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Aperiodical News Roundup – March 2025

By Katie Steckles. Posted April 2, 2025

Here’s a round-up of some mathematical and maths-related news that happened in March 2025 that we didn’t otherwise cover on the site.

Double Maths First Thing: Issue 13

By Colin Beveridge. Posted January 15, 2025

DMFT is significantly less perplexing than HMRC

Hello! My name is Colin and I am a mathematician on a mission to spread mathematical joy.

This week, I’ve made another contribution to the OEIS (currently in review) about the excellent puzzle #23 from Scroggsvent that Matt has written up here. I’ve also written a blog post I’m unusually pleased about: when Michelle Kwan skated last at the 1995 Figure Skating Worlds, she ended up fourth. However, her performance also caused Surya Bonaly (previously in third) to overtake Nicole Bobek (second) into the silver medal position. That’s… irregular.

This week’s links

Let’s do some silly things today. First of all, let’s answer a question you didn’t know you needed answering: Can you complete the game The Oregon Trail if you wait at a river for 14,272 years? The Oregon Trail wasn’t part of my childhood, but games like it were. I love this.

If you’ve read my blog for any length of time, you’ll know I enjoy the odd bit of ninja trickery, and this is a very odd bit of ninja trickery.

I also love correcting errors! Here’s Charles Petzold correcting some very wrong math[s] about flight times.

Brackett has pointed me at Public Math[s], If there’s one thing maths needs, it’s more zines. Make some! Don’t know how to? Maybe you should hire Hana.

Upcoming

Friend and hero Rob Eastaway is giving a free talk for the Historical Association on Tuesday 21st January at 5pm GMT about the introduction of Hindu-Arabic numerals to England. You can sign up to watch it here.

It’s MathsJam night around the world on Tuesday — find your local event here; I’ll be at the Weymouth one.

I’ve just caught up with the November Finite Group live-stream on combinatorial games. I watch them with my 11-year-old, who is forever pausing them to ask questions. It’s cheaper than many of the movies he wants to watch, and more engaging. The next one is on Thursday 23rd at 3pm GMT and is a crafternoon with Ayliean and Scroggs.

That’s all I’ve got for this week. If you have friends and/or colleagues who would enjoy Double Maths First Thing, do send them the link to sign up — they’ll be very welcome here.

If you’ve missed the previous issues of DMFT or — somehow — this one, you can find the archive courtesy of my dear friends at the Aperiodical.

Meanwhile, if there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website. You can also just reply to this email if there’s something you want to tell me.

Until next time,

A New Sequence!

By Barney Maunder-Taylor. Posted December 18, 2024

Or The Novice’s Guide To Achieving Mathematical Immortality

This is a guest post from Barney Maunder-Taylor.

A great way to achieve mathematical immortality is to solve an outstanding open question, like determining if $π + e$ is rational or irrational, or finding a counterexample to the Goldbach Conjecture. But for most of us, a more realistic approach would be to contribute a new sequence to the Online Encyclopaedia of Integer Sequences, the OEIS. This is the tale of one mathematician’s quest to do just that – with ideas to help YOU to contribute a sequence of your own.

Double Maths First Thing: Issue F

By Colin Beveridge. Posted December 18, 2024

Double Maths First Thing is the biscotti to your Wednesday morning coffee

Hello! My name is Colin and I am a mathematician on a mission to spread joy and delight in maths.

This week’s links

I have a difficult relationship with AI. I wrote about it here. tl;dr: it doesn’t fill me with joy and delight, although it can sometimes be useful. However, an interesting use case is in the ongoing and enormous project to formalise mathematics — using machine verification to find mistakes and gaps in proofs (or to say “yep, that’s legit!” when things do work). Via Harlan Carlens, here’s a piece about the use of AI in tackling the IMO. Something that strikes me as slightly more useful is formalising the proof of Fermat’s Last Theorem. Possibly, but not closely, related: an article about playing chess with God; I’m mainly disappointed about the lack of a “… moves in mysterious ways” joke.

