Subject: Re: [HM] Euclid and the unique factorization theorem
From: Franz Lemmermeyer (lemmerm@mpim-bonn.mpg.de)
Date: Wed Feb 02 2000 - 08:57:17 EST
It seems to me that the question I posed last time, namely whether Euclid's
proof of "unique factorization" has to be regarded as complete, is the wrong
question.
Here is the proposition IX.14:
If a number be the least that is measured by prime numbers, it will not be
measured by any other prime number except those originally measuring it.
Heath calls this simply the unique factorization theorem. However, in modern
lingo it says that the lowest common multiple of primes has no prime factors
different from these primes, and this is much less than the fundamental
theorem as we know it. In fact, it does not exclude the possibility that e.g.
pq^2 = p^2q.
Thus unless someone cares to correct my point of view, I'm taking sides with
those who do not attribute the fundamental theorem of arithmetic to Euclid.
franz
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