How long does it take to multiply very large integers? Using the algorithm you learned in elementary school, it takes O(n²) operations to multiply two n digit numbers. But for large enough numbers it pays to carry out multiplication very differently, using FFTs. If you’re multiplying integers with tens of thousands of decimal digits, the […]

# Author: John

## Stigler’s law of eponymy and human nature

Stigler’s law of eponymy states that no scientific discovery is named after the first person to discover it. Stephen Stigler acknowledged that he was not the first to realize this. Of course this is just an aphorism. Sometimes discoveries are indeed named after their discoverers. But the times when this isn’t the case are more […]

## Projecting Unicode to ASCII

Sometimes you need to downgrade Unicode text to more restricted ASCII text. For example, while working on my previous post, I was surprised that there didn’t appear to be an asteroid named after Poincaré. There is one, but it was listed as Poincare in my list of asteroid names. Python module I used the Python module unidecode […]

## Asteroids named after mathematicians

This evening I stumbled on the fact that John von Neumann and Fibonacci both have asteroids named after them. Then I wondered how many more famous mathematicians have asteroids named after them. As it turns out, most of them. I wrote a little script to search for the 100 greatest mathematicians (according to James Allen’s […]

## Why are dates of service on HIPAA’s Safe Harbor list?

The HIPAA Privacy Rule offers two ways to say that data has been de-identified: Safe Harbor and expert determination. This post is about the former. I help companies with the latter. Safe Harbor provision The Safe Harbor provision lists 18 categories of data that would cause a data set to not be considered de-identified unless […]

## Exponential sums in 2019

I’ve made a small change in my exponential sum page. I’ll need to give a little background before explaining the change. First of all, you can read exactly what these exponential sums are here. These plots can be periodic in two senses. The first is simply repeating the same sequence of points. The second is […]

## Goldbach’s conjecture, Lagrange’s theorem, and 2019

The previous post showed how to find all groups whose order is a product of two primes using 2019 as an example. Here are a couple more observations along the same line, illustrating the Odd Goldbach Conjecture and Lagrange’s four square theorem with 2019. Odd Goldbach Conjecture Goldbach’s Conjecture says that every even number greater […]

## Groups of order 2019

How many groups have 2019 elements? What are these groups? 2019 is a semiprime, i.e. the product of two primes, 3 and 673. For every semiprime s, there are either one or two distinct groups of order s. As explained here, if s = pq with p > q, all groups of order s are isomorphic if q is not a factor of p-1. […]

## Flattest US states

I read somewhere that, contrary to popular belief, Kansas is not the flattest state in the US. Instead, Florida is the flattest, and Kansas was several notches further down the list. How would you measure the flatness of a geographic region? The simplest approach would be to look at the range of elevation from the […]

## Naked eye view vs photos of the northern lights

My daughter Elizabeth recently photographed the northern lights (aurora borealis) in Tromsø, Norway, about 3° above the Arctic Circle. I haven’t seen the northern lights in person, and I didn’t know until she told me that the lights appear gray to the naked eye, like smoke. Sometimes the lights have a hint of color, but […]

## Check sums and error detection

The previous post looked at Crockford’s base 32 encoding, a minor variation on the way math conventionally represents base 32 numbers, with concessions for human use. By not using the letter O, for example, it avoids confusion with the digit 0. Crockford recommends the following check sum procedure, a simple error detection code: The check […]

## Base 32 and base 64 encoding

Math has a conventional way to represent numbers in bases larger than 10, and software development has a couple variations on this theme that are only incidentally mathematical. Math convention By convention, math books typically represent numbers in bases larger than 10 by using letters as new digit symbols following 9. For example, base 16 […]

## Most popular posts of 2018

Here are 10 of my most popular posts this year, arranged as pairs of posts in different areas. Astronomy Planets evenly spaced on a log scale Gravity, stars, and cows Programming Viability of unpopular programming languages Currying in various contexts Computer arithmetic The quadratic formula in low-precision arithmetic Eight-bit floating point Math Computing SVD and […]

## New prime record: 51st Mersenne prime discovered

A new prime record was announced yesterday. The largest known prime is now Written in hexadecimal the newly discovered prime is For decades the largest known prime has been a Mersenne prime because there’s an efficient test for checking whether a Mersenne number is prime. I explain the test here. There are now 51 known […]

## Multi-arm adaptively randomized clinical trials

This post will look at adaptively randomized trial designs. In particular, we want to focus on multi-arm trials, i.e. trials of more than two treatments. The aim is to drop the less effective treatments quickly so the trial can focus on determining which of the better treatments is best. We’ll briefly review our approach to […]

## Kepler and the contraction mapping theorem

The contraction mapping theorem says that if a function moves points closer together, then there must be some point the function doesn’t move. We’ll make this statement more precise and give a historically important application. Definitions and theorem A function f on a metric space X is a contraction if there exists a constant q with […]

## Trademark symbol, LaTeX, and Unicode

Earlier this year I was a coauthor on a paper about the Cap Score™ test for male fertility from Androvia Life Sciences [1]. I just noticed today that when I added the publication to my CV, it caused some garbled text to appear in the PDF. Here is the corresponding LaTeX source code. Fixing the […]

## RSA with one shared prime

The RSA encryption setup begins by finding two large prime numbers. These numbers are kept secret, but their product is made public. We discuss below just how difficult it is to recover two large primes from knowing their product. Suppose two people share one prime. That is, one person chooses primes p and q and the other chooses p […]

## Following an idea to its logical conclusion

Following an idea to its logical conclusion might be extrapolating a model beyond its valid range. Suppose you have a football field with area A. If you make two parallel sides twice as long, then the area will be 2A. If you double the length of the sides again, the area will be 4A. Following this […]

## Technological optimism

Kevin Kelly is one of the most optimistic people writing about technology, but there’s a nuance to his optimism that isn’t widely appreciated. Kelly sees technological progress as steady and inevitable, but not monotone. He has often said that new technologies create almost as many problems as they solve. Maybe it’s 10 steps forward and […]