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)
“Ay, look: high heaven and earth ail from the prime foundation;
All thoughts to rive the heart are here, and all are vain:
Horror and scorn and hate and fear and indignation—
Oh, why did I awake? When shall I sleep again?”
—A.E. (Alfred Edward)