14
$\begingroup$

Do primes become more or less frequent as you go further out on the number line? That is, are there more or fewer primes between $1$ and $1{,}000{,}000$ than between $1{,}000{,}000$ and $2{,}000{,}000$?

A proof or pointer to a proof would be appreciated.

  • 3
    Yes (as has been answered): the heuristic reason is that we can list the primes $p_1, p_2, \dots$. Each time you apply the sieve of Eratosnese you exclude the numbers divisible by $p_n$ and weed out "$(1-1/p_n)$" of the natural numbers (i.e., asymptotic density). So in the end you are left with $\prod ( 1 - 1/p_n)$ fraction of the natural numbers; since the sum $\sum \frac{1}{p_n} = \infty$, this is the proportion zero. So we should expect the primes to have density zero.2010-07-21
  • 2
    Check out the [Riemann-zeta function](http://en.wikipedia.org/wiki/Riemann_zeta_function) if you want something a bit more technical. It gives a pretty precise estimate of the distribution of primes, indirectly.2010-07-21

2 Answers 2

16

From the Wikipedia article about the prime number theorem:

Roughly speaking, the prime number theorem states that if a random number nearby some large number N is selected, the chance of it being prime is about 1 / ln(N), where ln(N) denotes the natural logarithm of N. For example, near N = 10,000, about one in nine numbers is prime, whereas near N = 1,000,000,000, only one in every 21 numbers is prime. In other words, the average gap between prime numbers near N is roughly ln(N).

3

The Sieve of Eratosthenes is a very intuitive visual representation of why the frequency of prime numbers goes down as you go further out on the number line.

  • 0
    This is really a less detailed version of Akhil's comment.2010-07-21
  • 2
    @BBischof Akhil's comment derives the answer to this question with mathematical rigor. My answer offers a link to a visual representation of the trend for the sake of gaining intuition. Its nice to have both the proof of the result and intuition about the result.2010-07-21
  • 0
    Agreed @Ami, it is a useful answer if not as rigorous.2010-07-21