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

5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103 (sequence A006562 in OEIS).

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 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.

### Other articles related to "prime, balanced prime":

... cube number 9025 – 952, centered octagonal number 9029 – Sophie Germain

**prime**9045 – triangular number 9059 – Sophie Germain

**prime**9072 – decagonal number 9077 – Markov number 9091. 9870 – triangular number 9871 –

**balanced prime**9880 – tetrahedral number 9887 – safe

**prime**9899 – ISO 9899 standard for C programming language 9901 ...

### Famous quotes containing the words prime and/or balanced:

“Vanessa wanted to be a ballerina. Dad had such hopes for her.... Corin was the academically brilliant one, and a fencer of Olympic standard. Everything was expected of them, and they fulfilled all expectations. But I was the one of whom nothing was expected. I remember a game the three of us played. Vanessa was the President of the United States, Corin was the British *Prime* Minister—and I was the royal dog.”

—Lynn Redgrave (b. 1943)

“[T]hat moment of evening when the light and the darkness are so evenly *balanced* that the constraint of day and the suspense of night neutralize each other, leaving absolute mental liberty. It is then that the plight of being alive becomes attenuated to its least possible dimensions.”

—Thomas Hardy (1840–1928)