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From the Fun Fact files, here is a Fun Fact at the Medium level:

Prime Number Theorem

Fix some number N. What fraction of the integers less than or equal to N are prime?

Thinking about it, we know that primes occur less and less often as N grows. Can we quantify this somehow?

Let Pi(N) denote the number of primes less than or equal to N that are prime. Then we expect that the fraction Pi(N)/N must change (decrease?) with N. In fact there is an amazing theorem called the Prime Number Theorem which says that


Pi(N)/N is asymptotic to 1/ln(N)

which means that the ratio of those two quantities approaches 1 as N goes to infinity! Thus Pi(N) is closely approximated by N/ln(N). In fact, a better estimate for Pi(N) is that it is very closely approximated by

INTEGRAL2 to x dt/ln(t) .

Presentation Suggestions:
Write out a table of Pi(N) for the first few values of N (or flash a transparency) just to give students a concrete feel for this function before telling them the answer.

The Math Behind the Fact:
The proof of the Prime Number Theorem requires some hard asymptotic analysis. For more primes and their frequency, see how many primes. Several people have proved various versions of the Prime Number Theorem; among them Chebyshev, Hadamard, de la Vallee Poussin, Atle, Selberg, although the theorem was suspected by Gauss (1791).

How to Cite this Page:
Su, Francis E., et al. "Prime Number Theorem." Mudd Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

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References:
    any book on analytic number theory

Keywords:    number theory, distribution of primes, how many primes
Subjects:    number theory
Level:    Medium
Fun Fact suggested by:   Francis Su
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