## Re: [PrimeNumbers] First order son primes

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• ... They have. Many of them can be proved to have a maximum length. Teske & Williams paper in LNCS 1838 is a nice treatment of consecutive prime values
Message 1 of 9 , Sep 4, 2008
On Tue, 2008-09-02 at 17:33 +0000, Robert wrote:
> First order son primes (p, 3p+2 prime) are more common than Sophie
> Germains (p,2p+1 prime): approx 36% more common.
>
> Why? - If we look at mod 3
>
> if p==1mod3 then 2p+1==0mod3
> if p==2mod3 then 2p+1==2mod3, 50% chance of a 2p+1 is not 0mod3
>
> if: p==1mod3 then 3p+2==2mod3
> if: p==2mod3 then 3p+2==2mod3, 100% chance that 3p+1 is not 0mod3
>
>
> Question: Why are chains of first order son primes not sought by prime
> hunters, as they might provide longer chains than SG, CC, despite the
> slight increase in magnitude?

They have. Many of them can be proved to have a maximum length.

Teske & Williams' paper in LNCS 1838 is a nice treatment of consecutive
prime values produced by iterating the mapping f(x) -> ax^2+b

I happen to know this paper because the authors could find chains for
(a,b) = (1, -17) of at most 5 primes. I found several longer ones
though none as large as the maximum possible, which is 16 for this
choice of (a,b). I can't now find the computational results which I
mailed off to Edlyn.

Paul
• ... Different variations have been sought but less than the better known Cunningham chains. Here are some prime sequences iterating ax+b:
Message 2 of 9 , Sep 4, 2008
Robert wrote:
> Why are chains of first order son primes not sought by prime hunters

Different variations have been sought but less than the better known
Cunningham chains.

Here are some prime sequences iterating ax+b:
http://www.research.att.com/~njas/sequences/?q=%22On+certain+chains+of+primes%22
You found the next term of one of them:
http://www.research.att.com/~njas/sequences/A083388

A page calling them generalized Cunningham chains:

A page saying "prime trees" about primes iterated with ax+/-b
where + and - can be mixed:
http://unbecominglevity.blogharbor.com/blog/_archives/2004/3/17/27759.html
A prime tree of depth 26 for 2x+/-308843535 starting at 177857809:
http://unbecominglevity.blogharbor.com/blog/_archives/2006/5/12/1952529.html

--
Jens Kruse Andersen
• ... http://unbecominglevity.blogharbor.com/blog/_archives/2004/3/17/27759.html ... http://unbecominglevity.blogharbor.com/blog/_archives/2006/5/12/1952529.html
Message 3 of 9 , Sep 4, 2008
--- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
<jens.k.a@...> wrote:

>
> A page saying "prime trees" about primes iterated with ax+/-b
> where + and - can be mixed:
>
http://unbecominglevity.blogharbor.com/blog/_archives/2004/3/17/27759.html
> A prime tree of depth 26 for 2x+/-308843535 starting at 177857809:
>
http://unbecominglevity.blogharbor.com/blog/_archives/2006/5/12/1952529.html

Gosh, did not realise, and such a sad story for the blogger. If he had
come here first he could have saved himself 2 years work !!!!!
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