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 , where n is its index in the ordered set of prime numbers,
The first few balanced primes are
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
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 2009[update] the largest known CPAP-3 with proven primes has 7535 digits found by David Broadhurst and François Morain:
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 , where k is its index in the ordered set of prime numbers,
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