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:


 * $m = \left\lceil{\log_{10} F_{1000} }\right\rceil$

Thus, by calculation:
 * $m = \left\lceil{208 \cdotp 64}\right\rceil = 209$

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

Hence the result.