How Many Prime Numbers Are There?
Is there a finite number of primes? Or is the number of prime numbers infinite?
Solution
The number of primes is infinite. Surprisingly, this is a very simple matter to prove, and the proof is accomplished by contradiction.
Assume that there is a finite number of prime numbers. Call them P1, P2,...Pn, where n is the number of primes.
Now multiply all these primes together, and add 1. This number is larger than all the primes, so it cannot be a prime. But it also cannot be written as a product
of prime numbers, as it leaves a remainder of one when divided by each element of the primes. This is a contradiction, from which we conclude that there
must be an infinite number of primes.
Isn't that slick?
"Slick Math" is written by Douglas Twitchell, and hosted at The Problem Site.
Contents copyright 2008 by Douglas Twitchell. Contents of this page may not be reproduced without permission of the author.
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