Subject: Re: [HM] Euclid and the unique factorization theorem
From: Franz Lemmermeyer (lemmerm@mpim-bonn.mpg.de)
Date: Wed Feb 09 2000 - 08:09:56 EST
On Wed, 2 Feb 2000, Martin Davis wrote:
> I'd like to quote from Hardy & Wright's famous THEORY OF NUMBERS (4th
> edition, p. 182):
>
> "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.
I knew their evaluation - if I recall it correctly, Bourbaki says something
very similar. On the other hand, I find that the last claim is misleading:
of course Euclid had special names for products of two or three factors, but
as his proof of the infinitude of primes shows, this does not mean that he
could not talk about products of arbitrary many numbers.
Moreover, as Marinus Taisbak says (if there's any preprint or tex file I'm
all ears), it is not at all obvious that it was _impossible_ for Euclid to
formulate the fundamental theorem.
Concluding, I'd like to mention R. Rashed's work on Arab mathematics, in
particular
The development of Arabic mathematics: between arithmetic and algebra,
Kluwer 1994
(a translation of his book in French from ca. 20 years ago), where he says
that three of al-Farisi's (14th century) propositions "were obviously
intended to establish the uniqueness of the factorization into prime factors".
A.G. Agarg\"un, C.R. Fletcher,
al-Farisi and the fundamental theorem of arithmetic,
Hist. Math. 21 (1994), 162--173
are more cautious, if I remember their paper well.
franz
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