• Corpus ID: 51774496

# Benford Behavior of Zeckendorf Decompositions

@article{Best2014BenfordBO,
title={Benford Behavior of Zeckendorf Decompositions},
author={Andr{\'e} Best and Patrick J. Dynes and Xixi Edelsbrunner and Brian McDonald and Steven J. Miller and Kimsy Tor and Caroline L. Turnage-Butterbaugh and Madeleine Weinstein},
journal={arXiv: Number Theory},
year={2014}
}
• Published 1 September 2014
• Mathematics
• arXiv: Number Theory
A beautiful theorem of Zeckendorf states that every integer can be written uniquely as the sum of non-consecutive Fibonacci numbers $\{ F_i \}_{i = 1}^{\infty}$. A set $S \subset \mathbb{Z}$ is said to satisfy Benford's law if the density of the elements in $S$ with leading digit $d$ is $\log_{10}{(1+\frac{1}{d})}$; in other words, smaller leading digits are more likely to occur. We prove that, as $n\to\infty$, for a randomly selected integer $m$ in $[0, F_{n+1})$ the distribution of the…
6 Citations

## Figures from this paper

Benford Behavior of Generalized Zeckendorf Decompositions
• Mathematics
• 2015
We prove connections between Zeckendorf decompositions and Benford’s law. Recall that if we define the Fibonacci numbers by $$F_1 = 1, F_2 = 2$$, and $$F_{n+1} = F_n + F_{n-1}$$, every positive
On Generalized Zeckendorf Decompositions and Generalized Golden Strings
Zeckendorf proved that every positive integer has a unique representation as a sum of non-consecutive Fibonacci numbers. A natural generalization of this theorem is to look at the sequence defined as
The Fibonacci Sequence and Schreier-Zeckendorf Sets
• H. Chu
• Mathematics
J. Integer Seq.
• 2019
The Fibonacci sequence is discovered by counting the number of subsets of $\{1,2,\ldots, n\}$ such that two consecutive elements in increasing order always differ by an odd number.
Fibonacci Sequence and Linear Recurrence Relations behind Schreier-Zeckendorf Sets
Schreier used the now-called Schreier sets to construct a counter-example to a question by Banach and Sak. A set is weak-Schreier if $\min S \ge |S|$, strong-Schreier if $\min S>|S|$, and maximal if
Difference in the Number of Summands in the Zeckendorf Partitions of Consecutive Integers.
Zeckendorf proved that every positive integer has a unique partition as a sum of non-consecutive Fibonacci numbers. We study the difference between the number of summands in the partition of two
On Zeckendorf Related Partitions Using the Lucas Sequence
• Mathematics
• 2020
Zeckendorf proved that every positive integer has a unique partition as a sum of non-consecutive Fibonacci numbers. Similarly, every natural number can be partitioned into a sum of non-consecutive

## References

SHOWING 1-10 OF 26 REFERENCES
Gaussian Behavior of the Number of Summands in Zeckendorf Decompositions in Small Intervals
• Mathematics
• 2014
Zeckendorf's theorem states that every positive integer can be written uniquely as a sum of non-consecutive Fibonacci numbers ${F_n}$, with initial terms $F_1 = 1, F_2 = 2$. We consider the
Gaussian Behavior in Generalized Zeckendorf Decompositions
• Mathematics
• 2014
A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of nonconsecutive Fibonacci numbers $$\{F_{n}\}_{n=1}^{\infty }$$; Lekkerkerker proved that the average
On the number of summands in Zeckendorf decompositions
• Mathematics
• 2010
Zeckendorf proved that every positive integer has a unique representation as a sum of non-consecutive Fibonacci numbers. Once this has been shown, it's natural to ask how many summands are needed.
A Generalization of Fibonacci Far-Difference Representations and Gaussian Behavior
• Mathematics
• 2013
A natural generalization of base B expansions is Zeckendorf's Theorem: every integer can be uniquely written as a sum of non-consecutive Fibonacci numbers $\{F_n\}$, with $F_{n+1} = F_n + F_{n-1}$
From Fibonacci numbers to central limit type theorems
• Mathematics
J. Comb. Theory, Ser. A
• 2012
Generalizing Zeckendorf's Theorem: The Kentucky Sequence
• Mathematics
• 2014
By Zeckendorf's theorem, an equivalent definition of the Fibonacci sequence (appropriately normalized) is that it is the unique sequence of increasing integers such that every positive number can be
MODULO ONE UNIFORM DISTRIBUTION OF THE SEQUENCE OF LOGARITHMS OF CERTAIN RECURSIVE SEQUENCES
Let {x.}°° be a sequence of real numbers with corresponding fractional parts {/3.}°°, where 0. = x. [x.] and the bracket denotes the greatest integer function. For each n > 1, we define the function
Representation of Natural Numbers as Sums of Generalised Fibonacci Numbers
The well-known observation of Zeckendorf is that every positive integer N has a unique representation N = u. +u. + • • • +u . , where (1) ij ^ 1 and i i ^ 2 for i 4= v < d , and ju {• is the
Differences of Multiple Fibonacci Numbers
Abstract We show that every integer can be written uniquely as a sum of Fibonacci numbers and their additive inverses, such that every two terms of the same sign differ in index by at least 4 and