15004 digits and was found by Serge Batalov. It is:

$p_{n}=2494779036241\times 2^{49800}+7,\quad p_{n-1}=p_{n}-6,\quad p_{n+1}=p_{n}+6.$ The value of n (its rank in the sequence of all primes) is not known.

## Generalization

The balanced primes may be generalized to the balanced primes of order n. A balanced prime of order n is a prime number that is equal to the arithmetic mean of the nearest n primes above and below. Algebraically, given a prime number $p_{k)$ , where k is its index in the ordered set of prime numbers,

$p_{k}={\sum _{i=1}^{n}({p_{k-i}+p_{k+i})} \over 2n}.$ Thus, an ordinary balanced prime is a balanced prime of order 1. The sequences of balanced primes of orders 2, 3, and 4 are given as sequences A082077, A082078, and A082079 in the OEIS respectively.