English

The Fibonacci Sequence is Normal Base 10

Number Theory 2022-02-21 v1

Abstract

In this paper, we show that the concatenation of the Fibonacci sequence is \textit{normal} in base 1010, meaning every string of a given length, kk, occurs as frequently as every other string of length kk (there are as many 11's as 22's and as many 704704's and 808808's). Although we know that almost every number is normal, we can name very few of them. It is still unclear if ee, π\pi, or 2\sqrt{2} are normal. We show that concatenating the Fibonacci sequence behind a decimal creates a normal number in every base of the form 5x×2y5^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}.

Keywords

Cite

@article{arxiv.2202.08986,
  title  = {The Fibonacci Sequence is Normal Base 10},
  author = {Brennan Benfield and Michelle Manes},
  journal= {arXiv preprint arXiv:2202.08986},
  year   = {2022}
}
R2 v1 2026-06-24T09:43:42.747Z