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...,

then

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

are

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

the

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

__________________________________________________

Do You Yahoo!?

Make a great connection at Yahoo! Personals.

http://personals.yahoo.com