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Re: Complex Lucas primes

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  • David Broadhurst
    ... Thanks for tidying up, Mike. Note that Q = -3 + 4*I = (1 + 2*I)^2 is the square of a reduced parameter, q, inside your small quadrant, which is where I
    Message 1 of 20 , Apr 27, 2009
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      --- In primenumbers@yahoogroups.com,
      "Mike Oakes" <mikeoakes2@...> wrote:

      > you are really just finding the probable-prime
      > norms of the Gaussian Lucas sequence
      > V(1,-3+4*I,n)

      Thanks for tidying up, Mike. Note that
      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) ... [1]

      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) ... [2]

      else he would never believe [1]. (As I recall, he was not
      best pleased when I derived [2] as a corollary of [1].)

      PS: I noted your neat pair of gigantic probable Gaussian primes

      V(1,2+2*I,19979)
      U(2+I,1-I,19979)

      Not as remarkable as the de Water and Noe probable prime pair

      V(1,-1,148091)
      U(1,-1,148091)

      but yours is still a notable collision.

      David
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