Cluster prime

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 $p$. 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


 * 97, 127, 149, 191, 211, 223, 227, 229, ...

It is not known if there are infinitely many cluster primes.

Properties

 * The prime gap preceding a cluster prime is always six or less. For any given prime number $n$, let $$p_n$$ denote the n-th prime number. If $$$$ ≥ 8, then $$p_n$$ − 9 cannot be expressed as the difference of two primes not exceeding $$p_n$$; thus, $$p_n$$ is not a cluster prime.
 * The converse is not true: the smallest non-cluster prime that is the greater of a pair of gap length six or less is 227, a gap of only four between 223 and 227. 229 is the first non-cluster prime that is the greater of a twin prime pair.
 * The set of cluster primes is a small set. In 1999, Richard Blecksmith proved that the sum of the reciprocals of the cluster primes is finite.
 * Blecksmith also proved an explicit upper bound on C(x), the number of cluster primes less than or equal to x. Specifically, for any positive integer $m$: $$C(x) < {x \over ln(x)^m}$$ for all sufficiently large x.
 * It follows from this that almost all prime numbers are absent from the set of cluster primes.