The Prime Number Theorem: The number of primes less than or equal to an integer n asymptotically approaches n / (ln n). That is, for sufficiently large values of n, there are about n / (ln n) prime numbers less than or equal to it.This theorem was originally conjectured by Gauss in 1792. It was finally proven, about simultaneously but independently, by Hadamard and de La Vallée Poussin in 1896. I will not present a proof here, for it is long and containing many concepts in advanced number theory and complex analysis. (I first saw the standard proof in a graduate level analytic number theory course. The proof, built from scratch, took the better part of a month of class time...) Out of the much effort spent in proving this theorem came another conjecture that has since become one of the great currently unsolved problems in mathematics - the Riemann’s hypothesis - about the behavior of a complex-valued function known as the Riemann zeta function.
While the majority of integers are composite numbers, the Prime Number Theorem says that prime numbers occur rather frequently, distributed fairly densely among the integers. Another immediate consequence of the PNT is that the value of the nth prime is approximately n(ln n), for sufficiently large values of n.
A pair of prime numbers that differ by 2 are called twin primes. The first few such pairs are 3 and 5, 5 and 7, 11 and 13, 17 and 19. There are many such pairs, as one can easily verify by examining, say, the list of the first few thousand prime numbers. However, it is still unknown that whether there are infinitely many such pairs among the integers.
Another interesting fact about the distribution of primes is given by Dirichlet's theorem, also known as the prime number theorem for arithmetic progression. It states that, given any nonzero integer a, and another integer k greater than or equal to 2, such that a and k do not have any common divisor other than 1, then the arithmetic progression a, a+k, a+2k, a+3k, a+4k,... contains infinitely many prime numbers. So, not only are there infinitely many primes overall, as we already knew. But there are infinitely many prime numbers in any arithmetic progression, as long as the initial term of the progression and the value of increment do not share a common divisor except the number 1.
Next time: By the numbers - some interesting facts of mathematics