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:
INTEGRAL_{2 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>.
