proudly presents  Carnival of Mathematics Number 253

Welcome to the 253rd edition of Carnival of Mathematics!

Carnival of Mathematics	All Carnivals at Theorem of the Day

Welcome to the 253rd edition of Carnival of Mathematics, entrusted to me by the sempiternally pulchritudinous aperiodical.com. This is the silly season edition, spanning two slow, hot months, holiday-rich, news-poor.

And 253 itself is poor (for I would cleave to the tradition of finding carnival edition number fun facts). It is poor in independent divisibility pairs.

I hasten to elaborate! Suppose you take a number, let's say 16, and two numbers strictly between 1 and 16, say 2 and 7. What is the probability that a random integer between 1 and 16 inclusive is divisible by 2? It's \(8/16=1/2\). What about divisible by 7? That's \(2/16 = 1/8\) since the random integer could be 7 or 14. And if we want both? Then only 14 will do and the probability is \(1/16\). Well, that makes 2 and 7 independent divisors for 16 because \(1/16=1/2\times 1/8\) and two events are independent if the probability of them both happening is the product of their individual probabilities. But that's it for 16. No other pair of integers from 2 to 15 have independent divisibility: 16 is poor. Take 2 and 3, say. The product of their individual probabilities is \(1/2\times 5/16=5/32\). But only 2 of the numbers up to 16 are divisible by both 2 and 3, and that gives probability \(1/8\). For every pair except 2 and 7 it's the same: 16 is poor. And my fun fact is, there is a little theorem that says the only poor numbers are: 6,8, 16 and, drum roll, products \(p(2p+1)\) for Sophie Germain primes \(p\). As in \(2\times 5=10\) and, longer drum roll, \(11\times 23=253\).

There's a big theorem about independent divisibility as well, it's all in a very accessible paper by two Pennsylvania mathematicians, Rosemary Sullivan and Neil Watling, in Vol. 13 of Integers, paper number 65 (65 is not poor, it has 4 pairs of smaller numbers with independent divisibility). Actually, even poorer are primes or squares of primes which have no independent pairs at all. But we don't want to get into a Four Yorkshiremen debate here. And 'poor' isn't even an official term, I just used it to fit my opening remarks. Silly season, see?

And after all that, these last two months have actually produced a groaning Carnival postbag! Maybe it's thanks to regular reminders from Double Maths First Thing which, by the time you read this, will have posted its 100th edition (100: very not poor, 21 independent divisibility pairs). Congratulations DMFT, 100 Wednesdays, week in week out, that's quite an achievement!

On the subject of weekly postings, Jean des Lunes submits Issue 46 of Mathematics News. "Each issue includes ten articles, which are News about international math competitions, new theorems and much more." So kudos to Jean des Lunes for approaching his semicentennial issue.

And on the subject of celebrations, Renaissance Mathematicus has been blogging about science history for 17 years. Thony Christie can, by his own admission, be intemperate, even ill-tempered. But for 17 years he has been always well-informed and never dull. He talks a little about Fermat numbers, of which 17 is one. I went and checked Carnival of Mathematics no. 17: Dave Marain, almost 19 years ago, helping early on to set the trend, also talked about \(2^{2^2}+1\) at MathNotations. (You can check out all the past Carnivals here—nearly all those permalinks are still holding firm!)

But to go back to Jean des Lunes newsletter, No. 46 reported Claude solving a physics problem "Quite quickly, Claude came up with an initial idea that was essentially correct. The answer was right there, and we simply hadn’t seen it." And actually, just these last two months seem to have been a watershed for AI in the mathematical sciences. Already Fractal Kitty's Carnival 252 had a section "LLMs and Mathematics" with three links. I feel I should follow her lead: these are the Carnivals you will remember for being 'when it all started to change'.

LLMs and Mathematics
The two months can be charted with three posts from Peter Woit's Not Even Wrong:
June 2: End of Civilization News about the Leiden Declaration on AI, which tries "to identify the new threats to the intellectual culture of the mathematics community and begin a discussion of what to do about them". Tim Gowers writes about the Declaration very insightfully in his blog.
July 15: Various and Sundry with news of the counterexample to the Jacobian conjecture, linking to a post by Kevin Buzzard:  Human mathematicians are being outcounterexampled. (And—oops!—news of the leak of the Fields medal winners, scraped—not by AI!—from hidden front-end code in the IMC website).
July 26: Requiem for a Field? inspired by the news that one of those Fields medal winners, Jacob Tsimerman, is joining OpenAI.

