Date: Nov 15, 2011 2:42 AM
Author: plutonium.archimedes@gmail.com
Subject: Chapt3 zone of algebraic completeness #1148 Correcting Math 3rd ed

On Nov 14, 4:36 pm, Transfer Principle <david.l.wal...@lausd.net>
wrote:
> On Nov 14, 1:22 am, Archimedes Plutonium
>
> <plutonium.archime...@gmail.com> wrote:
> > On Nov 13, 8:31 pm, Transfer Principle <david.l.wal...@lausd.net>
> > wrote:
> > If memory serves me with the Computer, only the number 101 is prime as
> > far as I know.
> > So that rather shocked my intuition to think that in the pattern
> > 101
> > 1001
> > 10001
> > That only the first one is a prime.
> > Transfer, I do not have a proof that only 101 is prime and the
> > Computer found no
> > primes out to a hundred of those (10^100)+1
>
> A prime that's one more than a power is called a Generalized
> Fermat prime.
>
> http://en.wikipedia.org/wiki/Fermat_number
>
> > So, question, is that pattern bearing of any more primes or is 101 the
> > only prime and whether (10^603)+1 is prime.
>
> 10^603+1 is not prime. Indeed, the mathematician Fermat proved
> that b^m+1 is composite for all b>1, unless m is a power of 2
> (i.e., m=2^n, so b^2^n+1 might be prime). The reason is a
> generalization of the sum of cubes polynomial factorization
> from high school Algebra II. A proof is given at the above
> Wikipedia link (for b=2, but generalizes to all b>1).
>
> In particular, if m=rs and s is odd, then 10^r+1 must divide
> 10^m+1 evenly. Since 603 has prime factors three and 67, we see
> that the following are all algebraic factors of 10^603+1:
>
> 10^1+1
> 10^3+1
> 10^9+1
> 10^67+1
> 10^201+1
>
> And they're all factors of 10^1809+1 as well. Indeed, one of the
> factors of 10^1809+1 is 10^603+1 itself!
>
> So it's possible for 10^512+1 and 10^1024+1 to be prime, but
> not 10^603+1 at all.
>
> Google gives results referring to large known Generalized Fermat
> primes, but most results give values of b other than 10. So I
> don't know whether there are any primes 10^2^n+1 other than n=1
> (but Wikipedia does state that for most b, we expect only finitely
> many primes in the sequence b^2^n+1).

Thanks for the information. It satisfies my curiousity.

I am going to drop the issue, and the more important issue of whether
I need a prime at 10^603 for Galois field theory. It is dangerous in
math
to hunt for a prime when not sure if a prime is even needed.

The zone of algebraic completeness serves as the Galois field.

So maybe I should focus more on the idea that a Galois group, ring and
field
are somewhat replaced by a Zone of Algebraic Completeness.

If it is found to be crucial that a prime exist in the region of
10^603 for
Galois field, then when I get the computer up and running, I should
explore both
pi and "e" in the 602 and 603 region for twin primes and for possible
primes of increased digit.

pi = 3.14159..32000
e  = 2.71828..17301

Now those as integers

314159..32000
271828..17301

Are the numbers 314159..32001 and 314159..31999 twin primes?
And is the number 271828..17299 a prime?

Is phi 1.618..861 as integer 1618..859 a prime?

If I knew for sure that a prime is needed, I would pursue this
further. But the best way
to pursue it would be "if a prime is required, how it is required
might suggest where to look for it"?

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