Associated Topics || Dr. Math Home || Search Dr. Math

### 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

Jacques Salomon Hadamard - MacTutor Math History Archive

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/
```
Associated Topics:
High School Number Theory

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search