Subject: Re: [HM] Euclid and the unique factorization theorem
From: Martin Davis (martin@eipye.com)
Date: Wed Feb 02 2000 - 19:37:51 EST
At 02:57 PM 2/2/00 +0100, Franz Lemmermeyer wrote:
> 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,
> ...it does not exclude the possibility that e.g. pq^2 = p^2q.
>
> Thus ... I'm taking sides with those who do not attribute the
> fundamental theorem of arithmetic to Euclid.
I'd like to quote from Hardy & Wright's famous THEORY OF NUMBERS (4th
edition, p. 182):
"The actual theorem which [Euclid] proves (vii 24) is 'if two numbers be
prime to any number, their product also will be prime to that number' i.e.
(a,c) = 1 & (b,c)=1 => (ab,c)=1"
They then point out that, by taking c to be a prime number p one gets the
theorem:
p|ab => p|a or p|b
which of course is the key lemma in the proof of the unique factorization
theorem. They continue:
"It might seem strange at first that Euclid, having gone so far, could not
prove [the unique factorization theorem] itself; but this view would rest
on a misconception. Euclid had no formal calculus of multiplication and
exponentiation, and it would have been most difficult for him even to state
the theorem. He had not even a *term* for the product of more than three
factors. The omission of [the unique factorization theorem] is in no way
casual or accidental; Euclid knew very well that the theory of numbers
turned on his algorithm, and drew from it all the return he could."
Martin in surprisingly sunny Berkeley.
Martin Davis
Visiting Scholar UC Berkeley
Professor Emeritus, NYU
martin@eipye.com
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http://www.eipye.com
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