```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 whetherI need a prime at 10^603 for Galois field theory. It is dangerous inmathto 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 andfieldare somewhat replaced by a Zone of Algebraic Completeness.If it is found to be crucial that a prime exist in the region of10^603 forGalois field, then when I get the computer up and running, I shouldexplore bothpi and "e" in the 602 and 603 region for twin primes and for possibleprimes of increased digit.pi = 3.14159..32000e  = 2.71828..17301Now those as integers314159..32000271828..17301Are 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 thisfurther. But the best wayto pursue it would be "if a prime is required, how it is requiredmight suggest where to look for it"?Archimedes Plutoniumhttp://www.iw.net/~a_plutoniumwhole entire Universe is just one big atom?where dots of the electron-dot-cloud are galaxies
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