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

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

Between 1028489 and 1028489 + 78087 there are no primes.

672276 is also palindromic in base 32 (A099165). It is KGGK32.