John D. Cook

The high-level explanation of a blockchain says that each block contains a cryptographic hash of the previous block. That's how the blocks are chained together. That's not exactly true, and it leaves out a lot of detail. This post will look in full detail at how Bitcoin blocks are chained together by inspecting the bits [...]The post How blocks are chained in a blockchain first appeared on John D. Cook.

I recently saw someone post [1] that 987654321/123456789 is very nearly 8, specifically 8.0000000729. I wondered whether there's anything distinct about base 10 in this. For example, would the ratio of 54321six and 12345six be close to an integer? The ratio is 4.00268, which is pretty close to 4. What about a larger base? Let's [...]The post 987654321 / 123456789 first appeared on John D. Cook.

The previous post looked at the basics of how to create a Smith chart. The Smith chart is the image of a Cartesian grid in the right half-plane under the function f(z) = (z- 1)/(z+ 1). At the end of the post I noted that evenly distributed grid lines in thez plane result in very [...]The post Spacing the circles on the Smith chart first appeared on John D. Cook.

The Smith chart from electrical engineering is the image of a Cartesian grid under the function f(z) = (z - 1)/(z + 1). More specifically, it's the image of a grid in the right half-plane. This post will derive the basic mathematical properties of this graph but will not go into the applications. Said another [...]The post How to make a Smith chart first appeared on John D. Cook.

The previous post looked at how to generate random points on a sphere, generating spherical coordinates directly. I wanted to include a demonstration that this generates points with the same distribution as the more customary way of generating points on a sphere, and then decided the demonstration should be its own post. I'll generate random [...]The post Generating random points in Colorado first appeared on John D. Cook.

The topic of generating random points on a unit sphere has come up several times here. The standard method using normal random samples generates points (x, y, z) in Cartesian coordinates. If you wanted points in spherical coordinates, you could first generate points in Cartesian coordinates, then convert the points to spherical coordinates. But it [...]The post Random spherical coordinates first appeared on John D. Cook.

One of the more subtle ideas to convey in an introductory statistics class is that statistics have distributions. Students implicitly think that when you calculate a statistic on a data set, say the mean, that then you have THE mean. But if your data are (modeled as) samples from a random variable, then anything you [...]The post Distribution of correlation first appeared on John D. Cook.

Almost Sure posted an interesting fact on X: If a point (x, y, z) is chosen at random uniformly on the unit sphere, then x, y, and z each have the uniform distribution on [-1, 1] and zero correlations (but not independent!) This follows from Archimedes' On the Sphere and Cylinder" published in 225BC. Archimedes [...]The post Distribution of coordinates on a sphere first appeared on John D. Cook.

I was calculating a correlation coefficient this afternoon and ran into something interesting. Suppose you have two uncorrelated random variablesX andY. If you draw, say, a thousand samples each from X andY and compute Pearson's correlation coefficient, you almost certainly will not get 0, though you very likely will get a small number. How do [...]The post ODE to Fisher's transform first appeared on John D. Cook.

Sometimes you'll hear that a process has so many nines of reliability or that the error rate is so many sigmas. A few years ago I wrote a post on converting between nines and sigmas. See that post for details, approximations, etc. Here I'd like to post a new graph that I believe is an [...]The post Quality metrics first appeared on John D. Cook.

In 2013, John Conway and Alex Ryba published a brief note [1] on how to convert identities involving sine and cosine into identities involving Fibonacci and Lucas numbers. Fibonacci and Lucas The Fibonacci numbers Fn are defined by F0 = 0, F1 = 1, and Fn+2 = Fn + Fn+1 for n > 1. Similarly, [...]The post Turning trig identities into Fibonacci identities first appeared on John D. Cook.

OpenAI Codex is now generally available (see here, here). I'm using the Codex extension in the Cursor code editor with my OpenAI account. Codex is very helpful for some tasks, such as complex code refactoring, implementing skeleton code for an operation, or writing a single small self-contained piece of code. Models have come a long [...]The post Experiences with GPT-5-Codex first appeared on John D. Cook.

