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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 n 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 (sequence A006562 in the 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. As of 2014 the largest known CPAP-3 has 10546 digits and was found by David Broadhurst. It is:[1]

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

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

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 sequence A082077 in the OEIS, sequence A082078 in the OEIS, and sequence A082079 in the 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

References

The Largest Known CPAP's. Retrieved on 2014-06-13.

Mathematics Encyclopedia

Hellenica World - Scientific Library

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