Benford Behavior of Generalized Zeckendorf Decompositions

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 220)


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 integer can be written uniquely as a sum of nonadjacent elements of this sequence; this is called the Zeckendorf decomposition, and similar unique decompositions exist for sequences arising from recurrence relations of the form \(G_{n+1}=c_1G_n+\cdots +c_LG_{n+1-L}\) with \(c_i\) positive and some other restrictions. Additionally, a set \(S \subset \mathbb {Z}\) is said to satisfy Benford’s law base 10 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 \(n\rightarrow \infty \) for a randomly selected integer m in \([0, G_{n+1})\) the distribution of the leading digits of the summands in its generalized Zeckendorf decomposition converges to Benford’s law almost surely. Our results hold more generally: One obtains similar theorems to those regarding the distribution of leading digits when considering how often values in sets with density are attained in the summands in the decompositions.


Zeckendorf decompositions Fibonacci numbers Positive linear recurrence relations Benford’s law 

MSC 2010:

11B39 11B05 60F05 (primary)11K06 65Q30 62E20 (secondary) 



This research was conducted as part of the 2014 SMALL REU program at Williams College and was supported by NSF grants DMS1265673, DMS1561945, DMS1347804, Williams College, and the Clare Boothe Luce Program of the Henry Luce Foundation. It is a pleasure to thank them for their support, and the participants at SMALL and at the 16th International Conference on Fibonacci Numbers and their Applications for helpful discussions.


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA
  2. 2.Department of Mathematical SciencesClemson UniversityClemsonUSA
  3. 3.Department of Mathematics and StatisticsWilliams CollegeWilliamstownUSA
  4. 4.Department of MathematicsUniversity of ChicagoChicagoUSA
  5. 5.Department of MathematicsPierre and Marie Curie University–Paris 6ParisFrance
  6. 6.Department of MathematicsDuke UniversityDurhamUSA
  7. 7.Department of MathematicsUniversity of California BerkeleyBerkeleyUSA

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