John D. Cook

Two families of orthogonal polynomials are named after Chebyshev because he explored their properties. These are prosaically named Chebyshev polynomials of the first and second kind. I recently learned there are Chebyshev polynomials of the third and fourth kind as well. You might call these posthumous Chebyshev polynomials. They were not developed by Mr. Chebyshev, [...]The post Posthumous Chebyshev Polynomials first appeared on John D. Cook.

I recently ran across an interesting family of Pythagorean triples [1]. You can verify that a^2 + b^2 = c^2 for all n. Sparseness When written in binary, a has only two bits set, and c has only four bits set. It's not as immediately obvious, but b has only two bits that are not [...]The post Sparse binary Pythagorean triples first appeared on John D. Cook.

The reactions to the new DeepSeek-R1 AI model in recent days seem limitless. Some say it runs so much faster than existing models that we will no longer need the billions of dollars in compute hardware that big tech is preparing to buy. Is that plausible? To get an answer, we need only look [...]The post DeepSeek-R1: Do we need less compute now? first appeared on John D. Cook.

The previous post discussed complex numbers, dual numbers, and double numbers. All three systems are constructed by adding some element to the real numbers that has some special algebraic property. The complex numbers are constructed by adding an element i such that i^2 = -1. The dual numbers add an element 0 with [...]The post Matrix representations of number systems first appeared on John D. Cook.

The complex numbers are formed by adding an element i to the real numbers such that i^2 = - 1. We can create other number systems by adding other elements to the reals. One example is dual numbers. Here we add a number 0 with the property ^2 = 0. Dual numbers have [...]The post Euler's formula for dual numbers and double numbers first appeared on John D. Cook.

The simplest trick for converting from one base to another is grouping. To convert between base b and base bk, group numbers in sets of k and convert one group at a time. To convert from binary to octal, for instance, group bits in sets of three, starting from the right end, and convert each [...]The post Tricks for radix conversion by hand first appeared on John D. Cook.

The previous post started by saying that rounding has a surprising amount of detail. An example of this is double rounding: if you round a number twice, you might not get the same result as if you rounded directly to the final precision. For example, let's say we'll round numbers ending in 0, 1, 2, [...]The post Double rounding first appeared on John D. Cook.

Rounding numbers has a surprising amount of detail. It may seem trivial but, as with most things, there is a lot more to consider than is immediately obvious. I expect there have been hundreds if not thousands of pages devoted to rounding in IEEE journals. An example of the complexity of rounding is what William [...]The post A magical land where rounding equals truncation first appeared on John D. Cook.

Abramowitz and Stegun has quite a few intriguing plots. The post will focus on the follow plot, Figure 9.4, available here. We will explain what the plot is and approximately reproduce it. The plot comes from the chapter on Bessel functions, but the caption says it is a plot of the Hankel function H0(1). Why [...]The post Duplicating Hankel plot from A&S first appeared on John D. Cook.

Last week Heron's formula came up in the post An Unexpected Triangle. Given the lengths of the sides of a triangle, there is a simple expression for the area of the triangle. where the sides are a, b, and c and s is the semiperimeter, half the perimeter. Is there an analogous formula for the [...]The post Area of a quadrilateral from the lengths of its sides first appeared on John D. Cook.

The purpose of this post is to expand on the following sentence [1]: Russian letters are created by entering [Ctrl-k followed by] a corresponding Latin letter followed by an equals sign -, or, in a few places, a percent sign %. The Russian alphabet has 33 letters, so there couldn't be a Latin letter for [...]The post Entering Russian characters in Vim with digraphs first appeared on John D. Cook.

It's not simple to transliterate Russian names to English. Sometimes there is a unique mapping, or at least a standard mapping, of a particular name, but often there is not. An example that comes up frequently in mathematics is Pafnuty Lvovich Chebyshev (1821-1894). This Russian mathematician's name has been transliterated at Tchebichef, [...]The post Chebyshev and Russian transliteration first appeared on John D. Cook.

