# Balanced prime

A balanced prime is a prime number that 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,

$p_n = {{p_{n - 1} + p_{n + 1}} \over 2}.$

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 OEIS).

For example, 53 is the sixteenth prime. The fifteenth and seventeenth primes, 47 and 59, add up to 106, half of which is 53, thus 53 is a balanced prime.

When 1 was considered a prime number, 2 would have correspondingly been considered the first balanced prime since

$2 = {1 + 3 \over 2}.$

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 the largest known CPAP-3 has 10546 digits found by David Broadhurst:[1]

$p_n = 1213266377 \times 2^{35000} + 2429,\quad p_{n-1} = p_n-2430,\quad p_{n+1} = p_n+2430.$

The value of n is not known.

When a prime is greater than the arithmetic mean of its two neighboring primes, it is called a strong prime. When it is less, it is called a weak prime.

## ... 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,

$p_k = { \sum_{i=1}^n ({p_{k - i} + p_{k + i})} \over 2n}.$

As such, the aforementioned primes are balanced primes of order 1. Other orders can be seen at 2 (sequence A082077 in OEIS), 3 (sequence A082078 in OEIS), and 4 (sequence A082079 in OEIS).

## References

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