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+--- Thread: A random question for mathematicians regarding i and the Fibonacci sequence. (/showthread.php?tid=685)
A random question for mathematicians regarding i and the Fibonacci sequence. - robo37 - 08/07/2011
What is the Fibonacci root of i, or to put it another way, what is the equation of x when
\( \Large \quad \quad {({1+\sqrt 5\over 2})^x-(-1)^x ( { 2\over 1+\sqrt 5})^x \over \sqrt 5}=n \)
?
All I need is a formular and I am capable of just entering the eqution and i straight into Google calculator to get a result.
I'm interested as decimal numbers entered into the Fibonacci function manly get complex results which shows a link between the function and i, and suggests that the Fibonacci root of i might not be complex and might be of some importance.
I know this isn't really to do with Tetration, but I've asked this across the internet and have not have any replies and I'll be greatfull for any help.
Thanks.
RE: A random question for mathematicians regarding i and the Fibonacci sequence. - Catullus - 06/27/2022
(08/07/2011, 11:17 PM)robo37 Wrote: What is the Fibonacci root of i, or to put it another way, what is the equation of x when ((1+sqrt 5)/2)^x-((-1)^x/((1+sqrt 5)/2)^x)))/sqrt 5=n?You did not rationalize your denominator.
The numerical solutions to Fibonacci(x)=i are
x ~ -.42709179162288762 - .22105741463869769 * I
and
x ~ 1.67226840550786450 + 3.2642638352534426 * I.
I do not know of any closed forms for those numbers.
For an inverse to the Fibonacci function, there are no result found in terms of standard mathematical functions.