Let J(x) be the function plotted below. This is the Bessel function J1, but we drop the subscript because it's the only Bessel function we're interested in for this post. You can think of J as a sort of damped sine. We can create versions of J with different frequencies by multiplying the argument x [...]The post An unexpected triangle first appeared on John D. Cook.

Many are voicing concern that the world is running out of data and that this will be a blocker to progress toward smarter AI models. One paper in fact projects timelines for when we will run out. AI researchers are looking for ways to adapt.Nvidia has trained a specific model to generate synthetic data for [...]The post Can AI Models Reason: Is Data All You Need? first appeared on John D. Cook.

Sometimes it's useful to apply dimensional analysis where it doesn't belong, to imagine things having physical dimension when they don't. This post will look at artificially injecting dimensions into equations involving factorials and related functions. Factorials The factorial of n is defined as the product of n terms. If each of these terms had units [...]The post Dimensional analysis for gamma function values first appeared on John D. Cook.

Yesterday I wrote about how the right notation could make Newton's interpolation theorem much easier to remember, revealing it as an analog of Taylor series. This post will do something similar for the binomial theorem. Let's start with the following identity. It's not clear that this is true, or how one might generalize it. But [...]The post Falling power analog of binomial theorem first appeared on John D. Cook.

Suppose one company would like to buy another company's client list, but only if the lists don't overlap too much. Neither company wants to hand over their list to the other before a sale takes place. What can they do? A low-tech solution would be for both parties to provide their client lists to a [...]The post Why eliminate trusted third parties? first appeared on John D. Cook.

Newton's interpolation formula looks awfully complicated until you introduce the right notation. With the right notation, it looks like a Taylor series. Not only is this notation simpler and more memorable, it also suggests extensions. The notation we need comes in two parts. First, we need the forward difference operator defined by and its [...]The post Discrete Taylor series first appeared on John D. Cook.

The previous post was an AI-generated podcast that I friend made by crawling my web site. I decided to create an actual podcast for posting occasional audio files. I expect to post very sporadically. I've posted two audio files, and I have one more in mind to post some day. Maybe that'll be the end [...]The post Podcast feed first appeared on John D. Cook.

A friend just sent me an audio file of a podcast about my consulting that he created by asking an AI to crawl my web site. The podcast is a remarkably natural-sounding conversation between two synthetic hosts. The only clues that the audio is automatically generated are a couple of mispronounced acronyms. DownloadThe post Consulting Podcast first appeared on John D. Cook.

A recent article reported on the successful factoring of a 512-bit RSA key. The process took $8 worth of rented computing power. What does that say about the security of RSA encryption? The following Python function estimates the computation steps required to factor a number b bits long using the best known algorithm. We're going [...]The post RSA security in light of news first appeared on John D. Cook.

We're awaiting the release of OpenAI's o3 model later this month. Its performance is impressive on very hard benchmarks like SWE-bench Verified, Frontier Math and the ARC AGI benchmark (discussed previously in this blog). And yet at the same time some behaviors of the frontier AI models are very concerning. Their performance on assorted math [...]The post Can AI models reason like a human? first appeared on John D. Cook.

One difficulty when working at a command line is navigating between directories, particularly between locations with long paths. There are several ways to mitigate this. One of the simplest is using cd - to return to the previous directory. Another is to use pushd and popd. Still another is to set the CDPATH variable. qcd [...]The post Quick change directory first appeared on John D. Cook.

The security of RSA encryption depends on the difficulty of factoring the product of two large primes. If you can factor large numbers efficiently, you can break RSA. But if can break RSA, can you factor large numbers? Sorta. It's conceivable that there is a way to break RSA encryption without having to recover the [...]The post Converse of RSA first appeared on John D. Cook.

Steganography attempts to prevent messages from being read by unintended recipients by hiding the messages rather than (or in addition to) encrypting them. Steganography is used when you not only want to keep your communication private, you want to hide the fact that you've communicated at all. Fun fact: The words steganography and stegosaurus are [...]The post Unicode Steganography first appeared on John D. Cook.

