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# 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,

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

The first few balanced primes are

5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103 .

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 $2 = \left\{\left(1 + 3\right) over 2\right\}.$

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:

$p_n = 197418203 times 2^\left\{25000\right\} - 1, p_\left\{n-1\right\} = p_n-6090, p_\left\{n+1\right\} = p_n+6090.$
The value of n is not known.