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:
“Perhaps if the future existed, concretely and individually, as something that could be discerned by a better brain, the past would not be so seductive: its demands would be balanced by those of the future. Persons might then straddle the middle stretch of the seesaw when considering this or that object. It might be fun.”
—Vladimir Nabokov (1899–1977)
“If one had to worry about one’s actions in respect of other people’s ideas, one might as well be buried alive in an antheap or married to an ambitious violinist. Whether that man is the prime minister, modifying his opinions to catch votes, or a bourgeois in terror lest some harmless act should be misunderstood and outrage some petty convention, that man is an inferior man and I do not want to have anything to do with him any more than I want to eat canned salmon.”
—Aleister Crowley (1875–1947)