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From: Archimedes Plutonium on 5 Jul 2010 06:31
A theorem sometimes called "Euclid's first theorem" or Euclid's
principle states that if is a prime and , then or (where means
divides). A corollary is that (Conway and Guy 1996). The
fundamental theorem of arithmetic is another corollary (Hardy and
Wright 1979).

Archimedes Plutonium wrote:
> 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

Better yet, all we need is a concept in Ancient Greek times for which
that concept
is dependent on knowing full well the Unique Prime Factorization of a
given number.

Now I think the concept of Perfect Numbers can not go anywhere without
the understanding
of Unique Prime Factorization.

What is the word for the concept of "unique" in ancient Greek? Did
Weil ever comb through
the Euclid Elements for the Greek concept of "unique"?

Wikipedia calls it Euclid's Lemma and Wolfram's calls it Euclid's
Theorem
for which Unique Prime Factorization is obtained. and Wolfram writes:"
The fundamental theorem of arithmetic is another corollary (Hardy
and Wright 1979)."

I do not know for what reason that Weil made that remark in his book,
and rather than
cast aspersion on Euclid's work, it casts aspersion on Andre Weil as
to whether
he was a competent in mathematics,
and casts doubt that any of Weil's mathematics is reliable. I dare
say, it is possible to
throw out all of Andre Weil's work in mathematics as "not true" and
not affect any of
mathematics as we know it, and all because of a flippant errant remark
about past
mathematics. Perhaps, Weil should have joined with his sister in
philosophy rather
than have entered in mathematics.

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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