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March 13th, 2010, 06:44 PM  #1 
Newbie Joined: Mar 2010 Posts: 6 Thanks: 0  Every integer can be expressed as a product of primes?
I'm reading What Is Mathematics?, 2ed. On page 22, it claims, "...every integer can be expressed as a product of primes...." This is clearly false. It fails, for instance, for every prime number, since 1 is not prime and, therefore, there can be no product (there's only one number), let alone a product of primes. This is very wellrespected book, so I assume I must be missing something. What is it? 
March 13th, 2010, 06:54 PM  #2 
Senior Member Joined: Nov 2008 Posts: 199 Thanks: 0  Re: Every integer can be expressed as a product of primes?
I believe the claim you say is made in the book is incorrect. 1 and 0 are not regarded as prime and are not expressible as a product of primes. There are no negative primes either so the claim is false for the negative integers. There's no problem with the primes themselves though as a product need not have 2 or more elements. In this case the 'products' in question are the primes themselves.

March 13th, 2010, 07:03 PM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,691 Thanks: 976 Math Focus: Elementary mathematics and beyond  Re: Every integer can be expressed as a product of primes? 
March 13th, 2010, 07:07 PM  #4 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Every integer can be expressed as a product of primes?
1 is the empty product of primes: 0 is not generally included in that statement (it's usually "every positive integer"). I sometimes like to think of it as , but that's just me. It does act as the top () element with respect to divisibility, though, so there's some sense to that notation. 
March 13th, 2010, 07:20 PM  #5  
Senior Member Joined: Nov 2008 Posts: 199 Thanks: 0  Re: Every integer can be expressed as a product of primes? Quote:
 
March 13th, 2010, 07:36 PM  #6  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Every integer can be expressed as a product of primes? Quote:
 
March 13th, 2010, 07:51 PM  #7 
Senior Member Joined: Nov 2008 Posts: 199 Thanks: 0  Re: Every integer can be expressed as a product of primes?
Does it have to be called something?

March 13th, 2010, 08:03 PM  #8  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Every integer can be expressed as a product of primes? Quote:
 
March 14th, 2010, 05:04 AM  #9 
Senior Member Joined: Nov 2008 Posts: 199 Thanks: 0  Re: Every integer can be expressed as a product of primes?
This is, of course, perfectly reasonable. Thanks.


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