Percentages of Prime Numbers

Date: 01/31/99 at 19:00:49
From: Denise Grayson
Subject: Percentages of prime numbers

I am an algebra II student, and I am doing a science fair project in 
which I am observing the percentages of primes. I have written a 
computer program that finds prime numbers and the percentage of them. 
Example: There are 4 prime numbers between 1 and 10, which is 40%. I 
print out a statement like this at every power of 10. I know that 
there are an infinite number of primes, and I was wondering if the 
percentage of primes will continue to decrease as you increase the 
span of your search, until you reach a certain point at which this 
percent will remain constant for the rest of infinity. I was wondering 
if you knew anything about past research on this subject, or if you 
could give me some suggestions on where to look.

Thank you very much,
Denise Grayson

Date: 02/01/99 at 13:02:39
From: Doctor Nick
Subject: Re: Percentages of prime numbers

You are correct in pointing out that there are an infinite number of 
primes. There is a very important theorem, proved in 1896 by Jacques 
Hadamard and a man named de la Vallee Poussin (working separately - 
they came up with different proofs at right about the same time) 
called the Prime Number Theorem. This theorem says that if p(x) is the 
number of primes less than x, then as x gets larger and large, the 
ratio p(x)/(x/log (x)) gets closer and closer to 1. This means that if 
x is large, there are about x/log(x) primes less than x. What this 
says is that the fraction of numbers less than x that are prime is 
about 1/log(x). (Here I am using log(x) to stand for the natural 
logarithm of x, sometimes written ln(x)). So, the percentage of primes 
among the first x integers gets closer and closer to 0 as x gets 
larger.

This subject has a very rich history, and the study of primes 
continues to this day. It is part of what's known as "analytic number 
theory." Almost any book on number theory will mention the Prime 
Number Theorem. I suggest trying to get your hands on any books on 
number theory. Though many will contain math that will take a long 
time for you to understand, there are others that are quite readable. 
I should mention, though, that the way the Prime Number Theorem has 
been proved is quite complicated. 

Also, try searching the Web for "prime number theorem" and see what 
you find. There is some interesting reading out there.

The function p(x), I've used above is usually written pi(x) with "pi" 
replaced by the Greek symbol for pi. It's read "pi of x" .

Here are a few places to start looking:

   The Prime Number Theorem - Jonas Wiklund
   http://www.math.umu.se/~jwi/articles/primenumberthm/primenumberthm.html   

   Jacques Salomon Hadamard - MacTutor Math History Archive
   http://history.math.csusb.edu/Mathematicians/Hadamard.html   

   De la Vallee Poussin - Technical University of Budapest
   http://www.vma.bme.hu/mathhist/Mathematicians/Vallee_Poussin.html   

Let me know if you need help understanding anything you come across, 
of if you have any other questions about prime numbers.

Have fun,

- Doctor Nick, The Math Forum
  http://mathforum.org/dr.math/