Different Kinds of Prime Numbers: Twin, Cousin and Sexy Primes
What is a prime number?
Before we start looking at special kinds of prime numbers, let's remind ourselves exactly what a prime number is.
A prime number is an integer (whole number) larger than 0 that has exactly two factors (integers that divide into it without a remainder). For example, the number 5 is a prime number because it can only be divided exactly by 1 and itself. Attempting to divide by any other integer would give a decimal answer.
Likewise, 29 is prime as it can only be divided by 1 and 29. It can be easily extrapolated from our examples so far that the two factors of a prime number must always be 1 and the prime itself. We know this as all integers must be divisible by 1 and themselves, and the primes are not allowed any further factors.
Note: 1 is not a prime number as it has exactly one factor, itself.
A prime number is an integer (whole number) larger than 0 that has exactly two factors (integers that divide into it without a remainder). For example, the number 5 is a prime number because it can only be divided exactly by 1 and itself. Attempting to divide by any other integer would give a decimal answer.
Likewise, 29 is prime as it can only be divided by 1 and 29. It can be easily extrapolated from our examples so far that the two factors of a prime number must always be 1 and the prime itself. We know this as all integers must be divisible by 1 and themselves, and the primes are not allowed any further factors.
Note: 1 is not a prime number as it has exactly one factor, itself.
Prime numbers and non-prime numbers
What are twin primes?
A twin prime is a prime number that is either 2 above or 2 below another prime number. For example, 29 and 31 are both prime numbers and are two apart, hence twin primes. A larger example of twin primes is 137 and 139.
The only number that is a prime twin twice is 5, which is twinned with both 3 and 7. We can prove this by considering the fact that all prime numbers larger than 3 can be written as 6n − 1 or 6n + 1 for some positive integer n. For a prime number other than 5 to have two twins, we would need three primes, with a gap of 2 in between each one. As two of our primes are 6n −1 and 6n + 1, the third one would need to be either 6n −3 or 6n + 3, both of which are divisible by 3 and hence not prime.
Interestingly, although much work has been done in number theory looking at the links between twin primes, it is still just a conjecture that there are infinitely many twin primes. Nobody has managed to prove or disprove this conjecture.
The only number that is a prime twin twice is 5, which is twinned with both 3 and 7. We can prove this by considering the fact that all prime numbers larger than 3 can be written as 6n − 1 or 6n + 1 for some positive integer n. For a prime number other than 5 to have two twins, we would need three primes, with a gap of 2 in between each one. As two of our primes are 6n −1 and 6n + 1, the third one would need to be either 6n −3 or 6n + 3, both of which are divisible by 3 and hence not prime.
Interestingly, although much work has been done in number theory looking at the links between twin primes, it is still just a conjecture that there are infinitely many twin primes. Nobody has managed to prove or disprove this conjecture.
Record-Breaking Twin Primes
The largest twin primes currently known are 2996863034895 × 21290000 ± 1 which were discovered in September 2016.
What are cousin primes?
We have already seen that two prime numbers differing by 2 are called twin primes. If they are slightly further apart with a gap of 4 we call them cousin primes. The smallest cousin primes are 3 and 7, other examples include 43 and 47, 613 and 617, and 937 and 941.
As with twin primes, there is only one prime number that is a cousin prime to two other numbers. In this case, it is 7, which is a cousin of both 3 and 11. We can prove this by considering that all cousin primes can be written as n and n + 4 for some integer n. For a set of three cousin primes to exist, they must be n, n + 4 and n + 8. However, it can be shown that for any n, at least one of these numbers must be divisible by 3 and hence not prime (except for the case of 3 itself, the only prime divisible by 3).
As with twin primes, there is only one prime number that is a cousin prime to two other numbers. In this case, it is 7, which is a cousin of both 3 and 11. We can prove this by considering that all cousin primes can be written as n and n + 4 for some integer n. For a set of three cousin primes to exist, they must be n, n + 4 and n + 8. However, it can be shown that for any n, at least one of these numbers must be divisible by 3 and hence not prime (except for the case of 3 itself, the only prime divisible by 3).
What are sexy primes?
Sexy primes are two numbers that differ by 6, their name coming from a play on the Latin word for six; sex. The first two sexy primes are, therefore, 5 and 11, followed by 7 and 13, and 11 and 17.
Unlike the twin and cousin primes we have already seen, there are many numbers that form sexy prime pairings with more than one other number. In fact, our definition can be extended to longer lists of sexy primes.
If we have three primes with a gap of 6 between each, i.e., of the form p, p + 6, p + 12, we call them sexy prime triplets. 5, 11 and 17 are sexy prime triplets, as are 941, 947 and 953.
We can extend this even further. If we have four primes with gaps of 6 between them, such as 41, 47, 53 and 59, we call them sexy prime quadruplets.
If we try to extend this to quintuplets, we hit a problem though. For a sequence of five numbers p, p + 6, p + 12, p + 18 and p + 24, one of the numbers must be a multiple of 5 due to 5 and 6 being co-prime (two numbers whose only shared factor is 1). Therefore, we cannot go any higher than sexy prime quadruplets.
Unlike the twin and cousin primes we have already seen, there are many numbers that form sexy prime pairings with more than one other number. In fact, our definition can be extended to longer lists of sexy primes.
If we have three primes with a gap of 6 between each, i.e., of the form p, p + 6, p + 12, we call them sexy prime triplets. 5, 11 and 17 are sexy prime triplets, as are 941, 947 and 953.
We can extend this even further. If we have four primes with gaps of 6 between them, such as 41, 47, 53 and 59, we call them sexy prime quadruplets.
If we try to extend this to quintuplets, we hit a problem though. For a sequence of five numbers p, p + 6, p + 12, p + 18 and p + 24, one of the numbers must be a multiple of 5 due to 5 and 6 being co-prime (two numbers whose only shared factor is 1). Therefore, we cannot go any higher than sexy prime quadruplets.
What are prime triplets?
If we have a pair of sexy primes that have another prime number in between them e.g 13, 17 and 19, then we call them prime triplets. All prime triplets are of the form p, p + 2 and p + 6 or p, p + 4 and p + 6, for some prime number p.
Contained within our prime triplets, we will always get a single pair of each of our special primes mentioned above. There will be twin primes (p and p + 2 or p + 4 and p + 6), cousin primes (p and p + 4 or p + 2 and p + 6) and sexy primes (p and p + 6).
Prime triplets are the closest together three primes can ever be, with the exception of our small primes 2, 3 and 5, and 3, 5 and 7. We can't have three primes of the form n, n + 2 and n + 4 for any n > 3, as for any three consecutive odd numbers or three consecutive even numbers, exactly one of these numbers would be divisible by 3, hence not prime.
Contained within our prime triplets, we will always get a single pair of each of our special primes mentioned above. There will be twin primes (p and p + 2 or p + 4 and p + 6), cousin primes (p and p + 4 or p + 2 and p + 6) and sexy primes (p and p + 6).
Prime triplets are the closest together three primes can ever be, with the exception of our small primes 2, 3 and 5, and 3, 5 and 7. We can't have three primes of the form n, n + 2 and n + 4 for any n > 3, as for any three consecutive odd numbers or three consecutive even numbers, exactly one of these numbers would be divisible by 3, hence not prime.
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