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From: Archimedes Plutonium on 10 Aug 2010 15:30


OwlHoot wrote:
> On Aug 10, 5:32 pm, Archimedes Plutonium
> <plutonium.archime...(a)gmail.com> wrote:
> >
> > So in words, the Euclid Infinitude of Primes proof, Indirect in
> >   short-
> >    form goes like this:
> >
> > 1) Definition of prime
> >    2) Hypothetical assumption, suppose set of primes 2,3,5,7,.. is
> >    finite with P_k the last and final prime
> >    3) Multiply the lot and add 1 (Euclid's number) which I call W+1
> >    4) W+1 is necessarily prime
>
> Not necessarily: W + 1 and W - 1, and for that matter any
> integer V +/- W where V and W are any integers whose product
> is divisible by all of 2, 3, 5, 7, .., P_k must be divisible
> only by primes larger than P_k.
>
> But these "new" primes can divide those integers V +/- W to
> a degree greater than 1, and there can be more than one of
> them.
>
> The rest of the proof works in the same way though:
>
> >    5) contradiction to P_k as the last and largest prime
> >    6) set of primes is infinite.
>
>
> Cheers
>
> John Ramsden

Hi John

I do not want to argue with you or make you feel bad in any way. But
the facts of your post
and your followup of coprime, that you are mistaken.

You must admit, that in all of your life of doing mathematics, you
were never able to do a Infinitude of Twin Primes proof. I am
certainly right on that, John.

But the reason you were never able to do a Infinitude of Twin Primes
proof, John Ramsden,
is because you were never able to do a valid Infinitude of Regular
Primes (Euclid style) Indirect Method. I do not mean to be hard on you
John, but you mixed up the Indirect with the
Direct method.

So what I am saying, John, if that you had been able to do a proper
and valid Euclid style
Infinitude of Regular Primes Indirect method, you would also be able
to prove the Infinitude of Twin Primes.

Your mistake, John is that in the Indirect Method, the allying of both
the definition of a prime number in step 1 with the hypothetical
assumption step 2, those two steps when allied, forces
W+1 and W-1 to be necessarily two new primes.

For example, if the only primes that exist are 3 and 5, then Euclid's
number of W+1 = 16 and
W-1 = 14 are necessarily two new primes under that hypothetical space.

If you cannot believe or understand what I am saying then try
listening to Karl Heuer of 1994:


Sun, 20FEB1994, 21:05:13 GMT sci.math
INCONSISTENT PEANO AXIOMS AND MATH PROFESSORS
Lines: 36
Sender: k...(a)spdcc.com (Karl Heuer)

k...(a)ursa-major.spdcc.com (Karl Heuer) writes:

In article (5JChA8g2...(a)jojo.escape.de>

det...(a)jojo.escape.de (Detlef Bosau) writes:
>Ludwig.Pluton...(a)dartmouth.edu meinte am 18.02.94
>>det...(a)jojo.escape.de (Detlef Bosau) writes:
>>>Wrong. Your two numbers are not necessarily prime
>>NO, YOU ARE WRONG. Those numbers are necessarily prime, due to
>>UPFAT, all the primes that exist in the finite set leave a remainder
>>of 1.
>I'll give you a lesson of elementary arithmetics. . .

I really shouldn't bother to get involved in this discussion again,
but
Ludwig is right. In logical terms, his key statement is "if P is a
finite set containing all the primes, then prod(P)+1 is prime." This
is
a true statement.

Let's step through your alleged counterexample:

>consider your set of primes to be: {2,3,5,7,11,13}, as I assert 13 to be
>the largest prime. [. . .] Now, you made the assertion, that
> > > > (2x3x5x11x13) + 1 [=30031] must be prime.

Yes, it's true that if 13 is the largest prime, then 30031 is prime.
Do
you disagree with that assertion?

>As you stated before, there exists an unique prime decomposition of
>30031. This is 59x509. It could be easily shown, that 59 and 509
>both are prime.

If 13 is the largest prime, then 59x509 is not a factorization of
30031.

--- end quoting Karl Heuer's post of 1994 ---


So, Mr. John Ramsden, the reason that in all your math career in doing
math, that you were
never able to do a proof of the Infinitude of Twin Primes is very
obvious why you could never do a Twin Primes proof, because you could
never do a proper valid proof of Infinitude of Regular Primes,
indirect method.

Now maybe you can teach me something, John Ramsden. Because here we
have a situation where mathematics offers us a proof of Infinitude of
Regular Primes both Direct and Indirect Method. So the question that
vexs me, is why would the Twin Primes only be amenable to a Indirect
Method? Do proofs of mathematics have a sort of Physics conservation
of energy principle attached. So that the energy of proving Regular
Primes are infinite is exhausted and the only energy left in the
entire logical system of mathematics is the energy for a "indirect
method for twin-primes, Mersenne primes, and all the other related
infinitude proofs? So I wonder if mathematics has a conservation
principle that says some proofs can have both direct and indirect, but
outlying conjectures must only have a indirect application? This is
vexing, but I am sure it must be linked to physics.

So, please, John Ramsden, please try to reconsider that you never, in
your life ever gave a valid Euclid IP indirect, for if you had done
so, perhaps in your youth, you would have immediately been able to
prove infinitude of twin primes.

Please, write out your indirect method of regular primes in a followup
post.

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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