John D. Cook

If three circles are all tangent to each other, you can find two more circles that are tangent to all three, and the equation for finding these new circles is remarkably elegant. This is Descartes' theorem. Two tangent circles To illustrate Descartes' theorem, we first need three mutually tangent circles. Drawing two mutually tangent circles [...]The post Finding tangent circles first appeared on John D. Cook.

A Pythagorean triangle is a right triangle with integer sides. Aprimitive Pythagorean triangle is one in which the sides have no factor in common. For example a triangle with sides (30, 40, 50) is a Pythagorean triangle but not a primitive Pythagorean triangle. It is possible for two primitive Pythagorean triangles to have the same [...]The post Primitive Pythagorean triangles with the same area first appeared on John D. Cook.

The previous post began by saying Bitcoin's Wallet Import Format (WIF) is essentially Base58 encoding with a checksum." More specifically, WIF uses Base58Check encoding. This post will fill in the missing details and show how to carry out computing Base58Check in Python. There are multiple ways to stub your toe doing this because it involves [...]The post Base58Check encoding in Python first appeared on John D. Cook.

Bitcoin's Wallet Import Format (WIF) is essentially Base58 encoding with a checksum. (See the next post for details.) It is meant to a human-friendly way to display cryptographic private keys. It's not that friendly, but it could be worse. The checksum (the first four bytes of SHA256 applied twice) is appended before the conversion to [...]The post Most legible font for WIF first appeared on John D. Cook.

In the process of writing the previous post, I ran across the Landau-Ramanujan theorem. It turned out to not be what I needed, but it's an interesting result. What portion of the numbers less thanN can be written as the sum of the squares of two non-negative integers? Edmund Landau discovered in 1908, and Srinivasa [...]The post Counting sums of squares first appeared on John D. Cook.

What are the solutions to the equation uxx + uyy = u on the unit square with the requirement that u(x, y) = 0 on the boundary? It's easy to see that the functions u(x, y) = sin(mx) sin(ny) are solutions with = (m^2 + n^2)^2 for non-negative integersm andn. It's not so obvious [...]The post Eigenvalues of the Laplacian on a square first appeared on John D. Cook.

The vibration of a thin membrane is often modeled by the PDE u + u = 0 whereu is the height of the membrane and is the Laplacian. Solutions only exist for certain values of , the eigenvalues of . You could think of u as giving the height of a vibrating drum head [...]The post The sound of drums that tile the plane first appeared on John D. Cook.

The sum of the firstn odd numbers isn^2. For example, 1 + 3 + 5 + 7 + 9 = 25 = 5^2 Here's another way to say the same thing. Start with the list of positive integers, remove every second integer from the list, and take the cumulative sum. The resulting list is the [...]The post Moessner's Magic first appeared on John D. Cook.

The elliptic curves Curve25519 and Ed25519 are both commonly used in applications. For example, Curve25519 is used in Proton Mail and Ed25519 is used in SSH. The two curves are related, as the numerical parts in their names suggest. The two curves are equivalent in some sense that we will describe below. An algebraic geometer [...]The post Equivalence between commonly used elliptic curves first appeared on John D. Cook.

Digital signatures often use elliptic curves. For example, Bitcoin and Ethereum use the elliptic curve secp256k1 [1]. This post will discuss the elliptic curve Ed25519 [2] using in Monero and in many other applications. Ed25519 has the equation y^2 - x^2 = 1 +dx^2 y^2 over the finite field Fq where q = 2255 - [...]The post Monero's elliptic curve first appeared on John D. Cook.

Bitcoin addresses include a checksum. The Base58Check algorithm uses the first four bytes of a hash function as a checksum before applying Base58 encoding. Ethereum addresses did not include a checksum, but it became apparent later that a checksum was needed. How can you retroactively fit a checksum into an address without breaking backward compatibility? [...]The post Retrofitting error detection first appeared on John D. Cook.

When I wrote about hyperinflation the other day I included an image of a 100 trillion dollar note from Zimbabwe. This is almost a cliche: everyone using this image when talking about hyperinflation. But Zimbabwe's 1014 dollar note was not the largest denomination ever used. In 1946, Hungary circulated at 100 quintillion (1020) peng note. [...]The post A bank note with 21 implicit zeros first appeared on John D. Cook.