A blog carnival is a way to discover new blogs. Writers on a given topic, such as math, take turns hosting the carnival, featuring recent posts from various writers. Blog carnivals were once much more common, but most have faded away. The Carnival of Mathematics, however, is one of the oldest carnivals and still active, [...]The post Carnival of Mathematics 235 first appeared on John D. Cook.

The previous post showed that there are 10 Abelian groups that have 2025 elements. Implicitly that means there are 10 Abelian groups up to isomorphism, i.e. groups that are not in some sense the same" even if they look different. Sometimes it is clear what we mean by the same" and there's no need to [...]The post Up to isomorphism first appeared on John D. Cook.

Every finite Abelian group can be written as the direct sum of cyclic groups of prime power order. To find the number of Abelian groups of order 2025 we have to find the number of ways to partition the factors of 2025 into prime powers. Now 2025 = 34 * 52. We can partition 34 [...]The post Abelian groups of order 2025 first appeared on John D. Cook.

Here's a visualization of how the day of the week for New Year's Day changes. The green diamonds represent leap years and the blue squares represent ordinary years. The day of the week for New Year's Day advances one day after each ordinary year and two days after each leap year, hence the diagonal stripes [...]The post Cycle of New Year's Days first appeared on John D. Cook.

RSA public key cryptography begins by finding a couple large primes. You essentially do this by testing random numbers until you find primes, but not quite. Filippo Valsorda just posted a good article on this. Suppose you're looking for a 1024-bit prime number. You generate random 1024-bit numbers and then test until you find one [...]The post Details of generating primes for cryptography first appeared on John D. Cook.

The previous post looked into the common definition of Unix time as the number of seconds since January 1, 1970 GMT" and why it's not exactly true. It was true for a couple years before we started inserting leap seconds. Strictly speaking, Unix time is the number of non-leap seconds since January 1, 1970. This [...]The post What exactly is a second? first appeared on John D. Cook.

The time it takes earth to orbit the sun is not a simple multiple of the time it takes earth to rotate on its axis. The ratio isn't even constant. The time it takes earth to circle the sun wobbles a bit, and the rotation of the earth is slowing down slightly. The ratio is [...]The post Unix Time and a Modest Proposal first appeared on John D. Cook.

I looked at Hacker News to see which posts on this site were most popular. I didn't look at my server logs, but generally the posts that get the most traffic are posts that someone submits to Hacker News. Older posts popular this year Two posts written earlier got a lot of traffic this year, [...]The post Most popular posts of 2024 first appeared on John D. Cook.

Stirling's asymptotic series for the gamma function is Now suppose you'd like to find an asymptotic series for the function 1/(z). Since the series for has the form f(z) times an infinite sum, it would make sense to look for a series for 1/ of the form 1/f(z) times an infinite sum. The hard [...]The post Series for the reciprocal of the gamma function first appeared on John D. Cook.

My nephew recently told me about being on a camping trip and seeing a long line of lights in the sky. The lights turned out to be Starlink satellites. It's fairly common for people report seeing lines of these satellites. Why would the satellites be in a line? Wouldn't it be much more efficient to [...]The post Starlink configurations first appeared on John D. Cook.

I've been playing around with Grok today, logging into some of my X accounts and trying out the prompt Draw an image of me based on my posts." [1] In most cases Grok returned a graphic, but sometimes it would respond with a text description. In the latter case asking for a photorealistic image made [...]The post Putting a face on a faceless account first appeared on John D. Cook.

OpenAI has just released its full o1 model-a new kind of model that is more capable of multi-step reasoning than previous models. Anthropic, Google and others are no doubt working on similar products. At the same time, it's hotly debated in many quarters whether AI models actually reason" in a way similar to humans. Emily [...]The post Can AI models reason: Just a stochastic parrot? first appeared on John D. Cook.

