In number theory, a cluster prime is a prime number such that every even positive integer ''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,

149 149 may refer to: *149 (number), a natural number * AD 149, a year in the 2nd century AD *149 BC __NOTOC__ Year 149 BC was a year of the Roman calendar, pre-Julian Roman calendar. At the time it was known as the Year of the Consulship of Censori ...

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,

127 127 may refer to: *127 (number), a natural number *AD 127, a year in the 2nd century AD *127 BC, a year in the 2nd century BC *127 (band), an Iranian band See also *List of highways numbered 127 Route 127 or Highway 127 can refer to multiple roads ...

191 Year 191 (Roman numerals, CXCI) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Apronianus and Bradua (or, less frequently, year 944 ' ...

211 Year 211 ( CCXI) was a common year starting on Tuesday of the Julian calendar. At the time, in the Roman Empire it was known as the Year of the Consulship of Terentius and Bassus (or, less frequently, year 964 ''Ab urbe condita''). The denomin ...

223 __NOTOC__ Year 223 ( CCXXIII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Maximus and Aelianus (or, less frequently, year 976 ' ...

227 Year 227 ( CCXXVII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Senecio and Fulvius (or, less frequently, year 980 ''Ab urbe condi ...

229 __NOTOC__ Year 229 ( CCXXIX) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Cassius (or, less frequently, year 982 '' ...

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

Properties

* The

prime gap A prime gap is the difference between two successive prime numbers. The ''n''-th prime gap, denoted ''g'n'' or ''g''(''p'n'') is the difference between the (''n'' + 1)-th and the ''n''-th prime numbers, i.e. :g_n = p_ - p_n.\ W ...

preceding a cluster prime is always six or less. For any given prime number , 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

, 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 (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin p ...

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 :

C(x) <

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 doesn't have the said property across all its ordered instances, but will after some instances have pa ...

x. ** It follows from this that

almost all In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathem ...

prime numbers are absent from the set of cluster primes.

References

External links

* {{MathWorld , urlname=ClusterPrime , title=Cluster Prime Classes of prime numbers