My Math Forum Is this proof of the fundamental theorem of arithmetic ok?

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 September 14th, 2013, 04:26 PM #1 Senior Member   Joined: Sep 2010 From: Oslo, Norway Posts: 162 Thanks: 2 Is this proof of the fundamental theorem of arithmetic ok? Hello there, I am a university student, and I was just trying to prove the fundamental theorem of arithmetic. I proved that there are an infinite number of primes. Now I have to prove that numbers can only be expressed uniquely. $n$ is the smallest product of primes which could be expressed in more than one way. It would be impossible to construct this with fewer than two primes: $n= p_1p_2$ and $n= p_3p_4$ This is my contradiction: $n-p_1p_3= p_1p_2-p_1p_3 = p_1(p_2-p_3)$ and $n-p_1p_3= p_3p_4-p_1p_3 = p_3(p_4-p_1)$ $n-p_1p_3$ is a smaller number than $n$, and it is possible to express this number in more than one way. But $n$ was the smallest number possible to construct from primes. This is the contradiction. What do you think? Criticism is most welcome. Thank you for your time. Kind regards, Marius
 September 19th, 2013, 11:17 AM #2 Senior Member   Joined: Sep 2010 From: Oslo, Norway Posts: 162 Thanks: 2 Re: Is this proof of the fundamental theorem of arithmetic o Anyone? M
September 19th, 2013, 10:48 PM   #3
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Re: Is this proof of the fundamental theorem of arithmetic o

You proof is not right. Here are some problems with it:

Quote:
 Originally Posted by king.oslo $n$ is the smallest product of primes which could be expressed in more than one way. It would be impossible to construct this with fewer than two primes: $n= p_1p_2$ and $n= p_3p_4$
n could be the product of more than 2 primes and a different number of primes in each case. E.g.:

$n= p_1p_2p_3$
and
$n= p_4p_5$

Quote:
 Originally Posted by king.oslo This is my contradiction: $n-p_1p_3= p_1p_2-p_1p_3 = p_1(p_2-p_3)$ and $n-p_1p_3= p_3p_4-p_1p_3 = p_3(p_4-p_1)$ $n-p_1p_3$ is a smaller number than $n$, and it is possible to express this number in more than one way. But $n$ was the smallest number possible to construct from primes. This is the contradiction.
You haven't shown that $n-p_1p_3$ is prime. It's simply a number less than n. It could even be 0.

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