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 , where n is its index in the ordered set of prime numbers,

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 (sequence A006562 in the OEIS).

## Infinitude

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 2014^{[update]} the largest known CPAP-3 has 10546 digits and was found by David Broadhurst. It is:^{[1]}

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 , where

*k*is its index in the ordered set of prime numbers,

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 (sequence A082077 in the OEIS), (sequence A082078 in the OEIS), and (sequence A082079 in the OEIS) respectively.

## See also

- Strong prime, a prime that is greater than the arithmetic mean of its two neighboring primes

## References

**^**The Largest Known CPAP's. Retrieved on 2014-06-13.