prime number theorem (idea) by jt

One of the more surprising results in mathematics, and a good example of a very deep result which is nevertheless simple enough that a six-year-old can understand its statement.

First, a succinct explanation for the mathematicians in the audience:

lim_n->oo pi(n)ln(n)/n = 1

And for the novices: for a start, go here to find out what a prime number is. Then go here to learn about logarithms, in particular those to base e. Now, what the prime number theorem says is this: to find out roughly how many primes there are less than a certain number, divide the number by its natural logarithm. The bigger the number, the more accurate the answer you get will be; in fact (to be very imprecise and annoy the pedantic readers) as the number gets "infinitely big", the approximation becomes "exact".

Amazed? Well, if you're not then you should be. Why the hell should the prime numbers' distribution have anything to do with e, which is defined using concepts which seemingly have nothing to do with number theory? The answer is long, and if I ever understand it I'll attempt a writeup.

sum of the reciprocals of the primes	Eratosthenes' Sieve	Riemann Hypothesis	prime number
Smallest number greater than 0	e	log2(x) = log10(x) + ln(x)	Riemann zeta function
If cryptography is outlawed, bayl bhgynjf jvyy unir cevinpl	Superstring Theory	natural logarithm	NORMA register machine
Skewes number	logarithmic integral	discrete mathematics	Counting proof that there are infinitely many prime numbers
Budapest Semesters in Mathematics	Jacques Hadamard	The E2 End of Summer Denial of School Northwest Gathering	Graham's Number
The Man who Loved Only Numbers	Euler Product Formula	Pi Function	Prime number language