The Fibonacci Sequence is Normal Base 10

TL;DR

This paper proves that concatenating Fibonacci numbers produces a normal number in certain bases, and provides evidence suggesting it may be absolutely normal in all bases, advancing understanding of normality in mathematical constants.

Contribution

It demonstrates the normality of Fibonacci concatenation in specific bases and offers evidence for its absolute normality across all bases, a novel result in number theory.

Findings

Fibonacci concatenation is normal in bases of the form 5^x * 2^y

Evidence suggests potential normality in all integer bases

Proposes Fibonacci concatenation as an absolutely normal number

Abstract

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}.