Balanced prime

In number theory, a balanced prime is a prime number with equal-sized prime gaps above and below it, so that it 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 Template:Mvar is its index in the ordered set of prime numbers,

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

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

Examples

The first few balanced primes are

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

Infinitude

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. Template:As of the largest known CPAP-3 has 10546 digits and was found by David Broadhurst. It is:

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

The value of n (its rank in the sequence of all primes) is not known.

Generalization

The balanced primes may be generalized to the balanced primes 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,

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

Thus, an ordinary balanced prime is a balanced prime of order 1.The sequences of balanced primes of orders 2, 3, and 4 are given as Template:OEIS, Template:OEIS, and Template:OEIS respectively.

See also

• Strong prime, a prime that is greater than the arithmetic mean of its two neighboring primes
• Interprime, a composite number balanced between two prime neighbours