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From: Archimedes Plutonium on 4 Jul 2010 17:27


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?

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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