Fibonacci Numbers/Examples/F1000

Example of Fibonacci Number

The Fibonacci number $F_{1000}$ is a number with $209$ decimal digits beginning with $4$.


Proof

By the corollary to the Euler-Binet Formula:

$F_{1000} \approx \dfrac {\phi^{1000} } {\sqrt 5}$

From Number of Digits in Number, the number of decimal digits $m$ in $F_{1000}$ is given by:

$m = \floor {\log_{10} F_{1000} } + 1$

Thus, by calculation:

$m = \floor {208 \cdotp 64} + 1 = 209$

and the first digit can be obtained by evaluating $10^{0 \cdotp 64} \approx 4 \cdotp 36$.

Hence the result.

$\blacksquare$


Sources

  • 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: Exercise $2$
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