John D. Cook

The previous post pointed out that the following code such as the following unexpectedly works. >>> from math import log, factorial >>> log(factorial(1000)) 5912.128178488163 If you don't find this unexpected, note that if you replace math.log with numpy.log the code will fail [1]. Functions like natural logarithm operate on real numbers. Real numbers are represented [...]The post Calculating log(1000!) first appeared on John D. Cook.

Last week I wrote a post on hiding cryptographic keys in decks of cards. I wrote some code for that post that shouldn't work, but before fixing I noticed that it in fact did work. The code computes logarithms for integers larger than the largest representable float. For example, the largest float is on the [...]The post The code that didn't break first appeared on John D. Cook.

A few days ago I wrote a post on counting rooted trees. That post looked at the sequence c(n) which counts the number of rooted trees with n nodes. Here one node is distinguished as the root, but the nodes below the root are not distinguished from each other; all that matters is how the [...]The post Enumerating trees and circles first appeared on John D. Cook.

After writing the previous post about metallic ratios, I thought about the analogy to alchemy and the attempt to make precious metals out of base metals. When can you make one metallic ratio out of another? Can you make the golden ratio out of the lead ratio? Before we can make gold out of lead, [...]The post Mathematical alchemy first appeared on John D. Cook.

The golden ratio is the first and best known of the metallic ratios. I've written about the silver ratio a few times, most recently here. And I've mentioned the bronze ratio a couple times. The metallic ratios after bronze don't have standard names. The nth metallic ratio M(n) is the number whose continued fraction representation [...]The post Ratio of metallic ratios first appeared on John D. Cook.

Yesterday I wrote that a lot of the special functions that pop up in mathematical physics are solutions to second order linear differential equations with polynomial coefficients. More generally, holonomic functions are defined to be those functions that are the solutions to linear differential equations, of any order, with polynomial coefficients. Most special functions are [...]The post Holonomic functions first appeared on John D. Cook.

In this post I mentioned two series which I denotedt(n) andc(n). The former is the number of unlabeled rooted trees withn nodes. The latter is the cumulative sum of the former, i.e. The sequencec(n) is also the number of constraints on an n-step Runge-Kutta method; that's how I became interested in it. Now thet(n) sequence [...]The post Estimating a cumulative sum first appeared on John D. Cook.

Second order linear differential equations with polynomial coefficients form their own area of study. This seems like a narrow class of equations, but it's very important in applications. This class of equations seems like a mathematically natural topic, but why is it so important in applications? I did a PhD in differential equations without ever [...]The post Why polynomial coefficients? first appeared on John D. Cook.

Combinatorial problems can be interesting for their own sake, but they are more interesting when there is a connection to a problem outside combinatorics, and the more unexpected the connection the better. Counting the number of unlabeled rooted trees [1] with n nodes is a pure mathematics problem. Designing numerical methods for solving differential equations [...]The post Counting rooted trees first appeared on John D. Cook.

The textbook version of the Runge-Kutta method for solving differential equations has 4 stages and has 4th order error. For lower order versions of RK the number of stages s also matches the order of the error p. But in order to achieve error on the order ofp >= 5, you need more than p [...]The post Runge-Kutta order versus stages first appeared on John D. Cook.

I was digging into the Runge-Kutta method for solving differential equations and a line from [1] piqued my curiosity. These calculations, which are not reproduced in Kutta's paper (they are however in Huen (1900)), are very tedious. The calculations are a set of eight constraints that the parameters of a fourth-order Runge-Kutta method must satisfy. [...]The post Solving the RK4 design equations first appeared on John D. Cook.

A couple years ago I wrote about how to compute the inverse of factorial. I used that code in writing the previous post because the post required solving the equation log2(n!) >= b given b. That is, given a number of bits b, find the smallest value of nsuch that n! >= 2b. What the [...]The post Inverse factorial improved first appeared on John D. Cook.

The previous post looked at the idea of storing a cryptographic key in the order of a deck of cards. A deck of 52 cards can store 225 bits of data because log2(52!) = 225. Here x isx rounded down to the nearest integer. If we want to store bigger keys, we're going to need [...]The post Cryptographic Keys and Decks of Cards first appeared on John D. Cook.

