The Fibonacci Sequence is Normal Base 10

Brennan Benfield, Michelle Manes

Published 2022-02-18Version 1

In this paper, we show that the concatenation of the Fibonacci sequence is \textit{normal} in base $10$, meaning every string of a given length, $k$, occurs as frequently as every other string of length $k$ (there are as many $1$'s as $2$'s and as many $704$'s and $808$'s). Although we know that almost every number is normal, we can name very few of them. It is still unclear if $e$, $\pi$, or $\sqrt{2}$ are normal. We show that concatenating the Fibonacci sequence behind a decimal creates a normal number in every base of the form $5^x\times2^y$. We then provide evidence that potentially extends our result to all integer bases, and claim that the Fibonacci concatenation is \textit{absolutely normal}.