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 OEIS|id=A006562.

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 2005 the largest known CPAP-3 has 7535 digits found by David Broadhurst and François Morain: [http://hjem.get2net.dk/jka/math/cpap.htm#k3] :$$p_n = 197418203 imes 2^{25000} - 1, p_{n-1} = p_n-6090, p_{n+1} = p_n+6090.The value of "n" is not known.

ee also

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.