Modular cycles in the Lucas and Fibonacci Series
- 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.
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