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A prime number is a number whose only positive divisors are itself and 1. Whereas composite
numbers (numbers which are not prime) can have many divisors (12, for example, has six
-- 1, 2, 3, 4, 6 and itself), prime numbers can only ever have two. The first ten primes
are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29, whereas the largest known prime has nearly
13 million digits. With the advent of supercomputers, new primes are being discovered all the time;
at the last count 2.5 x 1016 (25 million trillion) primes have been discovered. This number will
continue to grow since the number of primes is infinite, a proof first demonstrated by
the Greek mathematician Euclid around 300 BCE in his book Elements.
For a long time, it was assumed that prime numbers were important only in pure rather
than applied mathematics -- that is, in the branch of mathematics which relates to abstract
rather than concrete concepts. They are particularly important in the field of number theory (the
branch of pure mathematics dedicated to the study of integers). However, the development
in the 1970s of public key cryptography thrust primes into the spotlight, giving them an
opportunity for practical use. This system of cryptography is a kind of security system
which requires two separate keys. One key is used to encrypt (or 'lock') the plain text
(the original message) whilst another is needed to unlock (i.e. decrypt) the cyphertext (the
text in code). Neither key will perform both functions. Separate prime numbers are used
for both keys. These prime numbers may then be multiplied together, forming a very large
number, thus making it virtually impossible to break the code, since there is now known
quick and efficient way of identifying the prime factors of very large numbers.
The distribution of prime numbers appears at first glance to be largely random. According
to number theorist Don Zagier, primes "grow like weeds...seeming to obey no other law
than that of chance", and there is, as yet, no known formula for predicting the distribution
of primes. However, as Zagier adds, prime numbers also "exhibit stunning regularity
[and] that there are laws governing their behaviour, and that they obey these laws with
almost military precision."
Two special cases of primes should be noted in this discussion. Firstly, a well known
mathematical issue surrounding primes is Goldbach's conjecture, which posits that every even integer
greater than 2 can be expressed as the sum of two primes (e.g. 8=3+5) Certain numbers
are the sum two sets of primes (e.g. 14=3+11 or 7+7). This conjecture is widely assumed
to be true, but it has yet to be proved.
Secondly, there exists a special category of prime numbers called Mersenne primes. These
primes take the form 2p-1, wherein p is a known prime. The primes are named after Marin
Mersenne, a French monk who identified this type of prime in the 17th century. As of 2012,
47 Mersenne prime are known. 3 (22-1) and 31 (25-1) are examples, therefore of Mersennse
Primes. The largest is the result of the equation 243112609 -- 1. Mersenne Primes can be used
in so-called pseudorandom number generators, which again can be used in cryptography.
Prime numbers are a fascinating aspect of mathematics, and one which we still do not
properly understand. A line from Mark Haddon's book The Curious Incident of the Dog in the
Night-time captures this nicely, where the lead character says, "Prime numbers are what
is left when you have taken all the patterns away. I think prime numbers are like life.
They are very logical but you could never work out the rules, even if you spent all
your time thinking about them."