{"id":38,"date":"2004-10-10T18:42:36","date_gmt":"2004-10-10T18:42:36","guid":{"rendered":"http:\/\/www.sixthform.info\/maths\/?p=38"},"modified":"2004-10-10T18:42:36","modified_gmt":"2004-10-10T18:42:36","slug":"primes","status":"publish","type":"post","link":"https:\/\/www.sixthform.info\/maths\/?p=38","title":{"rendered":"Primes"},"content":{"rendered":"

Euclid’s<\/a> proof<\/a> that there are an infinite number of primes is a classic and as such appears as the first proof in Proofs from The Book<\/a>.
\nEqually well-known is the formula (known as The Prime Number Theorem<\/a>) which tells you that the number of primes \\pi(x) less than x is given by \\pi(x)\\sim \\dfrac{x}{\\log x} which means that the larger the value of x the closer (in a well-defined mathematical sense) \\dfrac{x}{\\log x} is to \\pi(x). This is quite hard to prove.<\/p>\n

An easier, but non-trivial result, is Bertrand’s postulate which says that there is always a prime between n and 2n.<\/p>\n

The fact that there are arbitrarily large gaps between successive primes is not difficult to prove. Suppose we want to find a gap between successive primes which is at least of size N. Then we look at the numbers<\/p>\n