William,

Now this makes more sense to me. Thanks 1M for the lesson. I will see what full sense I can make of this email. At the end I should be able to understand Aurifeuillian factors fully.

Thanks again...

elevensmooth <

elevensmooth@...> wrote:

Some details were still wrong. I got the middle

exponent of one Aurifeuillian factor wrong, and

I selected the wrong small Aurifeuillian factor

as a divisor of the large one. Here's yet

another try to get this right:

C=148330

X=2^C+1

N=X^4-2

N is PRP for 2, 29, 43, 101

(N+1) = (X^2+1)(X+1)(X-1)

(X^2+1)/2 = (2^296659 + 2^148330 + 1)

This is an Aurifeuillian factor of (2^593318 + 1)

593318=11*53938

(2^593318 + 1) has algebraic factors, including (2^53938 + 1).

(2^53938 + 1) has Aurifeuillian factors

(2^26969 - 2^13485 + 1) and (2^26969 + 2^13485 + 1)

(2^26969 - 2^13485 + 1) is a factor of (X^2+1)/2

To verify the small Aurifeuillian factor divides the

large one without using a big number package, let

z=2^13484. The large factor is

(2048z^22 + 64z^11 + 1). The small factor is

(2z^2 � 2z + 1). The polynomial factorization can be

confirmed, for example, at this web site:

http://icm.mcs.kent.edu/research/facdemo.html
Sorry for the continuing string of errors.

William

--

ElevenSmooth: Distributed Factoring of 2^3326400-1

http://www.ElevenSmooth.com
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