Balanced prime
Encyclopedia
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 .
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. 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.
. When it is less, it is called a weak prime.
, where k is its index in the ordered set of prime numbers,
As such, the primes abovementioned are balanced primes of order 1. Other orders can be seen at: 2, 3, 4.
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
that is equal to the arithmetic mean
Arithmetic mean
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...
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
53 (number)
53 is the natural number following 52 and preceding 54.-In mathematics:Fifty-three is the 16th prime number. It is also an Eisenstein prime....
, 157, 173
173 (number)
173 is the natural number following 172 and preceding 174.-In mathematics:* 173 is an odd number* 173 is a balanced prime* 173 is a deficient number* 173 is an Eisenstein prime with no imaginary part* 173 is an Odious number* 173 is a prime number...
, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103 .
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
Primes in arithmetic progression
In number theory, the phrase primes in arithmetic progression refers to at least three prime numbers that are consecutive terms in an arithmetic progression, for example the primes ....
is sometimes called a CPAP-3. A balanced prime is by definition the second prime in a CPAP-3. 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.
See also
When a prime is greater than the arithmetic mean of its two neighboring primes, it is called a strong primeStrong prime
In mathematics, a strong prime is a prime number with certain special properties. The definitions of strong primes are different in cryptography and number theory.- Definition in cryptography :...
. When it is less, it is called a weak prime.
... 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, given a prime number, where k is its index in the ordered set of prime numbers,As such, the primes abovementioned are balanced primes of order 1. Other orders can be seen at: 2, 3, 4.