I've written before about Bitcoin's elliptic curve and Monero's elliptic curve. In the Monero post I wrote Bitcoin and Ethereum use the elliptic curvesecp256k1." That's true, but it's incomplete. Ethereum does use the elliptic curve secp256k1 for digital signatures, as does Bitcoin, but Ethereum also uses a different elliptic curve for its consensus layer. Ethereum's [...]The post Ethereum's consensus layer elliptic curve first appeared on John D. Cook.

The inverse of the matrix is the matrix assumingad - bc 0. Also, the inverse of the bilinear function (a.k.a. Mobius transformation) is the function again assumingad - bc 0. The elementary takeaway is that here are two useful equations that are similar in appearance, so memorizing one makes it easy to memorize [...]The post Inverting matrices and bilinear functions first appeared on John D. Cook.

For reasons I don't understand, it's more common, at least in my experience, to see references on how to generate pointson a sphere thanin a sphere. But the latter is easily obtained from the former. The customary way to generate points on a sphere inn dimensions is to generate points from ann-dimensional normal (Gaussian) distribution [...]The post Generate random points inside a sphere first appeared on John D. Cook.

Exactly one month ago I wrote about sampling points from a triangle. This post will look at the three dimensional analog, sampling from a tetrahedron. The generalization to higher dimensions works as well. Sampling from a triangle In the triangle post, I showed that a naive approach doesn't work but a variation does. If the [...]The post Random samples from a tetrahedron first appeared on John D. Cook.

Many hope that AI will be smart enough" to make breakthrough scientific discoveries in the near future, such as find a cure for cancer. Some research efforts have sought to create an AI Scientist" that can make discoveries in an automated or semi-automated way; for a recent example, see [1]. Others [2] have called out [...]The post GPT-5 for AI-assisted discovery first appeared on John D. Cook.

This morning I took an old blog post and turned it into an X thread. I think the thread is easier to read. More expository and less rigorous. The post and thread look at generalizations of the fact that every integer and its fifth power end in the same digit. The conclusion is thatn andnk [...]The post More on Carmichael first appeared on John D. Cook.

Here's a differential equation from [1] that's interesting to play with. Even though it's a nonlinear system, it has a closed-form solution, namely where (a,b,c) is the solution att = 0 and = 1 + a^2 + b^2 + c^2. The solutions lie on the torus (doughnut). Ifm andn are coprime integers then the solutions [...]The post Differential equation on a doughnut first appeared on John D. Cook.

My interest in cryptocurrency is primarily technical, and privacy coins are the most technically interesting. The math behind coins like Monero and Zcash is far more sophisticated than the math behind Bitcoin. I'm also interested in the privacy angle since I work in data privacy. The zero-knowledge proof (ZKP) technology developed for and tested in [...]The post Zcash price doubled first appeared on John D. Cook.

Typically RSA public keys are the product of two large primes,n =pq. But there's no reason they couldn't be the product of say three primes,n =pqr, or more primes, as long as (n), or (n), is calculated correctly. Encryption is done the same way. Decryption could be done the same way, except there is the [...]The post RSA with multiple primes first appeared on John D. Cook.

In an inflationary economy, the purchasing power of your currency continually deteriorates. If you have an investment that grows slower than your purchasing power shrinks, you're actually losing value over time. The true rate of growth is the rate of change in the purchasing power of your investiment. If the inflation rate is r and [...]The post True growth rate accounting for inflation first appeared on John D. Cook.

An RSA public key is a pair of numbers (e,n) wheree is an exponent andn =pq wherep andq are large prime numbers. The original RSA paper said choose a private key d and computee. In practice now we choosee and computed. Furthermore, e is now almost always 65537 for reasons given here. So the public [...]The post A quiet change to RSA first appeared on John D. Cook.

Fermat numbers The nth Fermat number is defined by Pierre Fermat conjectured that the F(n) were prime for alln, and they are forn = 0, 1, 2, 3, and 4, but Leonard Euler factoredF(5), showing that it is not prime. Tangent numbers Thenth tangent number is defined by the Taylor series for tangent: Another way [...]The post Fermat primes and tangent numbers first appeared on John D. Cook.