A few days ago I wrote about now hyperinflation changes everything. I'd like to follow up with another example of hyperinflation breaking implicit assumptions. Something I was reading this week gave the approximation for present value v = 1/(1 + i) 1 - i + i^2 This implicitly assumes that the interest rate i [...]The post Taylor Series and the Argentine Peso first appeared on John D. Cook.

There are several numbers that are analogous to binomial coefficients and, at least in Donald Knuth's notation, are written in a style analogous to binomial coefficients. And just as binomial coefficients can be arranged into Pascal's triangle, these numbers can be arranged into similar triangles. In Pascal's triangle, each entry is the sum of the [...]The post Analogs of binomial coefficients first appeared on John D. Cook.

Actuarial mathematics makes frequent use of a notation that as far as I know isn't used anywhere else, and that is a bracket enclosing the northeast corner of a symbol. This is used as subscript of another symbol, such as ana or ans, and there may be other decorations such more superscripts or subscripts. For [...]The post Annuity and factorial notation first appeared on John D. Cook.

Base58 encoding and Base85 encoding are used to represent binary data in a human-friendly way. Base58 uses a smaller character set and so is more conservative. Base85 uses a larger character set and so is more efficient. There is a gotcha in that base" means something different in Base58 compared to Base85. More on that [...]The post Base58 versus Base85 encoding first appeared on John D. Cook.

My two latest blog posts have been about compound interest. I gave examples of interest rates of 6% up to 18% per year. Hyperinflation, with rates of inflation in excess of 50% per month, changes everything. Although many economists accept 50% per month as the threshold of hyperinflation, the world has seen worse. Zimbabwe, for [...]The post Hyperinflation changes everything first appeared on John D. Cook.

The previous post looked at the difference between continuously compounded interest and interest compounded a large discrete number of times. This difference was calculated using the following Python function. def f(P, n, r) : return P*(exp(r) - (1 + r/n)**n) where the function arguments are principle, number of compoundings, and interest rate. When I was [...]The post Numerical problem with an interest calculation first appeared on John D. Cook.

When I was a child, I heard an advertisement for a bank that compounded the interest on your savings account with every heartbeat. I thought that was an odd thing to say and wondered what it meant. If you have a rapid heart rate, does your money compound more frequently? I figured there was probably [...]The post Interest compounding with every heartbeat first appeared on John D. Cook.

Yesterday's post looked at the distribution of powers ofx mod 1. For almost allx > 1 the distribution is uniform in the limit. But there are exceptions, and the post raised the question of whether 3 + 2 is an exception. A plot made itlook like 3 + 2 is an exception, but that turned [...]The post Powers of 3 + 2 first appeared on John D. Cook.

For a positive real numberx, the expressionx mod 1 means the fractional part ofx: x mod 1 =x - x . This post will look at the distributions ofnx mod 1 and xn mod 1 for n = 1, 2, 3, .... The distribution of nx mod 1 is easy to classify. Ifx is [...]The post Multiples and powers mod 1 first appeared on John D. Cook.

AI models hallucinate, and always will [1]. In some sense this is nothing new: everything has an error rate. But what is new is the nature of AI errors. The output of an AI is plausible by construction [2], and so errors can look reasonable even when they're wrong. For example, if you ask for [...]The post Evaluating and lowering AI hallucination cost first appeared on John D. Cook.

Newton's method is a simple and efficient method for finding the roots of equations, provided you start close enough to the root. But determining the set of starting points that converge to a given root, or converge at all, can be very complicated. In one case it is easy to completely classify where points converge. [...]The post A Continental Divide for Newton's Method first appeared on John D. Cook.

Here's another little chess puzzle by Martin Gardner, taken from this paper. The task is to place all the pieces-king, queen, two bishops, two knights, and two rooks-on a 6 * 5 chessboard, with the requirement that the two bishops be on opposite colored squares and no piece is attacking another. Here is a solution.The post All pieces on a small chessboard first appeared on John D. Cook.

