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Re: Lehmer sequence puzzle

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  • David Broadhurst
    ... OK Boss, message received and understood. Sometimes such discoveries are best not cast as puzzles. Discovery: Mike Oakes has found a (Q,R) integer pair
    Message 1 of 33 , May 7, 2009
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      --- In primenumbers@yahoogroups.com,
      "Mike Oakes" <mikeoakes2@...> wrote:

      > I don't know at the moment how to fix my puzzle to stop
      > you guys giving the answer \infty straight away.
      ...
      > I posed the puzzle because I have found, experimentally,
      > examples where u(R,Q) is 1, 2 and even
      > (a great surprise to me!) 3.
      > I want to know if there are any examples >3 (and < \infty).
      > Help me, please, to both formulate the puzzle securely
      > and solve it!

      OK Boss, message received and understood.

      Sometimes such discoveries are best not cast as puzzles.

      Discovery:
      Mike Oakes has found a (Q,R) integer pair such that with
      x = sqrt(R)/2 + sqrt(R-4*Q)/2
      y = sqrt(R)/2 - sqrt(R-4*Q)/2
      (x^n+y^n)/(x+y) is a unit precisely 3 times for n > 1.

      Questions:
      Can anyone improve on Mike's tally of 3,
      in a situation where the number of units is finite?
      If not, can anyone prove that 3 is the maximum,
      in a situation where the number of units is finite?

      C'est ça?

      David
    • David Broadhurst
      ... Yes, Mike, that last form is by far the neatest, if one removes the unnecessary lucas , which should be well understood, when one speaks of U or V . 1)
      Message 33 of 33 , May 10, 2009
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        --- In primenumbers@yahoogroups.com, "Mike Oakes"
        <mikeoakes2@...> wrote [with unfailing courtesy]:

        > lucasU(338,25,11584)-5*lucasU(338,25,11583)
        > I guess Chris would expect the last of these as being the most
        > "canonical"? And I concur, it being also the shortest.

        Yes, Mike, that last form is by far the neatest,
        if one removes the unnecessary "lucas", which should
        be well understood, when one speaks of "U" or "V".

        1) François Édouard Anatole Lucas defined the integer sequence
        U(P,Q,n) = P*U(P,Q,n-1) - Q*U(P,Q,n-2),
        with U(P,Q,0) = 0 and U(P,Q,1) = 1,
        in terms of elementary arithmetic.

        2) I like to imagine that Derrick Norman Lehmer (1867-1938) saw that
        U(P,Q^2,2*k+1) = U(P,Q^2,k+1)^2 - (Q*U(P,Q^2,n))^2
        and told his son: "Go look at the factors for your Ph.D."

        3) Certainly, Derrick Henry Lehmer (1905-1991) did study
        U(P,Q^2,k+1) - Q*U(P,Q^2,k)
        being very well aware of the super-Lucasian extension to powers
        of algebraic numbers of degree 4, in the wider complex plane.

        4) Mike Oakes has found that the Lehmer number
        U(P,Q^2,k+1) - Q*U(P,Q^2,k)
        is probably prime, when P = 338, Q = 5, k = 11583.

        5) David Broadhurst is able to characterize this
        circumstance without using a dirty 4-letter word :-)

        Thank ye, kindly, Sir, for this interesting thread!

        David, pp SSSR
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