A couple days ago I analyzed the observation that repeatedly pressing the cosine key on a calculator leads to a fixed point. After about 90 iterations the number no longer changes. This post will analyze the same phenomenon a different way. Interval arithmetic Interval arithmetic is a way to get exact results of a sort [...]The post Interval arithmetic and fixed points first appeared on John D. Cook.

The previous post presented an approximation for -1 x 1 and said that it was related to a probability function. This post will make the connect explicit. LetX be a normally distributed random variable with mean and variance ^2. Then the CDF of X is So if = 0 and ^2 [...]The post Normal probability approximation first appeared on John D. Cook.

I recently ran across the fact that is a remarkably good approximation for -1 x 1. Since the integral above defines the error function erf(x), modulo a constant, this says we have a good approximation for the error function again provided -1 x 1. The error function is closely related to [...]The post Simple error function approximation first appeared on John D. Cook.

No matter what number you start with, if you press the cos key on a calculator repeatedly, the numbers eventually quit changing. This fact has been rediscovered by countless children playing with calculators. If you start with 0, which is likely the default when you turn on a calculator, you'll hit the final value after [...]The post Pressing the cosine key over and over first appeared on John D. Cook.

The previous post explained why the Gregorian calendar is the way it is, and that it consists of a whole number of weeks. It follows that the Gregorian calendar repeats itself every 400 years. For example, the calendar for 2025 will be exactly the same as the calendar for 1625 and 2425. There are only [...]The post Perpetual Calendars first appeared on John D. Cook.

The time it takes for the earth to orbit the sun is not an integer multiple of the time it takes for the earth to rotate on its axis, nor is it a rational number with a small denominator. Why should it be? Much of the complexity of our calendar can be explained by rational [...]The post Gregorian Calendar and Number Theory first appeared on John D. Cook.

Here's something I posted on X a couple days ago: There's no direct connection between AI and cryptocurrency, but they have a similar vibe. They both leave you wondering whether the emperor is sumptuously clothed, naked, or a mix of both. Maybe he's wearing a hospital gown with gold threads. In case you're unfamiliar with [...]The post Golden hospital gowns first appeared on John D. Cook.

Yesterday I needed to write a regular expression as part of a client report. Later I was curious whether an LLM could have generated an equivalent expression. When I started writing the prompt, I realized it wasn't trivial to tell the LLM what I wanted. I needed some way to describe the pattern that the [...]The post LLMs and regular expressions first appeared on John D. Cook.

A logarithmic scale is very useful when you need to plot data over an extremely wide range. However, sometimes even a logarithmic scale may not reduce the visual range enough. I recently saw a timeline-like graph that was coiled into a spiral, packing more information into a limited visual window [1]. I got to thinking [...]The post Coiled logarithmic graph first appeared on John D. Cook.

How many ways can you select six points in the plane so that every subset of three points forms the vertices of an isosceles triangle? This is a question asked by Erds and recently resolved. One solution is to choose the five vertices of a regular pentagon and the center. It's easy to verify that [...]The post Solution to a problem of Erds first appeared on John D. Cook.

This post was motivated by an exercise in [1] that says Prove that for the hyperbolic functions ... formulas hold similar to those in Section 2.3 with all the minuses replaced by pluses. My first thought was that this sounds like Osborn's rule, a heuristic for translating between (circular) trig identities and hyperbolic trig identities. [...]The post Multiple angles and Osborn's rule first appeared on John D. Cook.

The vis-viva equation greatly simplifies some calculations in orbital mechanics. It is reminiscent of how conservation of energy can sometimes trivialize what appears to be a complicated problem. In fact, the vis-viva equation is derived from conservation of energy, but the derivation is not trivial. Which is good: the effort required in the derivation implies [...]The post The vis-viva equation first appeared on John D. Cook.

The most efficient maneuver for transferring from one circular orbit to another circular orbit of roughly the same size is the Hohmann transfer orbit. It requires two burns: one to leave the initial circular orbit into an elliptical orbit, and another to leave the elliptic orbit for the new circular orbit. If the new orbit [...]The post Efficiently transferring to a much higher orbit first appeared on John D. Cook.