
Number Theory Number Theory Math Forum 
 LinkBack  Thread Tools  Display Modes 
March 13th, 2010, 05:44 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, 05:54 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, 06:03 PM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,930 Thanks: 1124 Math Focus: Elementary mathematics and beyond  Re: Every integer can be expressed as a product of primes? 
March 13th, 2010, 06:07 PM  #4 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 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, 06:20 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, 06:36 PM  #6  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 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, 06:51 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, 07:03 PM  #8  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 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, 04:04 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.


Tags 
expressed, integer, primes, product 
Search tags for this page 
Click on a term to search for related topics.

Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
How to simplify the product of functions on primes  kankan  Number Theory  9  March 23rd, 2014 12:52 PM 
Theorem that sum or product of integers result in an integer  king.oslo  Algebra  1  November 21st, 2013 09:20 AM 
Product of primes  proglote  Number Theory  21  July 8th, 2011 09:59 AM 
Infinite product of odd primes^2 divisible by Pi^2  Agno  Number Theory  1  May 9th, 2011 03:49 PM 