Euler-Binet Formula/Corollary 1

Corollary to Euler-Binet Formula

$F_n = \dfrac {\phi^n} {\sqrt 5}$ rounded to the nearest integer

where:

$F_n$ denotes the $n$th Fibonacci number
$\phi$ denotes the golden mean.


Proof

By definition of $n$th Fibonacci number, $F_n$ is an integer.

From Euler-Binet Formula:

$F_n = \dfrac {\phi^n - \hat \phi^n} {\sqrt 5} = \dfrac {\phi^n } {\sqrt 5} - \dfrac {\hat \phi^n} {\sqrt 5}$

But $\size {\dfrac {\hat \phi^n} {\sqrt 5} } < \dfrac 1 2$ for all $n \ge 0$.

Thus $\dfrac {\phi^n } {\sqrt 5}$ differs from $F_n$ by a number less than $\dfrac 1 2$.

Thus the nearest integer to $\dfrac {\phi^n } {\sqrt 5}$ is $F_n$.

$\blacksquare$


Sources

  • 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $5$
  • 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: $(15)$
  • 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5$
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