cluster prime

In number theory, a cluster prime is a prime number {{mvar|p}} such that every even positive integer k ≤ p − 3 can be written as the difference between two prime numbers not exceeding {{mvar|p}} ({{OEIS2C|id=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

:97, 127, 149, 191, 211, 223, 227, 229, ... {{OEIS2C|id=A038133}}

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

{{unsolved|mathematics|Are there infinitely many cluster primes?}}

Properties

The prime gap preceding a cluster prime is always six or less. For any given prime number {{mvar|n}}, let $p_n$ denote the n-th prime number. If ${{p_{n} - p_{n - 1}}}$ ≥ 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.{{cite journal |doi=10.2307/2589585|jstor=2589585|last1=Blecksmith|first1=Richard|last2=Erdos|first2=Paul|last3=Selfridge|first3=J. L.|title=Cluster Primes|journal=The American Mathematical Monthly|year=1999|volume=106|issue=1|pages=43–48}}
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 {{mvar|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.

References

External links

{{MathWorld | urlname=ClusterPrime | title=Cluster Prime}}

Category:Classes of prime numbers