balanced prime

The first few balanced primes are

5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903 {{OEIS|id=A006562}}.

{{unsolved|mathematics|Are there infinitely many balanced primes?}}

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|2023}} the largest known CPAP-3 has 15004 decimal digits and was found by Serge Batalov. It is:[http://primerecords.dk/cpap.htm#k3 The Largest Known CPAP's]. Retrieved on 2023-01-06.

:$p\_n\; =\; 2494779036241\; \backslash times\; 2^\{49800\}\; +\; 7,\backslash quad\; p\_\{n-1\}\; =\; p\_n-6,\backslash quad\; p\_\{n+1\}\; =\; p\_n+6.$

(The value of n, i.e. its position in the sequence of all primes, is not known.)

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, the $k$th prime number $p\_k$ is a balanced prime of order $n$ if

:$p\_k\; =\; \{\; \backslash sum\_\{i=1\}^n\; (\{p\_\{k\; -\; i\}\; +\; p\_\{k\; +\; i\})\}\; \backslash over\; 2n\}.$

Thus, an ordinary balanced prime is a balanced prime of order 1. The sequences of balanced primes of orders 2, 3, and 4 are {{nowrap|A082077}}, {{nowrap|A082078}}, and {{nowrap|A082079}} in the OEIS respectively.

• 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

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{{Prime number classes}}

Category:Classes of prime numbers

Category:Unsolved problems in number theory