Re: [PrimeNumbers] First order son primes
- On Tue, 2008-09-02 at 17:33 +0000, Robert wrote:
> First order son primes (p, 3p+2 prime) are more common than SophieThey 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.
- Robert wrote:
> Why are chains of first order son primes not sought by prime huntersDifferent variations have been sought but less than the better known
Here are some prime sequences iterating ax+b:
You found the next term of one of them:
A page calling them generalized Cunningham chains:
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:
Jens Kruse Andersen
- --- In email@example.com, "Jens Kruse Andersen"
> 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 !!!!!