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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 this integral:

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. 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).

See more Fun Facts about primes.

How to Cite this Page:
Su, Francis E., et al. "Prime Number Theorem." 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, calculus
Subjects:    number theory
Level:    Medium
Fun Fact suggested by:   Francis Su
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