Maths in a minute - prime pleasures 

Mathematicians love prime numbers, but why? Firstly, it's because they are like the atoms of number theory. They're divisible by nothing expect for 1 and themselves. Any other whole number bigger than 1 can be written as a product of primes in a unique way. For example, 6=2x3 and 18=2x3x3. That result is known as the fundamental theorem of arithmetic. It means that if you want to prove something about all whole numbers, it's often enough to prove it for the primes, as any number can decomposed into its prime factors.

But mathematicians also love primes because they are mysterious. The ancient Greek mathematician Euclid proved that there are infinitely many primes. His proof is simple: suppose that there are only finitely many primes, p₁, p₂, p₃, etc, up to some prime p_n. Let q=p₁ x p₂ x p₃ x ...p_n +1. So you q is the product of all the primes plus 1. It can't be prime itself because it's bigger than all the primes on the list. This means that q itself is divisible by some prime because of the fundamental theorem of arithmetic. But it's quite easy to show that none of the primes on our list can divide q, because if one did then it would also have to divide 1 and that's impossible. Hence there must be another prime beyond the primes on our list. That's a contradiction because we've assumed that all primes are on our list. Hence our initial assumption is wrong and there must be infinitely many primes. 

So we know that there are infinitely many primes, but how are they distributed among the other numbers? If you've got one prime, how long do you have to count up until you meet the next? One thing you notice when you look at primes is that they sometimes come in pairs that have a difference of 2, for example 3 and 5, 5 and 7, 11 and 13, 17 and 19, etc. Are there infinitely many such pairs? This seemingly simple question is still unsolved. It's known as the Twin Prime conjecture. 

One thing we do know is the asymptotic distribution of the primes. For any positive  number x, write P(x) for the number of primes less than or equal to x. Then (loosely speaking) as x gets bigger and bigger, P(x) gets closer and closer to x/ln(x). 

However, the exact distribution of the primes is still a mystery. It's the subject of one of the hardest open questions in maths, the Riemann hypothesis. 

There are other seemingly easy questions about primes that are still unanswered. An example is the Goldbach conjecture, which says that every even integer greater than 2 is the sum of two primes. 

So that's why mathematicians love primes. They are fundamental, yet mysterious. In recent years they have become popular with more practically-minded people too because they provide the codes that encrypt sensitive information sent over the internet.

You can find out more about primes on Plus:

The prime number lottery

The music of the primes

Mathematical mysteries: the Goldbach conjecture

Mind the gap

Elusive twins

All Plus content on primes

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