- --- In primenumbers@yahoogroups.com, "garychaffey2"

<garychaffey2@y...>

wrote:> I have been looking at the following chains of GFN. And would be

chain

> interested if anybody else has lloked at these and if a longer

is> known or could be found.

sequence

> Let b be an even number

> p(1)=b^2+1

> p(2)=(2*p(1))^2+1

> p(3)=(2*p(2))^2+1

> .

> .

> .

> p(n)=(2*p(n-1))^2+1

> I have found a few b such that the first three terms in this

> are prime but does a longer chain exist?

Hi Gary,

I've checked prime factors up to 23, and 13 is the only one which

gets in the way of a long string of primes. As it turns out, any

starting residue n of 13 when iterated through

(2*n)^2 +1

will eventually yield a zero residue . The maximum number of primes

which could be obtained is 5. (Based only on prime factors 23 and

less.)

It's interesting: The chain will never produce a factor of 3,7,11,19

or 23. It may or may not produce a factor of 5 or 17, depending on the

starting residue mod 5 or mod 17. But it will always produce a

factor of 13. Talk about an unlucky number :)

Mark - In an email dated 6/6/2005 5:43:37 pm GMT Daylight time, "garychaffey2" <garychaffey2@...> writes:

>I have been looking at the following chains of GFN. And would be

You might like to look at my "Quadratic-map prime chains" NMBRTHRY post of last year:

>interested if anybody else has lloked at these and if a longer chain is

>known or could be found.

>Let b be an even number

>p(1)=b^2+1

>p(2)=(2*p(1))^2+1

>p(3)=(2*p(2))^2+1

>.

>.

>.

>p(n)=(2*p(n-1))^2+1

>I have found a few b such that the first three terms in this sequence

>are prime but does a longer chain exist?

>Any comments welcome

http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0402&L=nmbrthry&F=&S=&P=922

-Mike Oakes - Mike Oakes wrote:

> You might like to look at my "Quadratic-map prime chains" NMBRTHRY post

Then also look at http://www.primepuzzles.net/puzzles/puzz_137.htm

> of last year:

> http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0402&L=nmbrthry&F=&S=&P=922

It says Yves Gallot found the smallest chain of 6 before Mike:

7072833120^1+1

7072833120^2+1

7072833120^4+1

7072833120^8+1

7072833120^16+1

7072833120^32+1

I have long considered going for 7 and maybe 8, but been occupied with other

projects.

A tip which has often been useful to me:

If you are interested in something related to a specific large number then

search the number at http://www.google.com

Searching Mike's 7072833120 quickly reveals Gallot was first.

--

Jens Kruse Andersen >A tip which has often been useful to me:

Thx for tip, Jens.

>If you are interested in something related to a specific large number then

>search the number at http://www.google.com

I was unaware of Yves's work, but it seems google is indeed getting quite "intelligent" these days, if it can respond to a single integer like that :-)

Must remember in future...

Mike