Intended for: September 27, 2014
Timetable
- First draft entered by Alonso del Arte on September 1, 2013 ✓
- Draft reviewed by Daniel Forgues on October 02, 2016 ✓
- Draft to be approved by August 27, 2014
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A192038: The Fibonacci “logarithm” for 4.*
-
What is the ten millionth
Fibonacci number? Without the
Binet formula, answering that question would require computing ten million Fibonacci numbers. Thanks to the Binet formula,
, where
is the
golden ratio, we can just plug in
and get the
th Fibonacci number with just a few operations instead of ten million or so additions. The thought then occurs: Must we plug in an integer? What happens if we put in any real number? Defining
f (x) := ϕ x − cos (π x) ϕ − x | 2√ 5 |
|
(a slight variation of the Binet formula,
Clark Kimberling tells us)** does not readily indicate any impediments to plugging a real number, and so, for example, we have
f (2 2√ 5 ) ≈ 3.84257859447. |
The next question then is: Can we find a value of
that will give us an integer that is not actually a Fibonacci number, like, say,
4? Indeed we can. Setting
we obtain
. That’s one solution, there are others, which are all negative numbers.
_______________
* Since the Fibonacci function
is strictly increasing only from
(
A171909), the Fibonacci “logarithm” has a unique value for
.
** Merve Özvatan and Oktay K. Pashaev, Generalized Fibonacci Sequences and Binet–Fibonacci Curves (July 2017), arXiv:1707.09151. (Cf. §5.1 Fibonacci Numbers of Real argument and §5.3 Fibonacci Spirals.)
Notes