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 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 well-respected 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: $1=\prod_{p\in\emptyset}p$ 0 is not generally included in that statement (it's usually "every positive integer"). I sometimes like to think of it as $2^\infty3^\infty5^\infty\cdots$, but that's just me. It does act as the top ($\top$) element with respect to divisibility, though, so there's some sense to that notation.
March 13th, 2010, 06:20 PM   #5
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Re: Every integer can be expressed as a product of primes?

Quote:
 Originally Posted by CRGreathouse 1 is the empty product of primes: $1=\prod_{p\in\emptyset}p$
I see. Out of interest, why is this defininition made?

March 13th, 2010, 06:36 PM   #6
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Re: Every integer can be expressed as a product of primes?

Quote:
 Originally Posted by pseudonym I see. Out of interest, why is this defininition made?
What else would you possibly call the empty product?

 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
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Re: Every integer can be expressed as a product of primes?

Quote:
 Originally Posted by pseudonym Does it have to be called something?
Sure, if you want 2^0 to make sense.

 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.

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