Distribution of the First Digits of Fibonacci Numbers

In a recent paper [1] , J. L. Brown, Jr., and R. L, Duncan showed that the sequence *{ QnFn I is uniformly distributed modulo 1 (u.d. mod 1), where c/7 denotes the natural logarithm and Fn is the/? Fibonacci number. In this paper we show that some modifications of these ideas have some interesting consequences concerning the distribution of the first digits of the Fibonacci numbers. This also answers a question raised in Problem H-125. It has been noticed, and proved in the probabiiitic or measure theoretic sense, that the proportion of physical constants whose first significant digit is less than or equal to a given digit a (in base 10), is log10 (1 + a). See [2] , [4] . We will show that a wide class of sequences, including the Fibonacci numbers, have a natural density satisfying a similar distribution. Hence, roughly speaking, a large percentage of the Fibonacci numbers have a small first digit. Let h be a given positive integer. All of our numbers will now be written in base b. Let j an I be a given sequence of positive numbers. For any digit d in base b, let x^ = number of n < x such that the first digit of an is < d. More generally, if