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From: Archimedes Plutonium on 4 Jul 2010 17:34


Archimedes Plutonium wrote:
> Archimedes Plutonium wrote:
> > Archimedes Plutonium wrote:
> > (snipped)
> > >
> > >
> > > quote of Weil's book "Number theory", 1984,
> > >  page 5: "Even in Euclid,
> > >  we fail to find a general statement about the uniqueness of the
> > >  factorization of an integer into primes; surely he may have been
> > > aware
> > >  of it, but all he has is a statement (Eucl.IX.14) about the l.c.m..
> > > of
> > >  any number of given primes. Finally, the proof for the existence of
> > >  infinitely many
> > >  primes (Eucl.IX.20).. "
> > >
> >
> > Maybe Weil was just being too exaggerating. Maybe all we need for the
> > historical record
> > is for an ancient text to show a sequence such as this:
> >
> > 1 = 1
> > 2 = 2
> > 3 = 3
> > 4 = 2x2
> > 5 = 5
> > 6 = 2x3
> > 7 = 7
> > 8 = 2x2x2
> > 9 = 3x3
> > 10 = 2x5
> > 11 = 11
> > 12 = 2x2x3
> > 13 = 13
> > 14 = 2x7
> > etc etc
> >
> > So that if in Euclid's writings we see some sequence like that then we
> > can say Euclid was
> > aware of UPFAT and that it was proven in his time. And that Gauss
> > would only later refine
> > the proof.
> >
> > Maybe Weil was just being overly harsh.
> >
>
> --- quoting Wikipedia on the proof of uniqueness for Fundamental
> theorem of Arithmetic ---
> A proof of the uniqueness of the prime factorization of a given
> integer proceeds as follows. Let s be the smallest natural number that
> can be written as (at least) two different products of prime numbers.
> Denote these two factorizations of s as p1···pm and q 1···qn, such
> that s = p1p2···pm = q 1q2···qn. By Euclid's proposition either p1
> divides q1, or p1 divides q 2···qn. Both q1 and q 2···qn must have
> unique prime factorizations (since both are smaller than s), and thus
> p1  =  qj (for some j). But by removing p1 and qj from the initial
> equivalence we have a smaller integer factorizable in two ways,
> contradicting our initial assumption. Therefore there can be no such
> s, and all natural numbers have a unique prime factorization.
> --- end quoting ---
>
> That is satisfying as a proof of UPFAT, to me. So I fail to see why
> Weil says what he
> says on page 5 of "Number theory"? Is Weil one to make spurious
> complaints?
>

Well, maybe this is more of a correcting of Andre Weil and his 1984
book
"Number Theory" then a correcting of Euclid. So why would Weil say
what
he says on page 5? Does he have some personal gripe on the Ancients of
Euclid?

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
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