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From: Archimedes Plutonium on 6 Jul 2010 02:29


Archimedes Plutonium wrote:
> I need to get back to my physics book, where I am in the middle of it
> with "missing mass".
>
> I was interrupted from that physics book by this:
>
> 2009 Mathematical Intelligencer magazine article:
> > [0] Michael *Hardy* and Catherine Woodgold,
> > "*Prime* *Simplicity*",  *Mathematical
> > Intelligencer<https://mail.google.com/wiki/
> Mathematical_Intelligencer>
>

Now from that Mathematical Intelligencer MI article Ore's
constructive proof was taken to be the same as Euclid's only
in modern day language.

--- quoting from Number Theory and Its History, Oystein Ore, 1948,
 page 65 ---
 Euclid's proof runs as follows: let a, b, c, . . ., k be any family
of
 prime numbers. Take their
 product P = ab x . . x k and add 1. The P+1 is either a prime or
not
a
 prime. If it is,
 we have added another prime to those given. If it is not, it must
be
 divisible by some prime
 p. But p cannot be identical with any of the given prime numbers a,
 b, . . ., k because then it
 would divide P and also P+1; hence it would divide their
difference,
 which is 1 and this is impossible. Therefore a new prime can always
be
 found to any given (finite) set of primes.
 --- end quoting Ore ----

As mentioned before, I have some complaints about the above Ore/Euclid
rendition in that set theory should have been spoken more of, such as
increasing
set cardinality.

I find the proof valid, but I disagree that the lemma is needed, for I
think the lemma
is excess baggage-- "hence it would divide their difference". In both
Ore and Euclid's
rendition, all that need to be stated was the Unique Prime
Factorization theorem which
would have rendered prime factors if P+1 was not prime itself and thus
no lemma of
contradiction need be posed.

Now there was a bit of fuss in this thread, when I read this passage
by Weil after looking in the
library for the Ore book:

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).. "

I have no idea as to why Weil felt he had to be deprecatory of Euclid.
For
all that Andre Weil had to do was put on his thinking cap and realize
that
many of the number theory concepts of ancient greek time could not
have
progressed the distance they did without knowing the Fundamental
theorem
of Arithematic--unique prime factorization theorem (UPFAT). The
concept of
perfect-numbers would not have occurred without UPFAT. And one must
keep
in mind that in Ancient Greek, they did not have the decimal number
notation
and that is why we see so often numbers in Euclid represented as
lengths of
line. And so the reason that Weil probaby never sees, nor Euclid ever
write
about unique prime factorization, is the difficulty of even writing
numbers
not in a decimal notation.

So this diversion caused by Weil flippant remark caused me to explore
whether the Greeks had UPFAT or whether Weil was in error. And as it
turns
out, Weil was mistaken. I do not know if there is any moral theme to
the Weil
diversion, perhaps when you call into question other mathematicians,
that your
own work is then in question also. For I have the feeling that once
mathematics
has defined the boundary between finite and infinite as 10^500, that
much or
maybe all of Weil's work in mathematics becomes untruthful and
irrelevant,
just as his remark of Greeks over unique prime factorization.

So I think the Euclid and Ore proofs are valid, albeit with excess
unneeded baggage
of a lemma. When all that was needed was to invoke UPFAT for the prime
factor search.

P.S. I need to remark about an item in that MI article saying words to
the effect that
any direct proof can be turned into an indirect method proof. For I
feel that is a
mistake and ironic that the MI article is to clear up errors and
errors of myth, and here
with this idea of turning any direct method into an indirect may
create a brand new
erroneous myth that future mathematicians have to straighten out.

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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