[ October 13, 2023 ] [ Last update – ]
Spotting palindromes 78087 and 672276
in the formula for indexing Fibonacci numbers.
By Patrick De Geest

The first few Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
A wealth of information is given in OEIS A000045.

The Fibonacci number is described in
https://www.geeksforgeeks.org/find-index-given-fibonacci-number-constant-time/ as

\(Fn = {1 \over sqrt(5)} * (a^n - b^n) \) where

\(a = {1 + sqrt(5) \over 2} ~~\) and \(~~\color{red}b = {1 - sqrt(5) \over 2} \)

On neglecting \(b^n\) which is very small because of the large value of n, we keep

\(Fn = {1 \over \sqrt{5}} * ({1 + \sqrt{5} \over 2})^n \)

Let us express this equation in function of index n using logarithm acrobacy

\(log(Fn) = log({1 \over \sqrt{5}}) + n * log({1 + \sqrt{5} \over 2}) \)

\(log(Fn) = -0.8047189562... +\ n\ *\ 0.4812118250... \)

\(log(Fn)\ +\ 0.8047189562... =\ n\ *\ 0.4812118250... \)

\(n\ =\ {{log(Fn)\ +\ 0.8047189562...}\over{0.4812118250...}} \)

\(n\ =\ {2.0780869212...\ *\ log(Fn)\ +\ 1.6722759381...} \)

The palindromes only show up when the accuracy is reduced to 6 decimals
and rounded which is more than enough for the objective.

\(n = round ( 2.0\color{blue}{78087}\color{black}{\ *\ log(Fn)\ +\ 1.}\color{blue}{672276} \color{black}{)} \)

where round means round to the nearest integer.
and log(Fn) means the natural logarithm of Fn.

What more is there to tell about these two peculiar palindromes ?