It’s been quite a year. We introduced a totally new mathematics degree course, redesigned from the previous version. In addition, restructures and people leaving meant there were fewer of us teaching on the maths degree. All this together means I’ve been teaching new topics aplenty this year. Now we’re at the end of teaching, I’m looking back (and definitely not just looking for things to do that aren’t the marking I must be getting on with…) and thought I would write another of those self-indulgent blog posts that are probably only of interest to me.
This year I have taught over 200 hours across level 3, level 4, and level 6. At a high level, I have taught proof and mathematical thinking, mathematical modelling and simulation, methods in various areas, research and project skills, and maths and society. More specifically, I have been covering the following topics (some in greater depth than others!) [those marked * are new topics for me to teach this year]:
In 1693, Christiaan Huygens was struggling to learn the new calculus developed by his former student Gottfried Leibniz. He wrote to Leibniz asking for “any important problems where they should be used, so that this give me desire to study them”. Ever since, ‘when will I ever use this?’ is a common refrain, especially among engineering students — right?
A study published in 2020 had found engineering students preferred pure problems without context, but we weren’t sure — it turns out defining when a problem is and isn’t placed in context isn’t as easy as we thought. We wrote some questions that were either just ‘solve this equation’ or were dressed up with an engineering context, and asked students what they preferred and why.
We found pretty split preferences between contextual and non-contextual problems, and learned a lot about why different students prefer different sorts of problems and how they solve them (the quotes in the title give a flavour of this). The resulting article has just been published in Teaching Mathematics and its Applications. Check it out!
There seem to be a lot of numerical coincidences bouncing around concerning the new year 2025. For example, it’s a square number: \( 2025 = 45^2 \). The last square year was \(44^2 = 1936\), and the next will be \(46^2=2116\).
The other one you have likely seen somewhere is this little gem: that 2025 equals both \((1+2+3+4+5+6+7+8+9)^2\) and \(1^3+2^3+3^3+4^3+5^3+6^3+7^3+8^3+9^3\).
A brief update about the state of social media. A couple of years ago, I wrote about the decline of Twitter and how I was going to consciously shift my activity to Mathstodon. Just over a year ago, we launched The Finite Group, a members’ club for mathsy people, partly inspired by the collapse of social media. We recently updated this to include a free membership.
Since then, I’ve ended up in an awful state where I’m somewhat engaging with X, Mastodon, Bluesky and Threads, but not engaging much with any. If I have something to broadcast, say a new blog post, I’ll put it on all four. If I have something to say or fancy a chat, I might put it on Mastodon. What’s happening just now is that more people seem to have decided to leave X for Bluesky. Will that stick? I don’t know, but it’s nice to see people who I used to see on Twitter being active, and for whatever reason those people haven’t got on with Mastodon, so it seems promising (from that point of view, though it’s not an ideal place to be).
Here’s my current position:
If I want to have a natter about something maths, the absolute best place for that is The Finite Group. You can join for free and get access to a friendly online chat community run via Discord. (There are also paid memberships where you get to watch livestreams.)
I’m going to stop looking at Threads. Things I post there get very little engagement, and I don’t see much I’m interested in. It’s also a regular little irritant because it keeps alerting me it has found something of interest to me, which turns out to be of no interest whatsoever.
I continue to log into X because there are some large organisations there whose updates I would like to receive. I have to wade through Elon’s thought of the day and crypto ads to get there and I hope in time this will stop being part of my life. Will I continue to post stuff I do there? I’m not sure, to be honest.
That is, the answer to each question can be made by treating the element in the first matrix as the first digit and the corresponding element in the second matrix as the second digit in the answer element. This is not how matrix multiplication works, and ought to be funny if I hadn’t totally over-explained the joke!
I saw one of these in a meme that Katie posted in the Finite Group chat and it got me thinking about how these work.
If we set up the matrices like this
\[ \begin{bmatrix} a & b\\ c & d \end{bmatrix} \begin{bmatrix} e & f\\ g & h \end{bmatrix} = \begin{bmatrix} 10a+e & 10b+f\\ 10c+g & 10d+h \end{bmatrix} \]
Then we establish four equations with eight unknowns.
Since there are more unknowns than equations, these don’t have a single solution. What I wanted was to find integer solutions with all values single-digits. I wrote some Python code to find these. I removed some that look overly symmetrical – either the rows of the matrix are identical, or the same matrix is repeated. This left 73 items.
From these 73 items, I wrote a second Python script that picks 20 of them at random and builds these into a LaTeX worksheet. For the Mastodon post I reformatted this into the shape and size that I thought would display better on social media, and added in one of the squared matrices for an extra hint something weird is up, hoping people might notice this isn’t just a boring post about matrix multiplication practice!
A few weeks ago I heard someone casually refer to ‘that formula of Euler’s that generates primes’. I hadn’t heard of this, but it turns out that in 1772 Euler produced this formula:
\[ f(x) = x^2 + x + 41\text{.} \]
Using this, \(f(0)=41\), which is prime. \(f(1)=43\), which is also prime. \(f(2)=47\) is another prime. In fact this sequence of primes continues for an incredible forty integer inputs until \(f(40)=41^2\). It might generate more primes for higher inputs, but what’s interesting here is the uninterrupted sequence of forty primes.
This got me wondering. Clearly \(f(0)\) is prime because 41 is prime, so that much will work for any function
\[ f(x) = x^2 + x + p \]
for prime \(p\), since \(f(0)=0^2+0+p=p\). Are there other values of \(p\) that generate a sequence of primes? Are there any values of \(p\) that generate longer sequences of primes?
I wrote some code to investigate this. Lately, I’ve taken to writing C++ when I need a bit of code, for practice, so I wrote this in C++.
I figured the cases where \(f(0)\) is prime but \(f(1)\) isn’t weren’t that interesting, since \(f(0)\) is trivially prime. In fact, \(f(x)=x g(x)+p=p\) when \(x=0\) for any prime \(p\), but saying so doesn’t seem worth the effort.
So I kept track of the primes \(p\) whose functions \(f(x)=x^2+x+p\) generate more than one prime, and the lengths of the sequences of primes generated by each of these. This produced a pair of integer sequences.
I put the primes that work into the OEIS and saw that I had generated a list of the smaller twin in each pair of twin primes. I was momentarily spooked by this, until I realised it was obvious. Since \(f(0)=p\) and \(f(1)=1^2+1+p=p+2\), any prime this works for will generate at least a twin prime pair \(p,p+2\).
What about the lengths of the sequences of consecutive primes generated? The table below shows the sequences of consecutive primes generated for small values of \(p\). Most primes that generate a sequence produce just two, and \(p=41\) definitely stands out by generating forty.
I was pleased to see this sequence of lengths of primes generated was not in the OEIS. So I submitted it, and it is now, along with the code I wrote. (I discovered along the way that the version where sequences of length one are included was already in the database.)
Anyway, I amused myself by having some C++ code published, and by citing Euler in a mathematical work. Enjoy: A371896.
You know how loads of things in maths are named for the wrong person? In 1996, a fun quiz appeared in The Mathematical Gazette based on history of maths misconceptions. It contained a series of questions where the obvious answer is not correct, such as “Who discovered Cramer’s rule?”, “Did Pascal discover the Pascal triangle?” and “Who first published Simpson’s rule?”
I was looking for a demo to show my students that generative AI programs are not producing accurate knowledge when I thought of this quiz. I put its questions to ChatGPT to see how it did. The point of the exercise is that these systems just parrot back words from their training data without any concept of truth, so if the training data is full of misconceptions, so too will be the responses. But these are misconceptions from the 1990s, so how much influence will they have on the responses?
