- --- In primenumbers@yahoogroups.com, "Robert" <robert_smith44@...> wrote:
>

Their 1mod3 sisters p, 3p-2, form first instance chains as follows:

> --- In primenumbers@yahoogroups.com, "Robert" <robert_smith44@> wrote:

>

> >

> > The earliest instance chains are:

> >

> > SPlen2 3,11

> > SPlen3 5,17,53

> > SPlen4 29,89,269,809

> > SPlen5 1129,3389,10169,30509,91529

> > SPlen6 10009,30029,90089,270269,810809,2432429

SPMinuslen2 starting 3

SPMinuslen3 3

SPMinuslen4 5

SPMinuslen5 61

SPMinuslen6 1171241 (huge jump)

SPMinuslen7 1197631

SPMinuslen8 25451791

Regards

Robert Smith - --- In primenumbers@yahoogroups.com, "Robert" <robert_smith44@...> wrote:
>

Their 1mod3 sisters p, 3p-2, form first instance chains as follows:

> --- In primenumbers@yahoogroups.com, "Robert" <robert_smith44@> wrote:

>

> >

> > The earliest instance chains are:

> >

> > SPlen2 3,11

> > SPlen3 5,17,53

> > SPlen4 29,89,269,809

> > SPlen5 1129,3389,10169,30509,91529

> > SPlen6 10009,30029,90089,270269,810809,2432429

SPMinuslen2 starting 3

SPMinuslen3 3

SPMinuslen4 5

SPMinuslen5 61

SPMinuslen6 1171241 (huge jump)

SPMinuslen7 1197631

SPMinuslen8 25451791

SPMinuslen9 25451791

Regards

Robert Smith - On Tue, 2008-09-02 at 17:33 +0000, Robert wrote:
> First order son primes (p, 3p+2 prime) are more common than Sophie

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

> 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?

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 - 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:

http://www.primenumbers.net/Henri/us/CunnGenus.htm

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 - --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"

<jens.k.a@...> wrote:

>

http://unbecominglevity.blogharbor.com/blog/_archives/2004/3/17/27759.html

> A page saying "prime trees" about primes iterated with ax+/-b

> where + and - can be mixed:

>

> 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 !!!!!