--- In

primenumbers@yahoogroups.com,

Phil Carmody <thefatphil@...> wrote:

> One caveat with two-for-the-price-of-one deals is that the

> two bits you get back might not actually be independent of

> each other, so you'd not be getting full value for money.

> It applies to all the Grantham one too of course.

Indeed. The decoupled version of CP Algorithm 3.5.9,

with Paul's preferred parameters, is

Lucas with parameters (P,Q) = (c,1);

Fermat with base d = 2*x+5, where x is the smallest

non-negative integer for which kronecker(x^2-4,N) = -1.

These are not chosen independently by Paul, since

c = (d^2 - 2*d + 9)/(4*d) mod N ... [1]

turns out to be the rather delicate condition for

Paul's "Lucas test with Fermat test for free".

It is hard to see why [1] might induce a

correlation between the probabilities for Lucas

pseudoprimality and Fermat pseudoprimality.

But Phil is correct in pointing out that we

cannot entirely discount that possibility.

When kronecker(5,N) = -1, Grantham suggests

using a Frobenius test for which the CP separation

gives c = 3, for Lucas, and d = 5, for Fermat,

while BPSW choose c = 3 and d = 2. We likewise

cannot discount the possibility of correlated

probabilities in such circumstance.

David