Our team started working within Nvidia in early 2009 at the beginning of the ORNL Titan project. Our Nvidia contacts dealt with applications, libraries, programming environment and performance optimization. First impressions were that their technical stance on issues was very reasonable. One obscure example: in a C++ CUDA kernel were you allowed to use enums," [...]The post Experiences with Nvidia first appeared on John D. Cook.

The previous post mentioned Legendre polynomials. This post will give a brief introduction to these polynomials and a couple hints of how they are used in applications. One way to define the Legendre polynomials is as follows. P0(x) = 1 Pk are orthogonal on [-1, 1]. Pk(1) = 1 for all k >= 0. The [...]The post Legendre polynomials first appeared on John D. Cook.

To first approximation, a satellite orbiting the earth moves in an elliptical orbit. That's what would get from solving the two-body problem: two point masses orbiting their common center of mass, subject to no forces other than their gravitational attraction to each other. But the earth is not a point mass. Neither is a satellite, [...]The post The biggest perturbation of satellite orbits first appeared on John D. Cook.

Suppose you have a radio transmitter T and a receiver R with a clear line of sight between them. Some portion the signal received atR will come straight fromT. But some portion will have bounced off some obstacle, such as the ground. The reflected radio waves will take a longer path than the waves that [...]The post Transmission Obstacles and Ellipsoids first appeared on John D. Cook.

The exponential sum of the day page on my site draws an image every day by plugging the month, day, and year into a formula. Details here. Today's image looks almost solid blue in the middle. The default plotting line width works well for most days. For example, see what the sum of the day [...]The post Zooming in on a fractalish plot first appeared on John D. Cook.

All integers are real numbers, but most computer representations of integers do not equal computer representations of real numbers. To make the statement above precise, we have to be more specific about what we mean by computer integers and floating point numbers. I'll use int32 and int64 to refer to 32-bit and 64-bit signed integers. [...]The post Most ints are not floats first appeared on John D. Cook.

Years ago I wrote about a fast way to test whether an integer n is a square. The algorithm rules out a few numbers that cannot be squares based on their last (hexadecimal) digit. If the the integer passes through this initial screening, the algorithm takes the square root of n as a floating point [...]The post Test whether a large integer is a square first appeared on John D. Cook.

What does an eighteenth century French mathematician have to do with the Ethereum cryptocurrency? A pseudorandom number generator based on Legendre symbols, known as Legendre PRF, has been proposed as part of a zero knowledge proof mechanism to demonstrate proof of custody in Ethereum 2.0. I'll explain what each of these terms mean and include [...]The post Legendre and Ethereum first appeared on John D. Cook.

The previous post looked at a function formed by patching together the function f(x) = log(1 +x) for positivex and f(x) =x for negativex. The functions have a mediocre fit, which may be adequate for some applications, such as plotting data points, but not for others, such as visual design. Here's a plot. There's something [...]The post Patching functions together first appeared on John D. Cook.

I saw a post online this morning that recommended the transformation I could see how this could be very handy. Often you want something like a logarithmic scale, not for the exact properties of the logarithm but because it brings big numbers closer in. And for big values ofx there's little difference between log(x) and [...]The post Log-ish first appeared on John D. Cook.

Suppose you have a system withn possible states. The entropy of the system is maximized when all states are equally likely to occur. The entropy is minimized when one outcome is certain to occur. You can say more. Starting from any set of probabilities, as you move in the direction of more uniformity, you increase [...]The post Uniformity increases entropy first appeared on John D. Cook.

The sinc function sinc(x) = sin(x) /x comes up continually in signal processing. Ifx is moderately small, the approximation sinc(x) (2 + cos(x))/3 is remarkably good, with an error on the order of x4/180. This could be useful in situations where you're working with the sinc function and thex in the denominator is awkward [...]The post Sinc function approximation first appeared on John D. Cook.

Four years ago I wrote a blog post about simple solutions to client problems. The post opens by recounting a conversation with a friend that ended with my friend saying So, basically you're recommending division." That conversation came back to me recently when I had a similar conversation during a deposition. I had to explain [...]The post Arithmetic for fun and profit first appeared on John D. Cook.

