For those who have been enjoying my recent series on unsolved mathematical problems, we have another entry today. This time featuring the properties of prime numbers. (And for those who prefer my physics and astronomy articles, I promise that I will try to write a few more of those as well in the coming weeks).

As most (and hopefully all) of you know, a prime number is a number which cannot be written as the product of two other numbers. It can only be properly divided by the number 1, and by itself. The first question one might ask about prime numbers is how many of them exist. Is there a largest prime number? In fact the answer to this question has been known since antiquity, and it turns out that the prime numbers are infinite. There is no such thing as a largest prime number.

However while that question has been answered, there are very closely related questions about the existence of largest prime numbers of specific types. The prime numbers can be classified based on their other properties, and it is quite possible that some of these classes of primes are bounded and contain a largest prime number of a specific type.

Prime Quadruplets: There can be large gaps between prime numbers, and yet there are an infinite number of pairs where p and p+2 are both prime numbers.  There are also a few sets of four prime numbers, of the form (p, p+2, p+6, p+8) throughout the natural numbers, but at this point no one has been able to either prove or disprove that there exists a largest prime quadruplet.

Sexy Primes & Cousins: While some mathematicians might consider all prime numbers to be sexy, in this case sexy primes are defined by number theorists as a pair of primes the differ by six. So p and p+6 are both prime numbers. In a similar manner, cousin primes are defined as a pair of prime numbers of the form p and p+4. We already know that there is no bound on pairs of prime numbers of the form (p, p+2), however no one has been able to prove if there is a largest pair of cousin primes or of sexy primes.

Germain Primes: Another form of prime number pair is the Germain prime numbers, in which p and (2p+1) are both prime numbers. Is there a maximum Germain prime pair?

Mersenne Primes: Mersenne primes are defined as prime numbers which can be written as p = 2n-1, and are the subject of one of the largest online mathematics projects in history with the Great Internet Mersenne Prime Search. The work of the GIMPS project has discovered some huge prime numbers, with the current record being (274,207,281 − 1), but mathematicians still do not know if there is a limit to how large a Mersenne prime can be.

Pierpoint Primes: A more general form of the Mersenne prime is the Pierpoint prime, which is defined as p=2n3m-1 where n and m are both integers. And just like the Mersenne primes, it is unknown if there exists an upper limit on Pierpoint primes.

Wagstaff Primes: There are a number of examples of prime number pairs in which q is a prime number, and p = (2q+1)/3 is also a prime number. (There are interesting mathematical reasons to explore Wagstaff primes, but that is beyond the scope of this article). It is currently unknown if there is any upper limit to the size of Wagstaff primes.

Cullen & Woodall Primes: Another two variations on the Mersenne primes are the Cullen prime, which can be written in the form p = n2n+1, and the Woodall prime written as p = n2n-1. (And as with the Wagstaff primes, they have interesting mathematical properties that are beyond the scope of this article). And as with all of the previous examples, it is still not known how many Cullen primes and Woodall primes exist, but it may be infinite.

It is an interesting fact that, although prime numbers are a simple concept and one that has been studied for millenia, they are also still so mysterious. We know more about the behaviour of the Universe thirteen billion years ago than we do about simple counting numbers that we all learned as children. 

So once more, I hope that you enjoy these mathematical articles, and I encourage you to think about these unsolved problems and maybe even have a try at solving a few of them!