Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math - Questions and Answers from our Archives
_____________________________________________
Associated Topics || Dr. Math Home || Search Dr. Math
_____________________________________________

Are Prime Numbers Infinite?


Date: 06/18/98 at 23:22:24
From: Vincent
Subject: Prime numbers

How do we know the number of primes is infinite? 

Thank you!


Date: 06/19/98 at 07:56:33
From: Doctor Allan
Subject: Re: Prime numbers

Hi Vincent,

The proof depends on the fact that every natural number has a unique
factorisation into primes. I assume that you are aware of this fact. 
If not, please write again.

Now assume that there is only a finite number of primes and let's call 
these p_1, p_2, ... , p_m.

Look at the number (p_1)*(p_2)*...*(p_m) + 1

Because of the unique factorisation, this number can be factored into 
our primes p_1, ... , p_m or it is prime itself.

You should now convince yourself that neither of these options is 
possible. Therefore our assumption that there is only a finite number 
of primes is incorrect, proving that there is indeed an infinite 
number of primes.

Sincerely,

-Doctor Allan,  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

[Privacy Policy] [Terms of Use]

_____________________________________
Math Forum Home || Math Library || Quick Reference || Math Forum Search
_____________________________________

Ask Dr. MathTM
© 1994-2013 The Math Forum
http://mathforum.org/dr.math/