Sexy prime
Encyclopedia
In mathematics
, a sexy prime is a prime number
that differs from another prime number by six. For example, the numbers 5 and 11 are both sexy primes, because they differ by 6. If p + 2 or p + 4 is also prime, then the sexy prime is part of a prime triplet
.
The term "sexy prime" stems from the Latin
word for six: sex.
) below 500 are:
, (7,13), (11,17), (13,19), (17,23), (23,29), (31,37), (37,43), (41,47), (47,53), (53,59), (61,67), (67,73), (73,79), (83,89), (97,103), (101,107), (103,109), (107,113), (131,137), (151,157), (157,163), (167,173), (173,179), (191,197), (193,199), (223,229), (227,233), (233,239), (251,257), (257,263), (263,269), (271,277), (277,283), (307,313), (311,317), (331,337), (347,353), (353,359), (367,373), (373,379), (383,389), (433,439), (443,449), (457,463), (461,467).
the largest known sexy prime was found by Ken Davis and has 11593 digits. The primes are (p, p+6) for
9001# = 2×3×5×...×9001 is a primorial
, i.e. the product of primes ≤ 9001.
, (17,23,29), (31,37,43), (47,53,59), (67,73,79), (97,103,109), (101,107,113), (151,157,163), (167,173,179), (227,233,239), (257,263,269), (271,277,283), (347,353,359), (367,373,379), (557,563,569), (587,593,599), (607,613,619), (647,653,659), (727,733,739), (941,947,953), (971,977,983).
the largest known sexy prime triplet, found by Ken Davis had 5132 digits:
representation (except for the quadruplet with p = 5). The sexy prime quadruplets below 1000 are :, (11,17,23,29), (41,47,53,59), (61,67,73,79), (251,257,263,269), (601,607,613,619), (641,647,653,659).
In November 2005 the largest known sexy prime quadruplet, found by Jens Kruse Andersen had 1002 digits:
In September 2010 Ken Davis announced a 1004-digit quadruplet with p = 23333 + 1582534968299.
of five terms with common difference 6, because 6>5 and the two numbers are relatively prime, one of the terms must be divisible by 5. Thus, the only sexy prime quintuplet is (5,11,17,23,29) with no longer sequence of sexy primes possible.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a sexy prime is a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
that differs from another prime number by six. For example, the numbers 5 and 11 are both sexy primes, because they differ by 6. If p + 2 or p + 4 is also prime, then the sexy prime is part of a prime triplet
Prime triplet
In mathematics, a prime triplet is a set of three prime numbers of the form or...
.
The term "sexy prime" stems from the Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
word for six: sex.
n# notation
As used in this article, n# stands for the product 2 · 3 · 5 · 7 · … of all the primes ≤ n.Sexy prime pairs
The sexy primes (sequences and in OEISOn-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences , also cited simply as Sloane's, is an online database of integer sequences, created and maintained by N. J. A. Sloane, a researcher at AT&T Labs...
) below 500 are:
, (7,13), (11,17), (13,19), (17,23), (23,29), (31,37), (37,43), (41,47), (47,53), (53,59), (61,67), (67,73), (73,79), (83,89), (97,103), (101,107), (103,109), (107,113), (131,137), (151,157), (157,163), (167,173), (173,179), (191,197), (193,199), (223,229), (227,233), (233,239), (251,257), (257,263), (263,269), (271,277), (277,283), (307,313), (311,317), (331,337), (347,353), (353,359), (367,373), (373,379), (383,389), (433,439), (443,449), (457,463), (461,467).
the largest known sexy prime was found by Ken Davis and has 11593 digits. The primes are (p, p+6) for
- p = (117924851×587502×9001#×(587502×9001#+1)+210)×(587502×9001#−1)/35+5.
9001# = 2×3×5×...×9001 is a primorial
Primorial
In mathematics, and more particularly in number theory, primorial is a function from natural numbers to natural numbers similar to the factorial function, but rather than multiplying successive positive integers, only successive prime numbers are multiplied...
, i.e. the product of primes ≤ 9001.
Sexy prime triplets
Sexy primes can be extended to larger constellations. Triplets of primes (p, p + 6, p + 12) such that p + 18 is composite are called sexy prime triplets. Those below 1000 are :, (17,23,29), (31,37,43), (47,53,59), (67,73,79), (97,103,109), (101,107,113), (151,157,163), (167,173,179), (227,233,239), (257,263,269), (271,277,283), (347,353,359), (367,373,379), (557,563,569), (587,593,599), (607,613,619), (647,653,659), (727,733,739), (941,947,953), (971,977,983).
the largest known sexy prime triplet, found by Ken Davis had 5132 digits:
- p = (84055657369 · 205881 · 4001# · (205881 · 4001# + 1) + 210) · (205881 · 4001# - 1) / 35 + 1.
Sexy prime quadruplets
Sexy prime quadruplets (p, p + 6, p + 12, p + 18) can only begin with primes ending in a 1 in their decimalDecimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
representation (except for the quadruplet with p = 5). The sexy prime quadruplets below 1000 are :, (11,17,23,29), (41,47,53,59), (61,67,73,79), (251,257,263,269), (601,607,613,619), (641,647,653,659).
In November 2005 the largest known sexy prime quadruplet, found by Jens Kruse Andersen had 1002 digits:
- p = 411784973 · 2347# + 3301.
In September 2010 Ken Davis announced a 1004-digit quadruplet with p = 23333 + 1582534968299.
Sexy prime quintuplets
In an arithmetic progressionArithmetic progression
In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant...
of five terms with common difference 6, because 6>5 and the two numbers are relatively prime, one of the terms must be divisible by 5. Thus, the only sexy prime quintuplet is (5,11,17,23,29) with no longer sequence of sexy primes possible.
See also
- Twin primeTwin primeA twin prime is a prime number that differs from another prime number by two. Except for the pair , this is the smallest possible difference between two primes. Some examples of twin prime pairs are , , , , and...
(two primes that differ by 2) - Cousin primeCousin primeIn mathematics, cousin primes are prime numbers that differ by four; compare this with twin primes, pairs of prime numbers that differ by two, and sexy primes, pairs of prime numbers that differ by six....
(two primes that differ by 4) - Prime k-tuplePrime k-tupleIn number theory, a prime k-tuple is an ordered set of values representing a repeatable pattern of prime numbers. A k-tuple is represented as to represent any set of values for all values of n. In practice, 0 is usually used for the lowest value of the k-tuple...