• I have been looking at the following chains of GFN. And would be interested if anybody else has lloked at these and if a longer chain is known or could be
Message 1 of 5 , Jun 6, 2005
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I have been looking at the following chains of GFN. And would be
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?
Gary Chaffey
• ... ... chain is ... sequence ... Hi Gary, I ve checked prime factors up to 23, and 13 is the only one which gets in the way of a long
Message 2 of 5 , Jun 6, 2005
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<garychaffey2@y...>
wrote:
> I have been looking at the following chains of GFN. And would be
> 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?

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
• ... You might like to look at my Quadratic-map prime chains NMBRTHRY post of last year:
Message 3 of 5 , Jun 7, 2005
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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
>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?

You might like to look at my "Quadratic-map prime chains" NMBRTHRY post of last year:
http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0402&L=nmbrthry&F=&S=&P=922

-Mike Oakes
• ... Then also look at http://www.primepuzzles.net/puzzles/puzz_137.htm It says Yves Gallot found the smallest chain of 6 before Mike: 7072833120^1+1
Message 4 of 5 , Jun 7, 2005
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Mike Oakes wrote:

> You might like to look at my "Quadratic-map prime chains" NMBRTHRY post
> of last year:
> http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0402&L=nmbrthry&F=&S=&P=922

Then also look at http://www.primepuzzles.net/puzzles/puzz_137.htm
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
Searching Mike's 7072833120 quickly reveals Gallot was first.

--
Jens Kruse Andersen
• ... Thx for tip, Jens. 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
Message 5 of 5 , Jun 7, 2005
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>A tip which has often been useful to me:
>If you are interested in something related to a specific large number then