Some probabilistic aspects of the terminal digits of Fibonacci numbers. (English) Zbl 0833.11034

Filipponi, Piero; Menicocci, Renato

Fibonacci Q. 33, No. 4, 325-331 (1995).

By terminal digits both the initial digit and the final digit are meant. The authors prove several probabilistic relationships between the initial and the final digits of the Fibonacci numbers \(F_k\), e.g., they give the joint probability that the initial and the final digit take fixed values. In this connection, even though not explicit stated, the indices of the Fibonacci numbers are assumed to be uniformly distributed in a large interval. The principal mean of reasoning is the statistical independence between the initial and the final digit which arises from the periodicity 60 of the sequence \(F_k \bmod 10\) and from the fact that every subsequence \(F_h, F_{60+ h}, F_{120+ h}, \dots\) obeys Benford’s law. Finally, the authors mention some related results on Lucas numbers.

Reviewer: P.Schatte (Freiberg)

MSC:

11K31	Special sequences
11B39	Fibonacci and Lucas numbers and polynomials and generalizations

Keywords:

initial digits; final digits; Lucas numbers; terminal digits; Fibonacci numbers; joint probability; Benford’s law