Re: Complex Lucas primes
- --- In email@example.com,
"Mike Oakes" <mikeoakes2@...> wrote:
> you are really just finding the probable-primeThanks for tidying up, Mike. Note that
> norms of the Gaussian Lucas sequence
Q = -3 + 4*I = (1 + 2*I)^2
is the square of a reduced parameter, q,
inside your small quadrant, which is where
I found these primes in the first place.
The general formula for squaring q by "renaming" is
V(p,q,2*n) = V(p^2-2*q,q^2,n)
That, in turn, is merely the m = 2 case of
V(p,q,m*n) = V(V(p,q,m),q^m,n) ... 
which is one of my favourite functional equations.
This bothered Bouk de Water when I showed him
a non-trivial example. He asked me to prove,
at the very outset, that
V(V(p,q,m),q^m,n) = V(V(p,q,n),q^n,m) ... 
else he would never believe . (As I recall, he was not
best pleased when I derived  as a corollary of .)
PS: I noted your neat pair of gigantic probable Gaussian primes
Not as remarkable as the de Water and Noe probable prime pair
but yours is still a notable collision.