The latest issue of Paged Out! has an article by Stephen Hewitt An off-line backup of your cryptographic key using playing cards." The idea is to use a deck of 52 to store a 128-bit cryptographic key. To erase the key, shuffle the deck. Hewitt gives his algorithm for embedding a key, one that can [...]The post Hiding data in permutations first appeared on John D. Cook.

My post from yesterday on permutation roots ends with a Mathematica code for finding the probability that a permutation of n elements has a kth root. This is done by finding the coefficient of xn in the generating function I wanted to say more about this, and look at implementing the same code in SymPy. [...]The post Counting permutations with roots first appeared on John D. Cook.

It's well known that you can convert the base 16 (hex) representation of an integer to the base 2 (binary) representation by simply converting each digit from hex to binary. For example, CAFEhex = 1100 1010 1111 1110two I imagine it's less well known that you can do the same thing with floating point numbers. [...]The post Printing floating point numbers in binary first appeared on John D. Cook.

Let be a permutation on n elements. If there is a permutation such that applying twice has the same effect on the list of elements as applying once, we say = ^2 and is a square root of . If we let our n elements be the integers 0 [...]The post Permutation roots first appeared on John D. Cook.

The function expq(x) is defined by taking the power series for exp(x) and keeping only the terms whose index is a multiple of q. For example, exp2(x) keeps only the even-numbered terms in the exponential power series and so equals cosh(x). In general, The first sum uses Iverson's bracket notation: a Boolean expression in brackets [...]The post exp_q first appeared on John D. Cook.

I was working with a wide spreadsheet from a client the other day and I had to convert between Excel column labels and column numbers. I had never paid attention to how Excel labels columns and implicitly thought it was base 26 using letters rather than digits. But then I realized that's not right. Excel [...]The post Excel column numbering first appeared on John D. Cook.

This post takes a more abstract view of the previous post. That post looked at the concrete question of whether a number ever has the same sine in radians as in degrees. The relation between radians and degrees is irrelevant except that /180 is an irrational number. Suppose and are two positive numbers [...]The post An almost periodic function first appeared on John D. Cook.

Ordinarily the sine of x radians and the sine of x degrees are very different numbers. Having your calculator in radian mode when it should be in degree mode, or vice versa, results in a major error. But sometimes it doesn't matter. A trivial example is when x = 0. A more interesting example is [...]The post When sine of x degrees equals sine of x radians first appeared on John D. Cook.

I recently had a project in which I had to reverse engineer a data analysis. There was some ambiguity regarding which of several possibilities someone chose for several of the variables, something analogous to the following example. Suppose you have three numbers with uncertain values with a known, or at least purported, sum. The first [...]The post Forensic accounting in Python first appeared on John D. Cook.

A couple days ago, Levent Alpoge, a mathematician working at Anthropic, discovered a counterexample to the Jacobian conjecture using Claude Fable 5. I was curious whether most mathematicians were trying to prove or disprove the conjecture, so I asked Claude. Before a counterexample to the Jacobian conjecture was found, did most mathematicians believe it was [...]The post Locally everywhere does not imply everywhere first appeared on John D. Cook.

The volume of a sphere of radius r is V = 4r^3 / 3 and the surface area is A = 4r^2 and so the ratio of volume to area is V / A = r / 3. Surprisingly, the same ratio holds for all regular solids ifr is the radius of the largest sphere [...]The post Volume to Area ratio for Regular Solids first appeared on John D. Cook.

I've written several posts about using Claude or ChatGPT to generate Prolog or Lean code to solve a chess puzzle. I didn't think Grok would be up to the task, though I didn't try it. I've heard good things about Grok 4.5, so I gave it a shot. It did great. Here's the problem, a [...]The post Solving a chess puzzle with Grok 4.5 first appeared on John D. Cook.

Suppose you want to search for a list of words. If you're using grep, you can add the -f flag provide a file of regular expressions, and you can add the -F to tell it that the regular expressions are in fact just words. I did something like this a couple days ago when searching [...]The post Fitting a regular expression to a list of words first appeared on John D. Cook.

