Sexy prime

In mathematics , prime numbers with a difference of 6 are called sexy prime numbers . For example, the numbers 5 and 11 are sexy prime numbers because one is 6 smaller than the other (or the other is 6 larger than the one). If p and p +6 are sexy primes and p +2 or p +4 are also, then the two sexy primes are part of a triplet of prime numbers .

The term sexy prime numbers comes from sex - the Latin word for six.

Types of sexy prime groups

Sexy prime twins

Sexy prime twins have the form . The sexy prime numbers under 500 are: ${\displaystyle (p,p+6)}$

(5.11), (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). (Series A023201 in OEIS ) and (Series A046117 in OEIS )

In October 2019, Peter Kaiser discovered the currently largest sexy prime number pair with 50,539 digits. The first prime number of the prime number pair ( p , p +6) is p

${\displaystyle p=(520461\cdot 2^{55931}+1)\cdot (98569639289\cdot (520461\cdot 2^{55931}-1)^{2}-3)-1}$

Sexy prime number triplets

Sexy prime numbers can be expanded into a larger constellation. Triples of prime numbers of the form are called sexy prime number triplets if p +18 is a composite number , i.e. not a prime number. The sexy prime triplets under 1000 are: ${\displaystyle (p,p+6,p+12)}$

(7,13,19), (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), (647,653,659), (727,733) 947,947) (971,977,983). (Sequence A046118 in OEIS ), (Sequence A046119 in OEIS ) and (Sequence A046120 in OEIS ).

On December 30, 2019, Norman Luhn and Gerd Lamprecht discovered the currently largest sexy prime number triplet with 10602 digits. The first prime number of the prime number triplet ( p , p +6, p +12) is p

${\displaystyle p=2683143625525\cdot 2^{35176}+1}$

Sexy prime quadruplets

Quadruples of prime numbers of the form are called sexy prime quadruplets . The first prime number p must end in its decimal representation with the digit 1 (except for the first four of a kind with p = 5). The sexy prime quadruplets under 1000 are: ${\displaystyle (p,p+6,p+12,p+18)}$

(5,11,17,23), (11,17,23,29), (41,47,53,59), (61,67,73,79), (251,257,263,269), (601,607,613,619), (641,647,653,659 ).

(Sequence A023271 in OEIS ), (Sequence A046122 in OEIS ), (Sequence A046123 in OEIS ) and (Sequence A046124 in OEIS ).

In November 2005, Jens Kruse Andersen discovered the then largest sexy prime number quadruplet with over 1000 digits (namely 1002 digits). From the prime quadruple ( p , p +6, p +12, p +18) the first prime number is p

${\displaystyle p=411784973\cdot 2347\#+3301}$

2347 # = 2 · 3 · 5 ·… · 2347 is a prime faculty , i. H. the product of all prime numbers ≤ 2347.

In October 2019, Gerd Lamprecht and Norman Luhn discovered the currently largest sexy prime number quadruple with 3025 digits. From the prime quadruple ( p , p +6, p +12, p +18) the first prime number is p

${\displaystyle p={\frac {121152729080\cdot 7019\#}{1729}}+1}$

Sexy prime quintuplets

Quintuples of prime numbers of the form are called sexy prime quintuplets . However, in an arithmetic sequence of five numbers, all of which have a difference of 6, one number must be divisible by 5. So the only sexy prime quintuplet is (5, 11, 17, 23, 29). ${\displaystyle (p,p+6,p+12,p+18,p+24)}$

There can be no longer sexy prime number sequence.

Summary

In order to clarify the differences between the various prime number tuples once again, a summary of the common names is given here:

(p, p + 2)	Prime twin
(p, p + 4)	Prime cousin
(p, p + 6)	Sexy prime twin
(p, p + 2, p + 6) and (p, p + 4, p + 6)	Prime number triplet
(p, p + 6, p + 12)	Sexy prime number triplet
(p, p + 2, p + 6, p + 8)	Prime quadruples
(p, p + 6, p + 12, p + 18)	Sexy prime quadruples
(p, p + 2, p + 6, p + 8, p + 12) and (p, p + 4, p + 6, p + 10, p + 12)	Prime quintuplets
(p, p + 6, p + 12, p + 18, p + 24)	Sexy prime quintuplets

Individual evidence

1. ↑ Serge Batalov, Congrats! Nice sexy pair, 50539 digits! . Retrieved December 4, 2019.
2. ↑ Mersenneforum. Retrieved January 27, 2020 . 
3. ↑ Gerd Lamprecht: Prime numbers (collection of interesting facts). The largest known sexy CPAP's ... Retrieved February 14, 2020 .  
4. ↑ Chris K Caldwell: The Top Twenty: Consecutive Primes in Arithmetic Progression. Prime Pages, accessed February 14, 2020 .  
5. ↑ 2683143625525 · 2 ³⁵¹⁷⁶ + 1 on Prime Pages
6. ↑ Jens Kruse Andersen, "Gigantic sexy and cousin primes" . Retrieved November 30, 2015.
7. ↑ Jens Kruse Andersen, http://www.primerecords.dk/cpap.htm#sexy . Retrieved December 4, 2019.

Web links

• Wolfram MathWorld, Sexy Primes. Retrieved November 30, 2015 . 
• James Grime: Sexy Primes (and the only sexy prime quintuplet) . In: Numberphile . Brady Haran .