In number theory, a cluster prime is a prime number p such that every even positive integer k ≤ p − 3 can be written as the difference between two prime numbers not exceeding p (OEIS:
A038134). For example, the number
23 is a cluster prime because 23 − 3 = 20, and every even integer from 2 to 20, inclusive, is the difference of at least one pair of prime numbers not exceeding 23:
5 − 3 = 2
7 − 3 = 4
11 − 5 = 6
11 − 3 = 8
13 − 3 = 10
17 − 5 = 12
17 − 3 = 14
19 − 3 = 16
23 − 5 = 18
23 − 3 = 20
On the other hand,
149 is not a cluster prime because 140 < 146, and there is no way to write 140 as the difference of two primes that are less than or equal to 149.
By convention, 2 is not considered to be a cluster prime. The first 23 odd primes (up to 89) are all cluster primes. The first few odd primes that are not cluster primes are
The
prime gap preceding a cluster prime is always six or less. For any given prime number n, let denote the n-th prime number. If ≥ 8, then − 9 cannot be expressed as the difference of two primes not exceeding ; thus, is not a cluster prime.
The converse is not true: the smallest non-cluster prime that is the greater of a pair of gap length six or less is
227, a gap of only four between 223 and 227. 229 is the first non-cluster prime that is the greater of a
twin prime pair.
The set of cluster primes is a
small set. In 1999, Richard Blecksmith proved that the sum of the reciprocals of the cluster primes is finite.[1]
Blecksmith also proved an explicit upper bound on C(x), the number of cluster primes less than or equal to x. Specifically, for any positive integer m: for all
sufficiently large x.
It follows from this that
almost all prime numbers are absent from the set of cluster primes.
References
^Blecksmith, Richard; Erdos, Paul; Selfridge, J. L. (1999). "Cluster Primes". The American Mathematical Monthly. 106 (1): 43–48.
doi:
10.2307/2589585.
JSTOR2589585.