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From: Archimedes Plutonium on 8 Jul 2010 16:27


Archimedes Plutonium wrote:
> Someone wrote:
>
> >>
> >> 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.
> >>
>
> Someone called asking me to critique the above garbled mess.
>

The above is not a proof, for it is an invalid attempt. I can even
shorten the above
to expose its flaws:

Another invalid attempt:
(1) Every n > 1 has at least one prime divisor
(2) Since the set N is infinite thus the prime divisors are infinite,
thus primes infinite
QED

Whoever wrote the above mess wanting to use prime divisors, rather
than writing out
a clear proof in long form, either could not write it out in long form
or did not want to
for it would expose the flaws.

The flaws of the above is that you have to actually produce a new
prime, whether
direct or indirect. So when you crank-dwell on every n>1 has at least
one prime
divisor, you never are able to actually produce a new prime not on the
starting list
and thus can never claim a contradiction.

The contradiction in the Indirect is that P+1 is larger than the
largest supposed prime.

So unless you produce a new prime, then all you have is just crank-
talk.


> I would bet that for every 1000 people that do a Infinitude of Primes
> proof, only 1 is
> going to see it clearly from start to finish and the other 999 are
> going to add excess baggage
> or even invalid steps.
>
> The corollary of every number larger than 1 has at least one prime
> divisor comes immediately
> from the definition of prime.
>
> So we have this.
>
> Corollary: Every number larger than 1 has at least one prime divisor.
> Proof: Definition of prime is a number divisible only by itself and
> one. Hence
> every number has at least one prime divisor.
>
> So the above gnarled mess is but one step excess baggage more than
> what any proof of Infinitude of Primes needs.
>
> Now I could have added that Corollary into my direct or indirect proof
> methods, but why bother, when the definition of prime says the same
> thing, but says it much more directly and
> clearly. I could add five alternative definitions of prime numbers to
> any of my proofs and would not have affected the validity of the
> outcome, but it sure would have made a mess.
>
> So there are going to be thousands of people who add excess nonsense
> into a Infinitude of Primes proof, rather than that one in a thousand
> clear mind that can do IP without nonsense
> and excess baggage.
>
>
> You still have to go through all these steps to reach a valid proof
> whether you add the Corollary or not:
>
> DIRECT Method (constructive method), long-form; Infinitude of Primes
>  Proof
>
>
> (1) Definition of prime as a positive integer divisible
>   only by itself and 1.
>
>
> (2) Statement: Given any finite collection of primes
>  2,3,5,7,11, ..,p_n possessing a cardinality n Reason: given
>
>
> (3) Statement: we find another prime by considering W+1 =(2x3x...xpn)
>   +1 Reason: can always operate on given numbers
>
>
> (4) Statement: Either W+1 itself is a prime Reason: Unique Prime
>  Factorization theorem
>
>
> (5) Statement: Or else it has a prime factor not equal to any of the
>   2,3,...,pn
>  Reason: Unique Prime Factorization theorem
>
>
> (6) Statement: If W+1 is not prime, we find that prime factor Reason:
>  We take the square root of W+1 and we do a prime search through all
>  the primes from 2 to
>  square-root of W+1 until we find that prime factor which
>  evenly divides W+1
>
>
> (7) Statement: Thus the cardinality of every finite set can be
>  increased. Reason: from steps (3) through (6)
>
>
> (8) Statement: Since all/any finite cardinality set can be increased
>  by one more prime, therefore the set of primes is an infinite set.
>  Reason: going from the existential logical quantifier to the
>  universal
>  quantification
>
>
> INDIRECT (contradiction) Method, Long-form; Infinitude of Primes
> Proof
>  and
>  the numbering is different to show the reductio ad absurdum
> structure
>  as
>  given by Thomason and Fitch in Symbolic Logic book.
>
>
> (1) Definition of prime as a positive integer divisible
>   only by itself and 1.
>
>
> (2) The prime numbers are the numbers 2,3,5,7,11, ..,pn,... of set S
>   Reason: definition of primes
>
>
> (3.0) Suppose finite, then 2,3,5, ..,p_n is the complete series set
>   with p_n the largest prime Reason: this is the supposition step
>
>
> (3.1) Set S are the only primes that exist Reason: from step (3.0)
>
>
> (3.2) Form W+1 = (2x3x5x, ..,xpn) + 1. Reason: can always operate and
>   form a new number
>
>
> (3.3) Divide W+1 successively by each prime of
>   2,3,5,7,11,..pn and they all leave a remainder of 1.
>   Reason: unique prime factorization theorem
>
>
> (3.4) W+1 is necessarily prime. Reason: definition of prime, step
>  (1).
>
>
> (3.5) Contradiction Reason: pn was supposed the largest prime yet we
>   constructed a new prime, W+1, larger than pn
>
>
> (3.6) Reverse supposition step. Reason (3.5) coupled with (3.0)
>
>
> (4) Set of primes are infinite Reason: steps (1) through (3.6)
>
>
>
> Hope that helps.
>
>
> 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 9 Jul 2010 19:58
On 8 July, 21:27, Archimedes Plutonium
<plutonium.archime...(a)gmail.com> wrote:
> Archimedes Plutonium wrote:
> > Someone wrote:
>
> > >> 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.
>
> > Someone called asking me to critique the above garbled mess.
>
> The above is not a proof, for it is an invalid attempt. I can even
> shorten the above
> to expose its flaws:

I can see you do not understand what is involved in a proof
by contradiction.

Obviously, the idea that every n>1 has at least one
prime divisor is too commplex for you.

On the other hand, you seem to concede sometimes
that every N>1 has a unique prime factorization

A) Every N>1 has a unique prime factorization.
B) No pair of consecutive naturals has a common factor
other than 1

A) and B) are theorems and so true.

H) Assume that p,q,r are the only primes

If H is true then by A) every natural must be some product of
p,q,r. The only primes that exist.

In sequence, the naturals could have prime factorizations

1, p, q, pp, r, pq, rr, ppq, ........., pppqqqqrrrr, .....
p= 1+1, q = p+1, pp= q+1, r = pp+1, etc

As no two consecutive naturals have a common factor other than 1

GCD(p,q) =1, GCD(pq,r) = 1 etc.

pqr, pqr+1 are consecutive naturals and so have
no common factor greater than 1

pqr+1 by unique factorization must be some product of p,q,r

But whatever unique product of primes pqr+1 equals
p,q,r,pq,pr,qr,pp, .., ppppppqqqrrrrrr, etc

pqr and pqr+1 must share a common factor other than 1.

This is a contradiction as a pair of consecutive naturals
has no common divisor other than 1

The assumption that the primes are finite in number is therefore
false.
The primes are infinite in number.

If the reasoning is false say which statement
or deduction is false or invalid.

If you cannot indicate which particular statement or deduction is
false or invalid, the proof must be valid

From: sttscitrans on 10 Jul 2010 04:56
On 10 July, 01:20, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> "sttscitr...(a)tesco.net" <sttscitr...(a)tesco.net> writes:
> > I can see you do not understand what is involved in a proof by
> > contradiction.
>
> That's entirely possible, but it's utterly pointless to formulate the
> standard proof of the infinitude of primes as an indirect proof.

Which proof is the standard proof ?
Who is trying to formulate the "standard proof" as an "indirect
proof ?

> > If you cannot indicate which particular statement or deduction is
> > false or invalid, the proof must be valid
>
> The validity of the proof in no way depends on
Archimedes Plutonium's
> abilities.

"you" = "one"

"If you don't train, you'll never be a top athlete"
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