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From: Archimedes Plutonium on 3 Jul 2010 02:09
> [0] Michael *Hardy* and Catherine Woodgold,
> "*Prime* *Simplicity*",  *Mathematical
> Intelligencer<https://mail.google.com/wiki/Mathematical_Intelligencer>

The above magazine article uses Ore's proof as what Euclid did as a
Direct/Constructive
proof of Infinitude of primes. I went and looked up this book by Ore.

--- 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 ----

I agree that Euclid did a Direct/Constructive proof of Infinitude of
Primes and I agree
the above is a valid proof.

But I have some minor issues with the above. The direct method is
increasing set cardinality
of any given finite set. So Ore begins by calling it a "family" of
prime numbers yet ends
with set theory of "given (finite) set of primes." So why not be
consistent and remove "family".

I understand Ore was trying to be as exacting to Euclid's own proof,
only given modern
day math language, and it is this lemma by contradiction "would divide
P and also P+1"
that concerns me.

I had always thought that Euclid had proved the Unique Prime
Factorization Theorem
(UPFAT), but reading Weil's book page 5, Euclid fell short of UPFAT.
So when Ore
goes into the sentence saying "it must be divisible by some prime p."
And the only
justification for that claim, as far as I can see is UPFAT.

So I guess Weil was correct, in that Euclid was unaware that the
factorization of any
number beyond 1 is a list of unique primes. For if Euclid had UPFAT,
then Euclid and
Ore could have skipped or eliminated this portion of the above: ". .
because then it
would divide P and also P+1; hence it would divide their difference,
which is 1 and this is impossible."

So if Euclid and then Ore writing a translation of the proof, had had
the UPFAT in
Ancient Greek times, then there would not be a need for a lemma of
contradiction.

Euclid and Ore could have said after forming P+1 that either P+1 was
prime or that
P+1 had a prime factor not on the original list, all due to UPFAT.

But I think it becomes more serious than this. I think when Euclid
wrote and Ore writes
"it must be divisible by some prime p." there is no justification for
that claim other than
UPFAT. So I think UPFAT is a necessary theorem in order to do the
Direct/Constructive proof.

So I take my words back, Euclid may have had a flaw that was a flaw
that cannot be
covered up. And if that is true, then Euclid did not have a valid
proof but came exceedingly
close to a valid proof with a missing theorem of unique prime
factorization required to do the
proof.

So I think that UPFAT is necessary to have in both the Direct and
Indirect methods and a valid
end result cannot accrue unless UPFAT is used. There seems to be no
justification for
"it must be divisible by some prime p." unless you know UPFAT and
invoke it at that juncture.

Maybe Weil is wrong and that Euclid really had UPFAT, because Eucl.IX.
14 preceded Eucl.IX.20

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).. "

Should we ask the question, can you have done Euclid's book containing
the infinitude
of primes and not know that the numbers have a unique prime
factorization?

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
From: sttscitrans on 3 Jul 2010 19:30
On 3 July, 07:09, Archimedes Plutonium
<plutonium.archime...(a)gmail.com> wrote:
> > [0] Michael *Hardy* and Catherine Woodgold,
> > "*Prime* *Simplicity*",  *Mathematical


> Should we ask the question, can you have done Euclid's book containing
> the infinitude
> of primes and not know that the numbers have a unique prime
> factorization?

This must have been pointed out to you hundreds of times ovet the
years.

If there are naturals >1 that have no prime divisors, you
cannot conclude that w+1 has prime divisors other than those of
w and so the prime divisors of w might be all the prime
divisors there are.

You don't even need uniqueness.
The fact that every N>1 has at least one prime
divisor is sufficient.

Maybe you have not realized that w and w+1 are consecutive integrs
with a greatest common measure of 1.

1 < w <w+1, if any prime p divides w then p does not divide w+1
If every prime that exists divides w then no prime divides w+1. This
contradicts the fact that every n>1 has a prime divisor.

Let 2 ,3, 5 are the only primes that exist
2, 3 and 5 divide 120. None of these primes divide
120+1 = 121 = 11*11 (or 120-1 =119=17*7). This a contradiction.
As every n>1 has a prime divisor and the only prime divors
assumed to exist are 2,3,5 at least one of them must divide
121 or 119. None does.
From: sttscitrans on 6 Jul 2010 17:47
On 3 July, 07:09, Archimedes Plutonium
<plutonium.archime...(a)gmail.com> wrote:
> > [0] Michael *Hardy* and Catherine Woodgold,
> > "*Prime* *Simplicity*",  *Mathematical

> I had always thought that Euclid had proved the Unique Prime
> Factorization Theorem
> (UPFAT), but reading Weil's book page 5, Euclid fell short of UPFAT.
> So when Ore
> goes into the sentence saying "it must be divisible by some prime p."
> And the only
> justification for that claim, as far as I can see is UPFAT.

The uniqueness part is superfluous.

The smallest divisor d >1 of M> 1 must be a prime.
If d >1 is composite, d has a divisor d', d'<d

This means every m>1 has a prime divisor, the smallest divisor
of m that exceeds 1.

1) Every n>1 has at least one prime diviosr
2) GCD(n, n+1) = 1
3) Assume the primes are finite in number
Let L= LCM of these primes
4) GCD(L,L+1) <>1

3) => 4), 4) is false, therefore 3) is false.

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