Bill Dubuque wrote:
> "Larry Hammick" <larryhammick@[EMAIL PROTECTED]
> wrote:
> >
> > Anyhow, here's a proof of IX.20 discovered by Filip Saidak in 2005 AD!
> > Define inductively a sequence (x(n), y(n)) of ordered pairs by
> > x(1) = 2, > y(1) = 3
> > and for n >= 2 :
> > x(n+1) = x(n)y(n)
> > y(n+1) = x(n)y(n) + 1
> > By induction, x and y stay positive. Also x(n) and y(n) are relatively
prime
> > by definition. So y(n) has a prime factor which x(n) lacks, and that
factor,
> > along with all the factors of x(n), appears in x(n+1). So by induction
x(n)
> > has at least n distinct prime factors, for arbitrary n.
>
> SIMPLER NN + N has more prime factors than N
>
> "Saidak's proof" is certainly not new.
>
> --Bill Dubuque
Not as neat or tidy or all-inclusive as the new Euclid Infinitude of
Primes
Proof that includes all the Natural Numbers. When Euclid or Saidak did
their proofs
of the Infinitude of Primes they were not aware of all the Natural
Numbers such as
.....999999 or ....666666777777. Where Natural Numbers = AP-adics.
So what is the easiest of all proofs that primes are infinite? And I
believe this proof is
the fastest, easiest.
We simply construct an Infinitude of Primes:
(1) 11 and 13 are prime
(2) the number ....1413121110987654321 is also prime as Champernownes
number (spelling)
(3) construct an infinite set of primes by simply attaching the "11"
and "13" on the far right
yielding ....141312111098765432111 and ....141312111098765432113 now
for the next
pair of twin primes we attach a 2 giving ....1413121110987654321211
and ....1413121110987654321213
now the next pair of twin primes we attach a 3 giving ....
14131211109876543213211 and ....14131211109876543213213
now for the next pair of twin primes we attach a 4 to the previous 3
and 2 and we keep doing this infinitely.
Now I not only proved the primes are infinite but that Twin-primes are
infinite. So this is the first
proof of Euclid Infinitude of Primes and Twin-primes in one proof.
Now I like to challenge Mr. Bill Dubuque to a case of "logical flow"
of proof coupled with brevity.
Brevity that Ian Stewart showed when he called it "multiply the lot
add 1". I
challenge Bill to give a Euclid Infinitude of Primes proof of Direct
Method and then Indirect Method.
Two proofs simultaneously, one of direct and the other indirect for
contrast. In the past,
mathematicians were never required to give both simultaneously. Maybe
if they had been required
that such a high percentage of failures 28/30 = 93% failure to render
a valid proof argument.
Are you up for a challenge Bill?
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
|
