### Best, A, Dynes, P, Edelsbrunner, X, McDonald, B, Miller, SJ, Tor, K, Turnage-Butterbaugh, C and Weinstein, M (2014)

#### Benford Behavior of Zeckendorf Decompositions

Fibonacci Quarterly 52(5), pp. 35–46.

**ISSN/ISBN:** Not available at this time.
**DOI:** Not available at this time.

**Abstract:** A beautiful theorem of Zeckendorf states that every integer can be written
uniquely as the sum of non-consecutive Fibonacci numbers {Fi}∞i=1. A set S ⊂ Z is said to
satisfy Benford’s law if the density of the elements in S with leading digit d is log (1 + 1 ). 10 d
We prove that, as n → ∞, for a randomly selected integer m in [0,Fn+1) the distribution of the leading digits of the Fibonacci summands in its Zeckendorf decomposition converge to Benford’s law almost surely. Our results hold more generally; instead of looking at the distribution of leading digits of summands in Zeckendorf decompositions, one obtains simi- lar theorems concerning how often values in sets with positive density inside the Fibonacci numbers are attained in these decompositions.

**Bibtex:**

```
@article {,
AUTHOR = {Best, Andrew and Dynes, Patrick and Edelsbrunner, Xixi and Mcdonald, Brian and Miller, Steven and Tor, Kimsy and Turnage-Butterbaugh, Caroline and Weinstein, Madeleine},
TITLE = {Benford Behavior of Zeckendorf Decompositions},
JOURNAL = {Fibonacci Quarterly},
YEAR = {2014},
VOLUME = {52},
NUMBER = {5},
PAGES = {35--46},
DOI = {},
URL = {https://www.fq.math.ca/Papers1/52-5/Best-Benford.pdf},
```

**Reference Type:** Journal Article

**Subject Area(s):** Number Theory