This document discusses different patterns that arise with prime numbers, including twin primes, cousin primes, and sexy primes. Twin primes are pairs of prime numbers that differ by 2, such as 3 and 5 or 11 and 13. Cousin primes differ by 4, like 3 and 7 or 43 and 47. Sexy primes have a gap of 6 between them and are sometimes grouped in constellations of three primes, such as 5, 11, and 17. The document provides examples of primes demonstrating these patterns and notes some unique properties of each type.

1. Twin Primes
Cousin Primes
Sexy Primes

2. Numbers hold a certain sense of intrinsic beauty that most
of us cannot deny. Even if you do fit in the cadre of people
who feel an aversion towards numbers, there are still
patterns that almost anyone can see. Today, we will be
looking at a few such patterns that arise in the series of
numbers known as primes. Now, before we step into
investigating these patterns, let’s first take a brief overview
of what prime numbers actually are.

3. Prime Numbers
A prime number is a number that can be defined as a natural
number. Without straying away too much from the topic at hand, a
natural number can simply be defined as those numbers that are used
for counting and ordering. To further elaborate, prime numbers are
whole numbers greater than 1 that cannot be obtained as the
product of any other whole number. It can only be obtained as the
product of one and the number itself.

4. Let’s take the example of two numbers—5 and 6. Now, 5
is a prime number because we cannot obtain it from the
multiplication of any other whole number preceding it,
such as 2 and 3. It can only be obtained by taking the
product of 1 and 5 (the number itself). Now, the number 6
does not follow the definition of a prime. It can be obtained
as the product of 2 and 3. It can also be obtained as the
product of 1 and 6. Such a number that has more multiples
apart from 1 and itself are known as composite numbers.

5. Prime vs Composite numbers

6. Twin Primes

7. A twin prime can be defined as a prime
number that is either two less or two more
than another prime number. Putting this in
more simple terms, they are two
primes that have a gap of two between
them. What makes them highly unique is the
fact that they become exceedingly rare as one
examines larger ranges.

8. Now, let’s take a look at the first set of twin
primes:
(3, 5), (5, 7), (11, 13), (17, 19), (29, 31),
(41, 43), (59, 61), (71, 73), (101, 103),
(107, 109), (137, 139), …

9. The unique property in the above pairs is that 5 is the
only number to appear in two distinct twin prime
numbers. We can only give a generalized form into
which all twin primes fall, and that is (6n-1,6n+1)
where n must be a natural number. The sum of all the
twin pair numbers are divisible by 12. This applies to
all twin pairs except the pair (3,5), as the sum of this
pair is not divisible by 12

11. Cousin Primes

12. Cousin primes are those primary numbers
that differ from another by a gap of 4. The
only prime belonging to two pairs of cousin
primes is 7. One of the numbers
in n, n+4, and n+8 will always be divisible by
3, which makes it in such a way where n=3 is
the only way all three numbers can be prime.

13. Some of the cousin primes below 1000 are:
(3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47), (67, 71),
(79, 83), (97, 101), (103, 107), (109, 113), (127, 131), (163,
167), (193, 197), (223, 227), (229, 233), (277, 281), (307,
311), (313, 317), (349, 353), (379, 383), (397, 401), (439,
443), (457, 461), (463,467), (487, 491), (499, 503), (613,
617), (643, 647), (673, 677), (739, 743), (757, 761), (769,
773), (823, 827), (853, 857), (859, 863), (877, 881), (883,
887), (907, 911), (937, 941), (967, 971)

15. Sexy Primes

16. In mathematics, sexy primes can be defined as
those primes that differ from another by a gap of
6. An example of this is 5 and 11. A fun fact about
sexy primes is that the name is a pun stemming
from the Latin word for six, which is sex! In fact,
unlike its predecessors, sexy primes come in what
are known as constellations. There are three types of
groupings for sexy primes.

17. The sexy prime pairs under 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)