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3698Modular cycles in the Lucas and Fibonacci Series

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  • John McNamara
    Oct 31, 2001
      By way of a preamble, it would seem that in different
      circles the "General Fibonacci Series" is variously
      known as "a Lucas Series" or "a Fibonacci Series".
      Dr Knott certainly uses the term Fibonacci as the
      general term, whereas this Forum seems to say Lucas,
      which I would have no trouble accepting, except that
      an Authority like Knott says otherwise. I will call it
      the GFS in this post, Lucas 1,3,4,7.... and Fibonacci
      1,1,2,3......., to avoid confusion, as I will be
      referring to all three ideas.

      Also as a preamble F(2n) = F(n)*L(n) is a well known
      relation between the Lucas and the Fibonacci.

      Not wishing to bore you all with another conjecture, I
      will simply say that as a result of some counsel
      (from, I suspect, David Broadhurst) as being very
      interesting to study, I have been examining the cycle
      of remainders when these series are divided by the
      primes. I have found something interesting in the
      early cases that I suspect may already be known. But I
      have never seen it expressly stated as such, but may
      be a kind of corollary to that already known fact. (I
      am very afraid of the law of small numbers, also.)

      It is simply that if a prime is 3mod10 or 7mod10 and
      is a factor of a term in the Lucas Series 1,3,4,7...,
      it seems always to be a regularly occuring factor of
      certain terms in every GFS.

      Concretely, 1,3,7,23,43,47 are, whereas 13,17,37,53
      not always factors.

      Another way of stating it is to say that the primes
      which appear as factors for the first time in the odd
      numbered Fibonacci Series terms are different in
      character from those which appear for the first time
      in the even numbered terms.

      Incidentally for 1mod10 or 9mod10 primes, the first
      occurence in an odd numbered term is 61 in term 15.

      In other words, there are two classes of primes > 3,
      (and it even includes 2 if one is "cute"), one class
      capable only of appearing as a factor in the even
      numbered terms of the Fibonacci series, and the rest
      making up the other class.

      I should be pleased to know of counterexamples, but I
      only operate from an internet cafe, and do not know
      mathematical programming languages, having to rely on
      a calculator, spreadsheet, pencil and paper, or my
      rapidly deteriorating "mental", as it was called in
      the middle of last century.

      John McNamara

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