In MathsJam-adjacent news, my esteemed friend Barney Maunder-Taylor took the puzzle “what is the smallest number n such that the digit sums of both n and n+1 are multiples of 7?” and turned it into an OEIS entry. I’ve done some proving around it, but have a play yourself. It’s nice!

In a rare concession to Christmas, here is a video about making cut-and-paste Christmas trees — there are PDFs linked, but the author asks that they not be distributed directly.

Lastly, there’s a live-stream about regexes for paid-up Finite Group members on Friday 20th, 8pm UK time. The Discord group is a lovely space (and I believe you can hang out there on the free tier). For me, it’s a Patreon worth supporting.

That’s all I’ve got this week! In the meantime, if you have friends and/or colleagues who would enjoy Double Maths First Thing, do send them the link to sign up — they’ll be very welcome here.

If you’ve missed the previous issues of DMFT or — somehow — this one, you can find the archive courtesy of my dear friends at the Aperiodical.

Until next time,

Row $\sqrt{- 1}$ of Pascal’s triangle

By Colin Beveridge. Posted July 31, 2024

Hi! My name is Colin, and I am a PROPER mathematician now. I’ve made a contribution to the Online Encyclopaedia of Integer Sequences.

Prime-generating functions

By Peter Rowlett. Posted May 9, 2024

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:

$f (x) = x^{2} + x + 41 .$

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.

$p$	$f (x)$	Primes generated	Number of consecutive primes generated
3	$x^{2} + x + 3$	3, 5	2
5	$x^{2} + x + 5$	5, 7, 11, 17	4
11	$x^{2} + x + 11$	11, 13, 17, 23, 31, 41, 53, 67, 83, 101	10
17	$x^{2} + x + 17$	17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257	16
29	$x^{2} + x + 29$	29, 31	2

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.

Primes, reversals and concatenations

By Peter Rowlett. Posted November 27, 2023

In the last Finite Group livestream, Katie told us about emirps. If a number p is prime, and reversing its digits is also prime, the reversal is an emirp (‘prime’ backwards, geddit?).

For example, 13, 3541 and 9999713 are prime. Reversing their digits we get the primes 31, 1453 and 3179999, so these are all emirps. It doesn’t work for all primes – for example, 19 is prime, but 91 is $7 \times 13$ .

In the livestream chat the concept of primemirp emerged. This would be a concatenation of a prime with its emirp. There’s a niggle here: just like in the word ‘primemirp’ the ‘e’ is both the end of ‘prime’ and the start of ’emirp’, so too in the number the middle digit is end of the prime and the start of its emirp.

Why? Say the digits of a prime number are $a_{1} a_{2} \dots a_{n}$ , and its reversal $a_{n} \dots a_{2} a_{1}$ is also a prime. Then the straight concatenation would be $a_{1} a_{2} \dots a_{n} a_{n} \dots a_{2} a_{1}$ . Each number $a_{i}$ is in an even numbered place and an odd numbered place. Now, since

$10^{k} (\mod 11) = {\begin{cases} 10, & if k is even; \\ 1, & otherwise, \end{cases}$

it follows that each $a_{i}$ contributes a multiple of eleven to the concatenation. A mismatched central digit breaks this pattern, allowing for the possibility of a prime.

I wrote some code to search for primemirps by finding primes, reversing them and checking whether they were emirps, then concatenating them and checking the concatenation. I found a few! Then I did what is perfectly natural to do when a sequence of integers appears in front of you – I put it into the OEIS search box.

Imagine my surprise to learn that the concept exists and is already included in the OEIS! It was added by Patrick De Geest in February 2000, based on an idea from G. L. Honaker, Jr. But there was no program code to find these primes and only the first 32 examples were given. I edited the entry to include a Python program to search for primemirps and added entries up to the 8,668th, which I believe is all primemirps where the underlying prime is less than ten million. My edits to the entry just went live at A054218: Palindromic primes of the form ‘primemirp’.

The 8,668th primemirp is 9,999,713,179,999.