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 (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 Number Theorem which says that

(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 (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 (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 Number Theorem requires some 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).

How to Cite this Page: 
Su, Francis E., et al. “ Number Theorem.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

References:
Any book on analytic

Fun Fact suggested by:
Francis Su

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