If you need to memorize a list of up to eight words, particularly for a short period of time, the most efficient method may be brute force: rehearse the words in sequence until you can remember them. But if you need to remember more words, or remember the list for a longer time, some sort [...]The post Memorizing a list of seed words first appeared on John D. Cook.

Several years ago I asked myself a couple questions. Which things, if I were 10x better at, would make little difference? Which things, if I were 10% better at, would make a big difference? I remember realizing, in particular, that if I knew 10x more about statistics, it wouldn't make a bit of difference. The [...]The post 10x vs 10% first appeared on John D. Cook.

The optimal choice of factoring algorithm depends on the size of the number you want to factor. For numbers larger than a googol (10100) the GNFS (general number field sieve) algorithm is the fastest known factoring algorithm, making GNFS the algorithm of choice for trying to factor public keys for RSA encryption The expected time [...]The post Time needed to factor large integers first appeared on John D. Cook.

This post will include some simple but useful calculations. Throughout this post, a and b are real numbers greater than 1. All logs are proportional For any two bases a and b, log base a and log base b are proportional, and in fact the proportionality constant is logb a. We have or to put [...]The post log log x first appeared on John D. Cook.

The previous post needed to fit a double exponential function to data, a function of the form By taking logs, we have a function of the form where thea in the latter equation is the log of thea in the former. In the previous post I used a curve fitting function from SciPy to find [...]The post Fitting a double exponential function to three points first appeared on John D. Cook.

In 2001, a quantum computer was able to factor 15. In 2012, a quantum computer was able to factor 21, sorta. They cheated by not implementing gates that they knew a priori would not be used. To this day, there still hasn't been a quantum computer to factor 21 without taking some shortcuts. Some reasons [...]The post Extrapolating quantum factoring first appeared on John D. Cook.

If and when practical scalable quantum computers become available, RSA encryption would be broken, at least for key sizes currently in use. A quantum computer could use Shor's algorithm factor n-bit numbers in time on the order of n^2. The phrase quantum leap" is misused and overused, but this would legitimately be a quantum leap. [...]The post Post-quantum RSA with gargantuan keys first appeared on John D. Cook.

John Conway discovered a right triangle that can be partitioned into five similar triangles. The sides are in proportion 1 : 2 : 5. You can make a larger similar triangle by making the entire triangle the central (green) triangle of a new triangle. Here's the same image with the small triangles filled in as [...]The post Conway's pinwheel tiling first appeared on John D. Cook.

Bitcoin transactions appear to be private because names are not attached to accounts. But that is not sufficient to ensure privacy; if it were, much of my work in data privacy would be unnecessary. It's quite possible to identify people in data that does not contain any direct identifiers. I hesitate to use the term [...]The post Silent Payments first appeared on John D. Cook.

In elementary algebra we learn how to solve ax2 + bx + c = 0 for roots r = r1, r2. But how about the inverse problem: given a real-valued r, find integer coefficients a, b and c such that r is a root of the corresponding quadratic equation. A simple approximation can be found [...]The post An inverse problem for the quadratic equation first appeared on John D. Cook.

The previous post looked at solving the equation which arises from the Mollweide map projection. Newton's method works well unless is near /2. Using a modified version of Newton's method makes the convergence faster when = /2, which is kinda useless because we know the solution is $theta; = /2 there. When [...]The post Inverting series that are flat at zero first appeared on John D. Cook.

Karl Brandan Mollweide (1774-1825) designed an equal-area map projection, mapping the surface of the earth to an ellipse with an aspect ratio of 2:1. If you were looking at the earth from a distance, the face you're looking at would correspond to a circle in the middle of the Mollweide map [1]. The part of [...]The post Mollweide map projection and Newton's method first appeared on John D. Cook.

There are four manmade object that have left our solar system: Pioneer 10, Pioneer 11, Voyager 1, and Voyager 2. Yesterday I wrote about the Golden Record which is on both of the Voyager probes. Some of the graphics on the cover of the Golden Record is also on plaques attached to the Pioneer probes. [...]The post 1420 MHz first appeared on John D. Cook.

