Description
Leading Digits of numbers in natural data sets are governed by Benford’s Law; which states that the leading digit has a higher probability of being smaller number, with 1 having the highest probability. Gelfand’s Question refers to a series of questions regarding the leading digits of powers of integers in base 10. The original question credited to Israel Gelfand asks if the leading digit of 2[superscript n] will ever equal 9. The remainder of the questions are concerned with what sequences of leading digits appear when looking at {2[superscript n], 3 [superscript n], ... , 9[superscript n]} simultaneously. These questions have all been answered for base 10; we will extend these results to an arbitrary base B. First we will examine a brief history of the question and summarize the results found in previous work. Next we show that for an arbitrary base B and positive integer ℓ < B, that most sets {ℓ[superscript n] : n ∈ N} satisfy Benford’s Law. For those ℓ[superscript n] that do not, we state the distributions of the leading digits explicitly. Finally, we subdivide the positive integers into two sets, prime powers and non-prime powers, and examine the leading digits of pairs and triples of integer powers. We are able to show that some of these pairs and triples are not dense over the set of possible digits. Using these results we are able to determine several sequences that cannot appear in the leading digits of the set {2ⁿ, 3ⁿ,...,(B - 1)ⁿ}.
