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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.

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    Check out the [Riemann-zeta function](http://en.wikipedia.org/wiki/Riemann_zeta_function) if $y$ou want something a bit more technical. It gives a pretty precise estimate of the distribution of primes, indirectly.2010-07-21

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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).

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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.

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    Agreed @Ami, it is a useful answer if not as rigorous.2010-07-21