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, the ${\displaystyle n}$th prime number ${\displaystyle p_{n))$ is a balanced prime if

${\displaystyle p_{n}=((p_{n-1}+p_{n+1)) \over 2}.}$

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

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^[update] the largest known CPAP-3 has 15004 decimal digits and was found by Serge Batalov. It is:^[1]

${\displaystyle p_{n}=2494779036241\times 2^{49800}+7,\quad p_{n-1}=p_{n}-6,\quad p_{n+1}=p_{n}+6.}$

(The value of n, i.e. its position 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, the ${\displaystyle k}$th prime number ${\displaystyle p_{k))$ is a balanced prime of order ${\displaystyle n}$ if

${\displaystyle p_{k}={\sum _{i=1}^{n}({p_{k-i}+p_{k+i})} \over 2n}.}$

Thus, an ordinary balanced prime is a balanced prime of order 1. The sequences of balanced primes of orders 2, 3, and 4 are A082077, A082078, and A082079 in the OEIS respectively.