Squares, high and low Let p be an odd prime number. Then half the numbers from 1 through p - 1 are squares and half are not. That is, for half of numbers 1 k < p, the equation x^2 = k mod p has a solution. The traditional name for these numbers is [...]The post Sum of low squares first appeared on John D. Cook.

Quick follow up to the previous two posts on ICD-10 codes and HCPCS codes. This post uses Python's squarify library to create treemaps visualizing how many codes begin with each letter. Here's the treemap for HCPCS codes. And here's the treemap for ICD-10 codes. The sizes of the squares are proportional to the number of [...]The post Visualizing Medical Code Hierarchy first appeared on John D. Cook.

Since I revisited my old post on ICD code matching, I thought I'd revisit by post on HCPCS codes too. HCPCS stands for Healthcare Common Procedure Coding System, and is pronounced hick picks." When most people say HCPCS, they technically mean HCPCS Level II, and that's what I mean here. The format of a HCPCS [...]The post Regular expressions for HCPCS codes first appeared on John D. Cook.

Seven years ago I wrote a post about regular expressions to match diagnosis codes. I wanted to revisit that post looking at speed and error rates. Regular expressions usually do not exactly match what you're looking for and nothing else. They have error false positives and false negatives. But they also have advantages, and context [...]The post Regular expression speed and error rates first appeared on John D. Cook.

I've been thinking about ICD-10 codes; they come up a lot in my work. The ICD-10-CM standard is divided into 21 chapters, which generally correspond to the first letter of a code. However, a chapter may contain blocks beginning with more than one letter, and codes starting with a single letter, namely D, can span [...]The post ICD-10 chapters and code letters first appeared on John D. Cook.

A few days ago I wrote a post entitled Does additional data always reduce posterior variance?. In a nutshell, the answer is no, not always. That led the previous post which looked at posterior means for three Bayesian models, showing how the posterior mean is a weighted average of the prior mean and the mean [...]The post Posterior variance first appeared on John D. Cook.

Common sense says that what you believe after seeing new data should be some sort of compromise between what you believed before and what the new data says. You don't want to ignore previous information or new information. How much should new data change your prior beliefs? When prior judgment and new information are in [...]The post Posterior mean first appeared on John D. Cook.

Years ago I wrote about Gilbreath's conjecture. It's a simple conjecture; you could explain it to anyone who understands what prime numbers are. See the linked post for a description of the problem. Gilbreath's conjecture is simple, but it's also kinda weird. As I wrote before, Paul Erds speculated that Gilbreath's conjecture is true but [...]The post Progress on Gilbreath's conjecture first appeared on John D. Cook.

I ran across a geometry theorem with the following diagram. The theorem corresponding to the diagram is interesting, but I found reproducing the diagram more interesting. The segmentAB is a diameter and the lineCD is perpendicular to the diameter. Assume the outer circle is a unit circle. I guessedC = (cos(1), sin(1)) and made the [...]The post Reproducing a geometry theorem diagram first appeared on John D. Cook.

I ran across the approximation e 2721/1001 recently. What makes this remarkable is its accuracy relative to the size of the denominator. You can create a trivial approximation just by truncating a decimal expansion e 2718/1000 but this is only good to four significant figures, but 2721/1001 is good to seven, almost eight, [...]The post e approximation first appeared on John D. Cook.

A discussion over lunch today brought up the fact that additional data does not always decrease the size of a confidence interval. This post will look at this from a Bayesian perspective. In general, new information reduces your uncertainty regarding whatever you're estimating. The posterior distribution becomes more concentrated as more data are collected. That's [...]The post Does additional data always reduce posterior variance? first appeared on John D. Cook.

This morning I wrote a post that included the central Delannoy numbers. Thenth central Delannoy numberDn counts the number of ways a king can move from one corner of a chessboard to the diagonally opposite corner without backtracking. The more general Delannoy numbers Dm,n are the analogy for an m * n rectangular board, not [...]The post DNA Sequence Alignment and Kings first appeared on John D. Cook.

