u/non-orientable

This article is about logic: specifically, how one goes about computing the Gödel number (which features prominently in Gödel's proof of his incompleteness theorems, but has utility beyond it). Usually, when one only sees the Gödel number worked out for only a very short mathematical sentence (no more than "2+1=3", say), and there is an excellent reason for that: even for quite basic theorems, the Gödel number quickly becomes completely unmanageable.

I was asked to compute the Gödel number of the Pythagorean theorem by someone who was likely unaware of this, and due to some perverse impishness, I was compelled to see it through. It was no easy task, but you can read the final result (for free) on Substack: The Gödel Number of a Non-Trivial Sentence.

How do you divide up a secret between a group of people such that no one person can reconstruct it, no two people can reconstruct it, but any group of three can? (In real life, more likely it will be servers, rather than people.) The answer uses mathematics that is entirely accessible to a good high school student… except for a little twist at the end, where you need some knowledge of number theory.

Read the full post (for free) on Substack: Polynomials and Secret Sharing.

A while back, I wrote an article exploring why so few video games take place on a sphere, and the torus is so much more common. But this leads to a natural question: is the torus the only surface that would pass the obstructions that we laid out? No, there is one more, >!the Klein bottle!<. We show that it could have been used as a world map, even though I don't know of any game that ever did. In the process, we discuss one of my common disagreements with how some math popularization is done.

Read the full post (for free) on Substack: An Alternative to Toroidal Games

What is the connection between group theory and cryptography? There are actually various ways in which it is used, but probably the single most common is the Diffie-Hellman key exchange. In this article, we’ll run through how it functions from a group-theoretic perspective, and then fill in some of the gory, number-theoretic details.

Read the full post (for free) on Substack: Groups and Diffie-Hellman

Set theory has a slightly odd place in mathematics education: it is essentially non-existent prior to a certain point (often something like an introduction to proofs class), and then completely ubiquitous. It is the framework that we use to express pretty much everything in modern mathematics. In this article, I have two goals:

1. show the basics of set theory and explain why it has this central position, and
2. show the drawbacks of using set theory as the central organizing principle.

For example, have you ever realized that, going by the standard set-theoretic definitions, the natural numbers are not a subset of the integers?

Read the full post (for free) on Substack: The Good, The Bad, The Set Theoretic.

In past posts, I proved and talked about some very classical results in number theory: that all primes that are 1 mod 4 are sums of squares; that there are infinitely many primes that are 1 mod 4, and so on. I wanted to write about something much more modern, but still recognizably in this same vein. Hence, the Friedlander-Iwaniec theorem: there are infinitely many primes that are the sum of a square and a 4th power.

This is a result simple enough that you could explain it to a middle schooler, and yet the proof is an entirely different league from the proofs mentioned above---it is almost 100 pages long, for a start! While I don't go into the proof (although I do show where you can find it for free, if you are interested), I do talk about its history and broader context, to give a sense of why it was such a big deal.

Read the full post (for free) on Substack: The Friedlander-Iwaniec Theorem

There's a common perception that mathematics is all about solving equations and working with numbers, which is almost entirely disconnected from the sort of work that mathematicians actually do. So, what is mathematics, actually?

This article is my own personal take on this question, that mathematics is the study of structure divorced from context. I'll define precisely what I mean by this, and we'll discuss some connections to questions of how generalizable or widely applicable mathematics should be, including Wigner's The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

Read the full post (for free) on Substack: What is Math?

It's weirdly common to hear myths about primes. (After you correct for the very low baserate of hearing about mathematics at all, of course.) I remember one of my high school math teachers telling us that you could get paid money for discovering new large primes; I'm not sure where that misconception came from, but it isn't remotely true. EDIT: as u/Eiim and u/will_1m_not point out, this probably originated from the fact that GIMPS offers $3k for each new Mersenne prime discovered, and will offer $50k to the first person to discover a prime larger than 10^(100,000,000). So there was truth to the claim after all!

In this post, I gather up all of the erroneous claims that I remember hearing and demonstrate why they are false, spending the most time on the claim that to find the n-th prime, you need to compute all of the preceding primes. We'll show that not only is that not true, but there exists an algorithm that computes the n-th prime (given n) faster than any algorithm that would compute all of the primes up to the n-th. This touches on the prime number theorem and some work by Meissel from the 1800s.

Read the full article (for free) on Substack: Debunking Prime Myths

First of all: last week, the Deranged Mathematician hit 1k subscribers. Thank you very much to all of you who made that possible!

Now, for the meat of this post: I have turned on paid subscriptions for the blog. Everything that was free before still is, and I only intend to keep about 50% of all posts behind a paywall. If you want the full details and/or to participate in a poll about what will be the first longer lecture series, see here: Updates to the Deranged Mathematician

Aside from providing access to the full archive, paid subscriptions also grant access to the draft of my book, A Projective Perspective on Linear Fractional Transformations. In some sense, this is a more "grown-up" version of my existing book, Linear Fractional Transformations: An Illustrated Introduction, published by Springer. The new book is aimed at an upper-level undergraduate audience and covers inversive, Minkowski, and hyperbolic geometry. It also has a chapter on Fuchsian groups and connections to number theory. This book is currently not in production; I just don't have the time and energy for it, and probably won't for a long time. But it seems like a waste for it to sit on a shelf gathering dust.

If you want to read a little bit about how that book happened (and access the draft itself if you have a paid subscription), see here: A Projective Perspective on Linear Fractional Transformations