# 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 […]

# Leading digits of powers of 2

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 […]

# 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 […]

# Technical notes on math and software

Table of contents Math diagrams Numerical computing Probability Differential equations 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 Physics Category theory definition […]