10+ mon, 2+ week ago (158+ words) 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. Download The post Consulting Podcast 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. John D. Cook, PhD My colleagues and I have decades of consulting experience helping companies solve complex problems involving data privacy, applied math, and statistics. Let's talk....

9+ mon, 6+ day ago (400+ words) When my children were little, I read the Little House on the Prairie books aloud to them and I naturally saw the books through the eyes of a child. Last night I started reading the books by myself for the first time and saw them very differently. Laura Ingalls Wilder wrote the Little House books [] The post Settlers versus Hipsters first appeared on John D. Cook. When my children were little, I read the Little House on the Prairie books aloud to them and I naturally saw the books through the eyes of a child. Last night I started reading the books by myself for the first time and saw them very differently. Laura Ingalls Wilder wrote the Little House books later in life, looking back at her childhood an early adult years. The events in the books took place in…...

10+ mon, 2+ week ago (220+ words) 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. 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 of it, or maybe I'll post more. The first file I posted was the one from the previous post. The second…...

1+ mon, 3+ week ago (260+ words) 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 that'n and'nk end in the same digit base'b if'b is square-free and'k = "(b) + 1. So, for example, 10 is square-free (i.e. not divisible by a non-trivial square) and "(10) + 1 = 5. Benjamin Clark replied suggesting looking at "(b) + 1 in addition to "(n) + 1. To back up a bit, "(n) is Euler's totient function, the number of positive integers less than and relatively prime to n. Robert Carmichael's totient function "(n) is a closely related function, one that replaced Euler's function in implementations of RSA encryption'more specifically in RSA decryption'something I wrote about earlier this week. Euler's generalization of Fermat's little…...

10+ mon, 4+ week ago (281+ words) 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 into prime powers 4 ways: And we can partition 52 two ways: A couple of notes here. First, we only consider positive powers. Second, two partitions are considered the same if they consist of the same factors in a different order. For example, 3 " 3 " 32 and 32 " 3 " 3 are considered to be the same partition. It follows that we can partition 2025 into prime powers 10 ways: we choose one of the five ways to partition 34 and one of the two ways to partition 52. Here are all the Abelian groups of order 2025: Given a prime number q, there are as many…...

10+ mon, 2+ week ago (413+ words) 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. We could also think of an nth rising power or nth falling power as an n-dimensional volume. If we do, then the dimensions in a Newton series cancel out, for example. Dimensional analysis for factorials and related functions could make it easier to remember identities, and easier to spot errors. And it could suggest the correct form of a result before the details of the result are filled in. In other words, artificial dimensional analysis can provide the same benefits of physically meaningful dimensional analysis. For integers n, "(n) = (n " 1)!, and so we could assign dimension n " 1 to "(n), and more generally assign dimension z " 1 to "(z) for…...

10+ mon, 4+ week ago (264+ words) 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 that's prime. You can immediately make this process twice as fast by setting the last bit to 1: it would be wasteful to generate a new number every time you happened to draw an even number. Most composite numbers have small factors, so you check for divisibility by 3, 5, 7 etc. before running more time-consuming tests. Probabilistic tests are far more efficient than deterministic tests, so in practice everyone uses probable primes in RSA. For details of how you apply these tests, and how many tests to run,…...

6+ mon, 4+ day ago (287+ words) There's an old joke about a drunk man looking for his keys under a lamppost. Someone stops and offers to help. He asks, "So, did you lose your keys here?" The drunk replies "No, I lost them over there, but here's where the light is." I routinely talk to people who have strong technical skills [] The post Looking for keys under the lamppost first appeared on John D. Cook. There's an old joke about a drunk man looking for his keys under a lamppost. Someone stops and offers to help. He asks, "So, did you lose your keys here?" The drunk replies "No, I lost them over there, but here's where the light is." I routinely talk to people who have strong technical skills and who want to go into consulting. They usually think that the main thing they need to…...

3+ mon, 2+ week ago (511+ words) A few days ago I wrote about the Jubjub elliptic curve that takes its name from Lewis Carroll's poem Jabberwocky. I've also written about a slightly smaller but still enormous curve named Baby Jubjub. This post introduces Tiny Jubjub, an elliptic curve with 20 elements, one designed to have some properties in common with its gargantuan counterparts, but small enough to work with by hand. As far as I know, the curve was created for the book The Moon Math Manual to zk-SNARKs and may not be known anywhwere outside that book. First, a word about the book. The math behind zero-knowledge proofs is complicated and unfamiliar to most, and has been called "moon math." The "zk" in the title stands for zero-knowledge. SNARK stands for "succinct non-interactive argument of knowledge." The book is an introduction to some of the mathematics…...

7+ mon, 1+ day ago (391+ words) In the latest two posts, we have needed to find the percentage points for a chi square random variable when the parameter ", the number of degrees of freedom, is large. In Volume 2 of Knuth's magnum opus TAOCP, he gives a table of percentage points for the chi square distribution for small values of [] The post Knuth's series for chi squared percentage points first appeared on John D. Cook. In the latest two posts, we have needed to find the percentage points for a chi square random variable when the parameter ", the number of degrees of freedom, is large. In Volume 2 of Knuth's magnum opus TAOCP, he gives a table of percentage points for the chi square distribution for small values of " and he gives an asymptotic approximation to be used for values of " > 30. where xp is the corresponding percentage point function…...