Dear Erlercw,

just speculating …

I think one method that might seem a little far-fetched, but maybe

worthwhile, if

you want to calculate big Fibonacci numbers via the formula of Wilf’s

book, could use continued fractions (see

*http://mathworld.wolfram.com/ContinuedFraction.html*

(http://mathworld.wolfram.com/ContinuedFraction.html) ).

The point is that the continued fraction representation [a_0,a_1, …]

of

1/2+sqrt(5)/2 is [1,1,1,…], so it’s periodic.

Now, in *ftp://ftp.inria.fr/INRIA/publication/publi-pdf/RR/RR-0760.pdf*

(ftp://ftp.inria.fr/INRIA/publication/publi-pdf/RR/RR-0760.pdf) ,

formulas (14) and (15), there is a method for potentiating and taking

logarithms of

arbitrary continued fractions.

I’d now replace the millionfold multiplication by

a multiplication of the logarithm, and exponentiate once for each of

the terms, and convert back, in the hope of finding some pattern

in the continued fraction (depending on the exponent).

My feeling is that there has to be some point where it’s more costly

to multiply on and on in contrast to such a method, but I don’t now

whether it’s already reached at F_{10**6}, and it’s clear that

you have to count implementation time for the methods also…

Best regards,

Axel