https://richardsholmes.com/mathematrec/ Recent content in Mathematrec on Rich Holmes Hugo -- gohugo.io en Sun, 28 Dec 2025 16:56:16 -0500 https://richardsholmes.com/mathematrec/2025-12-28_fortunately/ Sun, 28 Dec 2025 16:56:16 -0500 https://richardsholmes.com/mathematrec/2025-12-28_fortunately/ This Numberphile video talks about the Fortune conjecture, and as presented, it seems quite mysterious. But it’s less so once you think about it. Let’s start with Euclid’s proof that the number of primes is infinite. If you assume the contrary, that there are exactly $l n$ primes which we can denote $l p_1, p_2, p_3, …, p_n$, then there exists a number $l q=p_1p_2p_3…p_n+1$. That number isn’t divisible by any of $l p_1$ through $l p_n$; there is always a remainder of $l 1$. https://richardsholmes.com/mathematrec/2025-12-11_pythagorean-concatenation/ Thu, 11 Dec 2025 15:27:13 -0500 https://richardsholmes.com/mathematrec/2025-12-11_pythagorean-concatenation/ (This basically is me trying to work out an understanding of one of the comments on this Numberphile video.) Consider a right triangle whose legs are $l 588$ and $l 2353$. What’s its hypotenuse? Would you believe… $l \sqrt{5882353}$? Does it make sense? Well. Do those digits look at all familiar to you? Perhaps not, but the decimal representation of the rational number $l 1/17$ is $l 0.0588235294117647\ 0588235…$, a 16-digit repeating decimal. https://richardsholmes.com/mathematrec/2025-08-13_adventures-among-the-stupid-cheap-toroids/ Wed, 13 Aug 2025 17:44:31 -0400 https://richardsholmes.com/mathematrec/2025-08-13_adventures-among-the-stupid-cheap-toroids/ Remember Adventures Among the Toroids by B. M. Stewart? Way back in 2008 I borrowed a copy from the university library and built a number of its models, writing about them in a couple of blog posts. Ten years later I built some toroids virtually, using the Antiprism software, and I wrote some blog posts about them too. (The mentioned Google Drive folder is no more, but here is an un-curated folder full of toroid stuff, it probably has everything that was there: Toroids. https://richardsholmes.com/mathematrec/2025-08-11_website-move/ Mon, 11 Aug 2025 11:28:25 -0400 https://richardsholmes.com/mathematrec/2025-08-11_website-move/ Due to problems with the .xyz TLD, I’ve moved my website to https://richardsholmes.com. The old domain name is supposed to redirect there, though so far that isn’t working. previous: Street numbers next: Adventures Among the Stupid Cheap Toroids https://richardsholmes.com/mathematrec/2024-03-30_street-numbers/ Sat, 30 Mar 2024 10:45:09 -0400 https://richardsholmes.com/mathematrec/2024-03-30_street-numbers/ On the Fediverse, Christian Lawson-Perfect asks: Who can find a street with a number in its name that’s bigger than the number of any building on that street? Well, it’s not hard to find Million Street in Athens, Tenessee, for instance. Then there’s Billion Dollar Road in Conewango, New York. But that’s small change compared to Trillion Street NW in West Salem, Oregon, which is… probably one of the smallest streets in the world: https://richardsholmes.com/mathematrec/2023-12-22_multiply-perfect/ Fri, 22 Dec 2023 21:08:00 -0500 https://richardsholmes.com/mathematrec/2023-12-22_multiply-perfect/ There’s a fun mathematics video event called Clopen Mic Night and the December edition is available, for a limited time, for viewing on YouTube. One of the “acts” was Luna talking about multiperfect numbers — afterward I did some exploring of it myself. And I found this number: 518,666,803,200. (Not that I was the first to find it, of course, but I did find it on my own.) A perfect number sometimes is defined as a number whose aliquot factors — the ones not including the number itself — sum to that number. https://richardsholmes.com/mathematrec/2023-11-13_primeclassify/ Mon, 13 Nov 2023 21:34:36 -0500 