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/
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