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From: Archimedes Plutonium on 14 Jul 2010 21:51
Sorry about that unfinished post, I hit a wrong key which stopped me
from completing the
post so I shut the computer down and restarted it.

>
> I usually find that talking something to death achieves my goal.
>
> Okay, so I supposed in Math Induction that p_N was true, and so
> (2x3x5x7x. . .xp_N) was of the form 2^p
>
> Now when in p_N+1 I include that into the new Euclid Number as such:
>
> (2x3x5x7x. . .xp_N x p_N+1)
>
> what I am doing is squaring in the series 2,4,8, 16, 32, 64,
> 128, . . .
>
> So that we can see that 8 x 8 is
>

What I wanted to say was that the Math Induction gives me a proof of
the Infinitude of Mersenne Primes and Infinitude of Perfect Numbers
when entering the Math Induction along
with the Indirect Euclid Infinitude of Primes proof.

The way Mathematical Induction helps is that it guarantees the Euclid
Number to be of form
(2^p) where I finally either add 1 or subtract 1.

How does Math Induction make that guarantee? As I tried to write above
that with the Supposition of case N and to show for case N+1 that the
supposition N case has Euclid's
Number in the form of 2^p and then when I enter into case N+1, it is
easy to see that I
squared the Euclid Number, for example if N was 8, then N+1 was 64.

So that the 2^p pattern of 2,4,8,16,32,64..... where the 2^p must be
one of those
numbers out of that pattern, well when I do the N+1, I am guaranteed
in staying within that
2^p pattern.

Yes, indeed. So by coupling Indirect Infinitude of Primes Euclid with
Mathematical Induction
I can prove the infinitude of Mersenne primes which yields Infinitude
of Perfect Numbers.

I have not seen many proofs in mathematics that had a genuine need of
mathematical induction. I have seen alot of cases of work problems to
keep students busy with applying
math induction. So I wonder if there are other examples of genuine
important math proofs that required Mathematical Induction, as seen in
this case where it is a valuable tool.

Now in the case of primes of form (2^p)+1 that Math Induction ceases
beyond 5, and shows us that Math Induction of initial cases should
always have several initial cases. So primes
of form (2^p)+1 are finite. Now whether I can prove primes of form
(2^n)+1 and (2^n)-1 using
the Indirect method with Math Induction are infinite sets is a good
exercise.

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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