The first digit of a power of 2 is a 1 more often than any other digit. Powers of 2 begin with 1 about 30% of the time. This is because powers of 2 follow Benford’s law. We’ll prove this below. When is the first digit of 2n equal to k? When 2n is between […]
Table of contents Math diagrams Numerical computing Probability Differential equations Category theory Cryptography Logic Python Probability approximations Regular expressions C++ Special functions Typesetting: TeX, HTML, Unicode Emacs R Miscellaneous math My notes on cryptography have their own page. Math diagrams Diagram of probability distribution relationships Modes of convergence Topological properties diagram Category Relationships in Mathematical […]
My previous post looked at a problem that requires repeatedly finding the first digit of kn where k is a single digit but n may be on the order of millions or billions. The most direct approach would be to first compute kn as a very large integer, then find it’s first digit. That approach […]
Gelfands’s question asks whether there is a positive integer n such that the first digits of jn base 10 are all the same for j = 2, 3, 4, …, 9. (Thanks to @republicofmath for pointing out this problem.) This post will explore Gelfand’s question via probability. The MathWorld article on Gelfand’s question says that […]
Suppose you take factorials of a lot of numbers and look at the leading digit of each result. You could argue that there’s no apparent reason that any digit would be more common than any other, so you’d expect each of the digits 1 through 9 would come up 1/9 of the time. Sounds plausible, […]
