malc...@[EMAIL PROTECTED]
wrote:
>
> Hi,
>
> You say my synopsis is wrong as a whole? Please, explain.
>
Thanks for participating, because I think we can unravel much more
when
we actually do a Euclid IP rather than we commenting on the sidelines
on Euclid IP. As
an old pragmatist that I am and pragmatist saying-- The learning is in
the doing, and not the peripheral commenting.
> It also seems you want me to provide a proof... Ok, here it goes.
>
> We say a natural number is prime iff it is greater than 1 and doesn't
> have any divisor other than 1 and itself.
> By the fundamental theorem of arithmetic, every number greater than 1
> has a prime divisor.
>
> Let us suppose that the subset of all prime numbers is finite.
> Then the product of all primes plus one, since it doesn't belong to
> the subset of all prime numbers (it is greater than any of them) is
> not a prime, but it has a prime divisor, since it is greater than 1.
> But in fact, it doesn't have any divisor from the list of all primes.
> Contradiction.
>
There is a problem with this statement in your above.
"Let us suppose that the subset of all prime numbers is finite."
I doubt it is a mathematical statement but the assemblage of
contradictions in terms. Like saying "Let us suppose infinity is
finite."
Malcolm, can you write your above and parallel each statement with a
numbers example.
Here is one for my proof of Euclid IP.
(1) Definition of Prime number
(2) Suppose the set of all primes is Finite
(2*) Suppose 3 and 5 were all the primes that exist
(3) Multiply the lot and add 1
(3*) (3x5) +1 =3D 16
(4) This new number is necessarily a new prime since we revert to our
definition in (1)
and all the primes divided into this new number leave a remainder of 1
(4*) 3 and 5 are all the primes that exist and when divided into 16
leave a remainder
of 1 and by the definition of prime from (1), that 16 is indeed a new
prime
(5) Contradiction
(6) Set of all primes is infinite
The essence of the correction of math professors such as Conway,
Courant, Hardy, Niven,
Montgomery, Zuckerman, is that they could not understand that 16 is
prime in the Indirect
Method. No matter what the number "multiply the lot and add 1" is, no
matter what it is, whether
it is 16 or (3x5x7) +1 =3D 106, that 106 is a new prime when the
universe of all primes is 3,5,7.
This is my correction of the math communities horrible habit of not
giving a valid Euclid Infinitude
of Primes proof, as they seem unable to understand the logic and they
mix the direct with the indirect.
Malcolm, you see how I parallel the statements with a numbers example.
Your above is not a proof because you can never run a numbers example
parallel to your statements
for some of your statements are not even mathematical.
> I keep on believing that the key idea in this proof is the
> construction of the number "the product of all primes plus one", and
> not the strict application of the rules of logic, which are supposed
> to work when applied correctly.
>
The crux of the proof is the synchronized working arrangement between
two elements in the proof-- the definition of prime at the start and
the "multiply the
lot and add 1" Yours above Malcolm does not even use the definition of
prime.
I am hoping that Bill Dubuque offers up his version so that I can dive
into
the internal logic of the direct versus the indirect. There is much
mathematics
in the what I call "Meta-Euclid Infinitude of Primes Proof" The
analysis of the
logical underpinning of the direct and indirect.
> For some other proofs of the infinitude of primes I recommend:
>
> Number Theory. An Introduction via the Distribution of Primes
> Benjamin Fine
> Gerhard Rosenberger
> Birkh=EF=BF=BDuser
>
> Cheers.
Thanks, will look them up next time in a library and see if they
managed to give a valid
Euclid IP proof. The statistics for math professors is not running
good since only 2 out of
30 have managed to give a valid Euclid IP.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
|