If AI can do part of your job, you're likely to either be fired or promoted. AI will always have some error rate, and if it can do your job at an acceptable error rate, you'll need to find another job. But if the error rate is not acceptable, the ability to identify and fix [...]The post The AI fork in the road first appeared on John D. Cook.

Yesterday I wrote a post outlining mental math posts I'd written over the years. I don't write about mental math that often, but I've been writing for a long time, and so the posts add up. In the process of writing the outline post I noticed a small gap. In the post on mentally calculating [...]The post Day of the week centuries from now first appeared on John D. Cook.

Suppose you're learning a new language and want to boost your vocabulary in a very time-efficient way. People have many ways to learn a language, different for each person. Suppose you wanted to improve your vocabulary by reading books in that language. To get the most impact, you'd like to pick books that cover as [...]The post Learning languages with the help of algorithms first appeared on John D. Cook.

I've written quite a few posts on mental math over the years. I think mental math is important/interesting for a couple reasons. First, there is some utility in being able to carry out small calculations with rough accuracy without having stop and open up a calculator. Second, the constraints imposed by mental calculation make you [...]The post Mental math posts first appeared on John D. Cook.

The two Voyager probes, launched in 1977, are now in interstellar space beyond our solar system. Each carries a Golden Record, a recording of sounds and encoded images meant to represent Earth and human civilization. I've long intended to listen to the record and yesterday I did. One of the cuts is a 12-minute collage [...]The post Morse code beyond the solar system first appeared on John D. Cook.

Last week I wrote about a mental random number generator designed by George Marsaglia. It's terrible compared to any software RNG, but it produces better output than most people would if asked to say a list of random digits. Marsaglia's RNG starts with a two-digit number as a seed state, then at each step replacesn [...]The post Cycles in Marsaglia's mental RNG first appeared on John D. Cook.

I wrote a couple posts last month about the seed phrase words used by Bitcoin and other cryptocurrencies. There are 2048 words on theBIP39 list. Monero uses a different word list, one with 1626 words [1]. You can find Monero's list here. Why 1626 words? It's not hard to guess why the BIP 39 list [...]The post Monero's seed phrase words first appeared on John D. Cook.

Let f(z) = (az + b)/(cz + d) where = ad - bc 1. Iff has no singularity inside the unit disk, i.e. if |d/c| > 1, then the image of the unit disk under f is another disk. What is the area of that disk? The calculation is complicated, but the result [...]The post Area of the unit disk after a Mobius transformation first appeared on John D. Cook.

Yesterday I wrote about a triangle inequality discovered by Paul Erds. LetPbe a point inside a triangleABC. Letx,y,zbe the distances fromPto the vertices and letp,q,r, be the distances to the sides. Then Erds' inequality says x+y+z>= 2(p+q+r). Using the same notation, here are four more triangle inequalities discovered by Oppenheim [1]. px + qy + [...]The post More triangle inequalities first appeared on John D. Cook.

LetD be the unit disk in the complex plane and letf be a univalent function on D, meaning it is analytic and one-to-one on D. There is a simple way to compute the area off(D) from the coefficients in its power series. If then The first equality follows from the change of variables theorem for [...]The post Area of unit disk under a univalent function first appeared on John D. Cook.

Ted Dunning left a comment on my post on random sampling from a triangle saying you could extend this to sampling from a polygon by dividing the polygon into triangles, and selecting a triangle each time with probability proportional to the triangle's area. To illustrate this, let's start with a irregular pentagon. To pick a [...]The post Random samples from a polygon first appeared on John D. Cook.

Plane geometry has been studied since ancient times, and yet new results keep being discovered millennia later, including elegant results. It's easy to come up with a new result by proving a complicated theorem that Euclid would not have cared about. It's more impressive to come up with a new theorem that Euclid would have [...]The post A triangle inequality by Erds first appeared on John D. Cook.

If you have a triangle with verticesA,B, andC, how would you generate random points inside the triangleABC? Barycentric coordinates One idea would be to use barycentric coordinates. Generate random numbers , , and from the interval [0, 1]. Normalize the points to have sum 1 by dividing each by their sum. Return A + [...]The post Randomly selecting points inside a triangle first appeared on John D. Cook.