In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, a cluster prime is a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
such that every even positive
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
''k'' ≤ p − 3 can be written as the difference between two prime numbers not exceeding (). 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 , 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
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair or In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''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
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is every element of .
Dually, a lower bound or minorant of is defined to be an element of that is less ...
on C(x), the number of cluster primes less than or equal to x. Specifically, for any positive integer :
for all
sufficiently large
In the mathematical areas of number theory and analysis, an infinite sequence or a function is said to eventually have a certain property, if it does not have the said property across all its ordered instances, but will after some instances have ...
x.
** It follows from this that
almost all
In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning o ...
prime numbers are absent from the set of cluster primes.
References
External links
* {{MathWorld , urlname=ClusterPrime , title=Cluster Prime
Classes of prime numbers