https://richardsholmes.com/mathematrec/2023-11-13_primeclassify/ That business with the beeping machine pushed me down a rabbit hole over the weekend, and when I came out I was hanging on to a brand new Python library of functions for checking what classes a prime number belongs to. It’s on PyPi here, and the source code is on GitLab here. Note it doesn’t find primes in these categories, it checks whether your prime is in them. You can’t use it to find the next, larger-than-the-largest-known Mersenne prime, for instance. https://richardsholmes.com/mathematrec/2023-11-09_prime-time/ Thu, 09 Nov 2023 07:20:01 -0500 https://richardsholmes.com/mathematrec/2023-11-09_prime-time/ On the Fediverse, mcc posted: A machine that beeps once every time the current UTC timestamp is a prime number which led to a number of responses; one was this from Christian Lawson-Perfect. I’m pretty sure the linked website pre-existed but the date-tracking functionality was newly added. Unfortunately in my browser the beeping doesn’t work. A few thoughts. First, to explain, the Unix time, which is almost certainly used by your computer or phone, is the number of non-leap seconds since 00:00:00 UTC on Thursday, 1 January 1970. https://richardsholmes.com/mathematrec/2023-08-08_nonextendable-knights-paths-part-4/ Tue, 08 Aug 2023 13:01:44 -0400 https://richardsholmes.com/mathematrec/2023-08-08_nonextendable-knights-paths-part-4/ Note: There are errors here, see revised article for corrected analysis. The same approach as was used for the 5x5 torus can apply to other tori. For each possible offset between the termini, count up the number of distinct neighbors and the number of neighbor clusters. Their sum is a lower limit on the minimal number of moves for a non-extendable path. If a cluster doesn’t have a Hamiltonian path, though, extra moves are needed; likewise if a single intermediate position isn’t enough to get from one cluster to another. https://richardsholmes.com/mathematrec/2023-08-07_nonextendable-knights-paths-part-3/ Mon, 07 Aug 2023 18:02:25 -0400 https://richardsholmes.com/mathematrec/2023-08-07_nonextendable-knights-paths-part-3/ Note: There are errors here, see revised article for corrected analysis. How about non-extendable paths on toroidal boards? It’s easy enough to modify my script to do that. Change two lines. Boom. What’s not so easy is running the modified script, because it takes a lot longer. Analyzing a 3x3 torus takes about 40 seconds, versus 0.2 seconds for the ordinary 3x3 board. 4x4? Takes more than two hours. Granted, that’s on my computer, which is not the fastest in the world, and it’s a Python script, and Python is not especially fast. https://richardsholmes.com/mathematrec/2023-08-05_non-extendable-knights-paths-part-2/ Sat, 05 Aug 2023 08:44:47 -0400 https://richardsholmes.com/mathematrec/2023-08-05_non-extendable-knights-paths-part-2/ Thinking about it some more, I believe all the ingredients are there to prove the minimum on an infinite board is 23 moves, or 24 positions. If the termini are adjacent, we must reach 17 positions from the start. 12 positions are in two 6-cycles, so clearly we need 12 moves to reach them plus 1 move to get from one cycle to the other. There are four more positions that require 2 moves each to reach. https://richardsholmes.com/mathematrec/2023-08-03_non-extendable-knights-paths/ Thu, 03 Aug 2023 03:12:10 -0400 https://richardsholmes.com/mathematrec/2023-08-03_non-extendable-knights-paths/ 8x8 board On the Fediverse, @sam_hartburn@mathstodon.xyz asks: What’s the shortest sequence of knight’s moves you can make on a chessboard such that the path is blocked at both ends (i.e. from either end of the sequence there are no squares you can move to that you haven’t already landed on)? and provides a candidate path, one with 10 positions linked by 9 moves from the starting square to the ending