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March 13th, 2010, 06:44 PM   #1
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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?
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March 13th, 2010, 06:54 PM   #2
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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.
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March 13th, 2010, 07:03 PM   #3
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Re: Every integer can be expressed as a product of primes?

Fundamental theorem of arithmetic
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March 13th, 2010, 07:07 PM   #4
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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.
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March 13th, 2010, 07:20 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:
I see. Out of interest, why is this defininition made?
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March 13th, 2010, 07:36 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?
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March 13th, 2010, 07:51 PM   #7
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Re: Every integer can be expressed as a product of primes?

Does it have to be called something?
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March 13th, 2010, 08:03 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.
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March 14th, 2010, 04:04 AM   #9
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Re: Every integer can be expressed as a product of primes?

This is, of course, perfectly reasonable. Thanks.
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