Imagine the following dialog. Professor:f is a function of a real variablex that takes a real parameterk. Student: What's a parameter? Professor: It's a constant that can vary. Student: Then if it can vary, isn't it a variable? Professor: Sorta, but no not really. This conversation plays out over and over, and unfortunately it often [...]The post Distinguishing variables from parameters first appeared on John D. Cook.

Golden rectangles The defining property of golden rectangle is that if you stick a square on its longer side, you get another golden rectangle. The smaller vertical rectangle is similar to the larger horizontal rectangle. This means / 1 = (1 + ) / which tells us ^2 = 1 + and [...]The post Silver Rectangles and the Ways of Kings first appeared on John D. Cook.

Here's kind of a strange problem with an interesting solution: find a functionf such that the derivative off equals the inverse off for all positive x. f'(x) =f-1(x) This is a differential equation, but a very unusual one, one that cannot be solved using any of the techniques taught in a class on differential equations. [...]The post Derivative equals inverse first appeared on John D. Cook.

The calculator utility bc has a minimal math library. For example, there's no tangent function because you're expected take the ratio of sine and cosine. (The Gnu version of bc does have a function for tangent, but the POSIX version does not.) And yet bc includes support for Bessel functions J(x). The bc function j [...]The post Who you gonna believe: Grok or the docs? first appeared on John D. Cook.

Here's a crazy bash one-liner I found via an article by Peter Krumins: echo {w,t,}h{e{n{,ce{,forth}},re{,in,fore,with{,al}}},ither,at} This prints 30 English words: when, whence, whenceforth, where, wherein, wherefore, wherewith, wherewithal, whither, what, then, thence, thenceforth, there, therein, therefore, therewith, therewithal, thither, that, hen, hence, henceforth, here, herein, herefore, herewith, herewithal, hither, hat This post will explain how [...]The post Brace expansion tree first appeared on John D. Cook.

The previous post looked at how many digits are in the reduced fraction for thenth harmonic number. I was curious about how long the cycle of digits in a harmonic number might be. I wrote about the period length for the digits of fractions almost a decade ago. This post includes code so I can [...]The post When will the decimals in a/b repeat? first appeared on John D. Cook.

The previous post looked at writing the harmonic numbers as reduced fractions and estimating the number of digits in the numerator and denominator based on asymptotics. This is a follow up post with plots. We'll choose our baseb to be 2. And we'll look at the total number of bits in both the numerator and [...]The post Height of harmonic numbers first appeared on John D. Cook.

Thenth harmonic number is the sum of the reciprocals of the firstn positive integers. Hn = 1 + 1/2 + 1/3 + 1/4 + ... + 1/n The product of all the denominators isn!, so you could write Hn as a fraction Hn = p/q wherep =n! Hn is an integer and q =n!. While [...]The post Writing down harmonic numbers first appeared on John D. Cook.

Hart's theorem says If a triangle be formed by the arcs of three circles, the inscribed and the three escribed circles are all tangent to a new circle or line. Here triangle" means a three-sided figure whose sides are portions of a circle. The inscribed circle is the largest circle that can fit inside the [...]The post Hart's theorem first appeared on John D. Cook.

This post will reveal the connection between my two previous posts: one on the Star Trek lemma and one on Pythagorean triples. In the process of writing the latter, I looked at the Wikipedia article on Pythagorean triples and noticed this curious paragraph. In every Pythagorean triangle, the radius of theincircleand the radii of the [...]The post Incircles and Excircles of Pythagorean triangles first appeared on John D. Cook.

In this post we find all Pythagorean triples that contain consecutive numbers, all Pythagorean triples (a,b,c) such thata + 1 =b orb + 1 = c. a + 1 =b George Osborne wrote a paper [1] addressing the question of when the squares of two consecutive numbers is also a square. Geometrically this is asking [...]The post Consecutive Pythagorean triangle sides first appeared on John D. Cook.

I was reading an article this evening and saw a footnote to a book by Arthur Baragar [1]. This caught my eye because he was my officemate at UT for a year. I found his book on Archive.org and was surprised to see The Star Trek Lemma" in the table of contents. What could this [...]The post The Star Trek lemma first appeared on John D. Cook.