A couple days ago I wrote about the the problem that Bitcoin requires to be solved as proof-of-work. In a nutshell, you need to tweak a block of transactions until the SHA256 double hash of its header is below a target value [1]. Not all cryptocurrencies use proof of work, but those that do mostly [...]The post Why use hash puzzles for proof-of-work? first appeared on John D. Cook.

Suppose you have a list of positive data points y1, y2, ..., yn and you wanted to find a value that minimizes the squared distances to each of the ys. Then the solution is to take to be the mean of the ys: This result is well known [1]. The following variation is [...]The post Minimize squared relative error first appeared on John D. Cook.

In order to prevent fraud, anyone wanting to add a block to the Bitcoin blockchain must prove that they've put in a certain amount of computational work. This post will focus on what problem must be solved in order produce proof of work. You'll see the proof of work function described as finding strings whose [...]The post What is the Bitcoin proof-of-work problem? first appeared on John D. Cook.

This post looks at whether you should delete names or replace names when deidentifying personal data. With structured data, generating synthetic names does not increase or decrease privacy. But with unstructured data, replacing real names with randomly generated names increases privacy protection. Structured data If you want to deidentify structured data (i.e. data separated into [...]The post Deleting vs Replacing Names first appeared on John D. Cook.

I went down a rabbit hole this week using two symbols in LaTeX. The first was the Russian letter Sha (, U+0248), and the second was the currency symbol for Bitcoin (, U+20BF). Sha I thought there would be a LaTeX package that would include as a symbol rather than as a Russian letter, [...]The post Typesetting Sha and Bitcoin first appeared on John D. Cook.

Years ago I wrote a post Golden powers are nearly integers. The post was picked up by Hacker News and got a lot of traffic. The post was commenting on a line from Terry Tao: The powers , 2, 3, ... of the golden ratio lie unexpectedly close to integers: for instance, 11= 199.005... is [...]The post Golden powers revisited first appeared on John D. Cook.

Suppose you need to write software to compute the sine function. You were told in a calculus class that this is done using Taylor series-it's not, but that's another story-and so you start writing code to implement Taylor series. How many terms do you need? Uh, ..., 20? Let's go with that. from math import [...]The post Naively computing sine first appeared on John D. Cook.

The Euler-Mascheroni constant is defined as the limit So an obvious way to try to calculate would be to evaluate the right-hand side above for large n. This turns out to not be a very good approach. Convergence is slow and rounding error accumulates. A much better approach is to compute It's not obvious [...]The post Computing the Euler-Mascheroni Constant first appeared on John D. Cook.

It is possible to express every positive integer as a sum of powers of the golden ratio using each power at most once. This means it is possible to create a binary-like number system using as the base with coefficients of 0 and 1 in front of each power of . This system [...]The post Golden ratio base numbers first appeared on John D. Cook.

Sometimes a paper comes out that has the seeds of a great idea that could lead to a whole new line of pioneering research. But, instead, nothing much happens, except imitative works that do not push the core idea forward at all. For example the McCulloch Pitts paper from 1943 showed how neural networks could [...]The post Scientific papers: innovation ... or imitation? first appeared on John D. Cook.

I just stumbled across the binomial number system in Exercise 5.38 of Concrete Mathematics. The exercise asks the reader to show that every non-negative integer n can be written as and that the representation is unique if you require 0 a <b <c. The book calls this the binomial number system. I skimmed a paper [...]The post Binomial number system first appeared on John D. Cook.

Casting out nines is a well-known way of finding the remainder when a number is divided by 9. You add all the digits of a number n. And if that number is bigger than 9, add all the digits of that number. You keep this up until you get a number less than 9. This [...]The post Additive and multiplicative persistence first appeared on John D. Cook.

Logarithms give a way to describe large numbers compactly. But sometimes even the logarithm is a huge number, and it would be convenient to take the log again. And maybe again. For example, consider googolplex = 10googol where googol = 10100. (The name googol is older than Google," and in fact the former inspired the [...]The post Iterated logarithm first appeared on John D. Cook.