odigious calculator) told a friend that whenever he had a spare 15 minutes he would spend it in counting the primes in a 'chiliad' (a range of 1000 numbers). By the end of his life it is estimated that he had counted all the primes up to about 3 million. Both Legendre and Gauss came to the conclusion that for large n the density of primes near n is about 1/log(n). Legendre gave an estimate for (n) the number of primes n of (n) = n/(log(n) - 1.08366) while Gauss's estimate is in terms of the logarithmic integral (n) = (1/log(t) dt where the range of integration is 2 to n. You can see the Legendre estimate and the Gauss estimate and can compare them. The statement that the density of primes is 1/log(n) is known as the Prime Number Theorem. Attempts to prove it continued throughout the 19th Century with notable progress being made by Chebyshev and Riemann who was able to relate the problem to something called the Riemann Hypothesis: a still unproved result about the zeros in the Complex plane of something called the Riemann zeta-function. The result was eventually proved (using powerful methods in Complex analysis) by Hadamard and de la Valle Poussin in 1896. There are still many open questions (some of them dating back hundreds of years) relating to prime numbers.Some unsolved problemsThe Twin Primes Conjecture that there are infinitely many pairs of primes only 2 apart. Goldbach's Conjecture (made in a letter by C Goldbach to Euler in 1742) that every even integer greater than 2 can be written as the sum of two primes. Are there infinitely many primes of the form n2 + 1 ?(Dirichlet proved that every arithmetic progression : {a + bn | n N} with a, b coprime contains infinitely many primes.) Is there always a prime between n2 and (n + 1)2 ?(The fact that there is always a prime between n and 2n was called Bertrand's conjecture and was proved by Chebyshev.) Are there infinitely many prime Fermat numbers? Indeed, are there any prime Ferma...
