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