Benford’s law

This page is an index to articles on the site about Benford’s law. Benford’s law and probability distributions Pareto distribution Weibull distribution Cauchy distribution Benford’s law and number theory Leading digits of powers of two Gelfand’s question Leading digits of factorials Benford’s law and computer science Collatz 3n+1 conjecture Benford’s law and statistics Benford’s law […]

The 3n+1 problem and Benford’s law

This is the third, and last, of a series of posts on Benford’s law, this time looking at a famous open problem in computer science, the 3n + 1 problem, also known as the Collatz conjecture. Start with a positive integer n. Compute 3n + 1 and divide by 2 repeatedly until you get an odd […]

Cauchy, Benford, and a problem with NHST

Introduction Samples from a Cauchy distribution nearly follow Benford’s law. I’ll demonstrate this below. The more data you see, the more confident you should be of this. But with a typical statistical approach, crudely applied NHST (null hypothesis significance testing), the more data you see, the less convinced you are. This post assumes you’ve read the […]

Benford’s law and SciPy

Imagine you picked up a dictionary and found that the pages with A’s were dirty and the Z’s were clean. In between there was a gradual transition with the pages becoming cleaner as you progressed through the alphabet. You might conclude that people have been looking up a lot of words that begin with letters […]

Collatz conjecture skepticism

The Collatz conjecture asks whether the following procedure always terminates at 1. Take any positive integer n. If it’s odd, multiply it by 3 and add 1. Otherwise, divide it by 2. For obvious reasons the Collatz conjecture is also known as the 3n + 1 conjecture. It has been computationally verified that the Collatz […]

Progress on the Collatz conjecture

The Collatz conjecture is for computer science what until recently Fermat’s last theorem was for mathematics: a famous unsolved problem that is very simple to state. The Collatz conjecture, also known as the 3n+1 problem, asks whether the following function terminates for all positive integer arguments n. def collatz(n): if n == 1: return 1 […]

Physical constants in Python

You can find a large collection of physical constants in scipy.constants. The most frequently used constants are available directly, and hundreds more are in a dictionary physical_constants. The fine structure constant α is defined as a function of other physical constants: The following code shows that the fine structure constant and the other constants that […]