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 .

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:

$p\_n\; =\; 197418203\; times\; 2^\{25000\}\; -\; 1,\; p\_\{n-1\}\; =\; p\_n-6090,\; p\_\{n+1\}\; =\; p\_n+6090.$

The value of n is not known.

See 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.