Write a C code to find out the following in given two numbers:
- Sexy prime number pair
- Sexy prime number triplet
- Sexy prime number quadruplet
- Sexy prime number quintuplet
What is Sexy Prime Number? Sexy primes are prime numbers that differ from each other by 6. For example, the numbers 5 and 11 are both sexy primes, because 11-5=6
Types of groupings
- Sexy prime pairs: The sexy primes (sequences OEIS: A023201 and OEIS: A046117 in OEIS) below 500 are:
(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). As of April 2019, the largest known pair of sexy primes was found by S. Batalov and has 31,002 digits. The primes are:
(p, p+6) = (187983281 × 251478 + 4)×(5 × 251478 - 1) + 2 {\displaystyle \pm } \pm 3.[1]
- 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 (OEIS: A046118, OEIS: A046119, OEIS: A046120):
(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), (727,733,739), (941,947,953), (971,977,983). As of 2019 the largest known sexy prime triplet, found by Peter Kaiser had 6031 digits:
p = 10409207693*2^20000-1.[2]
- Sexy prime quadruplets: Sexy prime quadruplets (p, p + 6, p + 12, p + 18) can only begin with primes ending in a 1 in their decimal representation (except for the quadruplet with p = 5). The sexy prime quadruplets below 1000 are (OEIS: A023271, OEIS: A046122, OEIS: A046123, OEIS: A046124):
(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). In November 2005 the largest known sexy prime quadruplet, found by Jens Kruse Andersen had 1002 digits:
p = 411784973 · 2347# + 3301.[3] In September 2010 Ken Davis announced a 1004-digit quadruplet with p = 23333 + 1582534968299.[4]
In May 2019 Marek Hubal announced a 1138-digit quadruplet with p = 15672379112677# + 3301 + 6n.[5]
In June 2019 Peter Kaiser announced a 1534-digit quadruplet with p = 19299420002127 * 25050 + 17233 + 6*n
- Sexy prime quintuplets: In an arithmetic progression of five terms with common difference 6, one of the terms must be divisible by 5, because 5 and 6 are relatively prime. Thus, the only sexy prime quintuplet is (5,11,17,23,29); no longer sequence of sexy primes is possible.