one. https://richardsholmes.com/mathematrec/2023-07-07_here_we_are_again/ Fri, 07 Jul 2023 16:39:46 -0400 https://richardsholmes.com/mathematrec/2023-07-07_here_we_are_again/ Well. I’m trying something else for my blogs, again. I found some faults with WriteFreely and write.as. For one thing I could not get ActivityPub federation to work properly — that is, it worked more or less correctly on the writing.exchange Mastodon server, but on vmst.io and mastodon.social I could follow my blogs but new content simply did not appear in my Home feed. For another thing, I use MathJax, but in WriteFreely it’s basically impossible to proofread MathJax — because it doesn’t render it until you publish. https://richardsholmes.com/mathematrec/2023-07-04_p2-polynomials-pascaled/ Tue, 04 Jul 2023 14:34:59 +0000 https://richardsholmes.com/mathematrec/2023-07-04_p2-polynomials-pascaled/ I’ve now watched that 3Blue1Brown video and it explains how the Moser circle problem’s solution for $l n$ points is just the sum of the first 5 numbers in the $l n^{th}$ row of Pascal’s triangle, or $l R(n,5) = \sum_{i=0}^{4}{{n}\choose{i}}$, and how this explains why the first five numbers as well as the tenth one are powers of 2, since the $l n^{th}$ row of Pascal’s triangle sums to $l 2^n$. https://richardsholmes.com/mathematrec/2023-07-02_a-line-and-a-fractal-and-some-miscellaneous-points/ Sun, 02 Jul 2023 23:33:13 +0000 https://richardsholmes.com/mathematrec/2023-07-02_a-line-and-a-fractal-and-some-miscellaneous-points/ I’m still thinking about $l (n-1)^{th}$ degree polynomials that generate the first $l n$ powers of 2 at the start. Which is rather a mouthful so I’m just going to call them p2 polynomials from here on. For instance, $l p2(5) = \frac{1}{24}(n^4-2n^3+11m^2+14m+24)$ generates the sequence 1, 2, 4, 8, 16, 31, 57, 99… (By the way, this series of posts originally was prompted in part by this recent video by 3Blue1Brown which mentions the circle division problem I started off with — Moser’s circle problem, it’s called. https://richardsholmes.com/mathematrec/2023-07-01_not-small-numbers-but-differences/ Sat, 01 Jul 2023 13:56:56 +0000 https://richardsholmes.com/mathematrec/2023-07-01_not-small-numbers-but-differences/ I guess it’s just finite differences, right? If you have numbers generated by a polynomial of degree $l n$ then the $l n^{th}$ order differences are constant. For instance, take $l g(n)=n^3+6n^2-4n-3$. Plugging in 0, 1, 2… you get -3, 0, 21, 66, 141, 252, 405, 606… Take differences between successive numbers and you get 3, 21, 45, 75, 111, 153, 201… And repeat 18, 24, 30, 36, 42, 48… https://richardsholmes.com/mathematrec/2023-07-01_circles-and-small-numbers-and-meta-small-numbers/ Sat, 01 Jul 2023 01:46:43 +0000 https://richardsholmes.com/mathematrec/2023-07-01_circles-and-small-numbers-and-meta-small-numbers/ Here’s a rather well known mathematical gotcha. What’s the next number in this series: 1, 2, 4, 8, 16… ? Answer, of course, 31. Draw a circle, mark some points on the circumference, connect every such point to every other, and count the number of regions the circle’s been divided into. With one point there are no lines and the circle consists of 1 region: With two points there is one line and 2 regions: https://richardsholmes.com/mathematrec/2023-06-14_here-we-are-p3t9/ Wed, 14 Jun 2023 23:30:43 +0000 https://richardsholmes.com/mathematrec/2023-06-14_here-we-are-p3t9/ Having reached the breaking point with wordpress.com (the straw being their introduction of an AI garbage content generator), I’ve moved my blogs here, including Mathematrec. It’s powered by WriteFreely. That means it can publish via ActivityPub and be followed from other Fediverse platforms, such as Mastodon. At least in theory. I haven’t seen that working yet but it’s early days. I’ve set it up with a custom domain, richholmes.xyz, and there’s intended to be an actual though minimal website at that URL.