No Largest Prime Number

Date: 8/19/96 at 12:24:1
From: Anonymous
Subject: Largest Prime Number

What's the largest prime number?

Thanks in advance for your answer,
     Xander from the Netherlands

Date: 8/21/96 at 14:39:27
From: Doctor Mike
Subject: Re: Largest Prime Number

Hello Xander,
   
The answer to your question is that there is NO largest prime number.
   
There is a pretty easy proof of this fact.  Suppose temporarily that
there are only a finite number of prime numbers.  The smallest one
would be 2, the next is 3, then come 5, 7, 11 and so on.  We could 
give them symbolic names like p1 = 2, p2 = 3, p3 = 5, p4 = 7, p5 = 11, 
etc. Because we are temporarily assuming that there is only a certain
finite number of them, let pL stand for the biggest one of them all.
Remember we are assuming that p1, p2, ... pL is the COMPLETE list.
   
From these prime numbers, imagine another VERY large number that you
would get by multiplying all these "L" prime numbers together and then
adding one.  In symbols it is : 
   
       Q  =  (p1)*(p2)*(p3)*(p4)*(p5)*(p6)*...*(pL) + 1 
   
This is a very interesting number, because whenever you divide it by
a prime number (that is, whenever you divide it by a pN) you get a
remainder of 1.  But what it means to be a prime number is exactly 
that it has no prime numbers that divide into it evenly.  So "Q" is 
prime, or has prime factors larger than pL.  
  
The situation now is that by assuming there are only a limited number
of prime numbers we can show that there must others not in the 
original list. This is obviously impossible, so our original assumption is 
false, too. That means then that there must be an infinite number of 
primes and that there is no largest prime. 
  
The proof method I have used here is called "Proof by contradiction"
or "reductio ad absurdum" in Latin, and it is very commonly used in
mathematics and philosophy.  You assume the opposite of what you 
really want to prove, and then show an absurdity to which this 
assumption leads when it is carried to its logical conclusion.  
Pretty useful! 
  
I hope this helps. 

-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/