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.
Famous quotes containing the words balanced and/or prime:
“I balanced all, brought all to mind,
The years to come seemed waste of breath,
A waste of breath the years behind
In balance with this life, this death.”
—William Butler Yeats (1865–1939)
“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)