Meanwhile, July 12, David Woods posted on mathstodon: "OpenAI claims to have proven the cycle double cover conjecture". A claim that held up: within a week, Sang-il Oum had a lovely post: "A proof of the cycle double cover conjecture by OpenAI: An exposition". This is a proof which would have been career-making in the 1980s when all its ingredients had in fact already been assembled. Now it belongs to 'good old-fashioned graph theory' and people don't care so much but it is still a spectacular achievement.

Wither/Whither the ACM, posted by Lance Fortnow on the Computational Complexity blog, joins the conversation about what it means to be a computer scientist in the evolving AI age. A requiem for another field? Tim Gower in his thoughts about Leiden (linked above) remarks "I have invested a lot of thought into automatic theorem proving of a more traditional kind. One of my main motivations for that was the hope that the work I put into it would extend the state of the art, measured by which problems a computer can solve ... That ship has sailed now, and that saddens me" One might feel the same about an ongoing Terence Tao project to bring humans and AI together to address big mathematical questions. It's latest iteration tackles the inverse Galois problem. No-one is more up-to-speed than Tao on AIs and mathematics but if an LLM, single-handedly, produced a complete resolution of this problem tomorrow who could really be surprised? Update 11/08/26: \(M_{23}\), the last of the sporadic finite simple groups not known to be a Galois group, has been shown to be the Galois group of a finite Galois extension of \(\mathbb{Q}\). AI was used but only in an administrative capacity (see the Acknowledgements section of the paper).

And over this last weekend, OpenAI struck again: A serious challenge to equity in mathematical research posted by Jon Bannon on mathoverflow.net. Ten problems, including some big ones, solved. And news that OpenAI LLMs will be made available "to researchers at eligible institutions". Which means not available to the large majority, or course. "What a time to be alive," whoops Kevin Buzzard (linked above), from the towers, of course, of an extremely, I may almost say an ostentatiously, eligible institution.

Fractal Kitty had another subheading which I would now like to borrow:

The Journey, Not Just the Destination
This, in various forms, is how to react to the LLM mathematics onslaught. For instance, Brian M. Sutin's Skewray Research is pursuing a deep exploration of stable probability distributions. A first report appeared in Carnival number 250; now there are two updates: here and here. A journey towards understanding something: not the headline and investment-grabbing stuff that LLM factories are so keen on. Skewray also have this on entropy and 'surprisals' (which is a word, I find). "Am I smarter than Claude Shannon?" asks Brian M. Sutin. "No," he replies. "Is he smarter than Claude by Anthropic?" I ask. "Yes," I reply—he's on a journey to understanding; Claude will never understand anything except how to autocomplete its next sentence (see this great 3Blue1Brown video).

Or what about the Tappy Math(s) project, reported on Blue Sky by Sam Hartburn, an engaging and original way to take young people on a musical journey to understanding mathematical concepts. "'A pair and a pair’ started with a research paper about the difficulties children experience identifying shapes when those shapes are not presented in the canonical form that they’re used to seeing." (To quote from the lyrics: "My rectangle is tilted but there's no need to be scared." Indeed!) More at tappymathematics.com.

A submission from hexsumgame.com links to a new educational game that also looks engaging. This came with the comment "I saw that submitting puzzles is acceptable, but if I misunderstood then apologies in advance if this is not the type of content you were looking for!" On the contrary, if it's new and mathematical then it's definitely Carnival-worthy. But hexsumgame also have a blog and the June (eligible!) entry, AI-AI-AI, offers an interesting and commendable clarification of their stance on AI in game development.

A mathstodon post by Mark Dominus starts a discussion about algebra vs geometry which is very much about human understanding and appreciation of mathematics. For us humans there's a fascinating and instructive interaction between these different faces of mathematical reasoning, well explored in this post. I couldn't help being reminded of a talk I gave about Pascal's triangle back in May 2023. I included some erroneous ChatGPT responses in a rather sniggering sort of way but it was already clear that this was absolutely Michael Atiyah's algebra devil incarnate. I include a picture I made, which is already of historical interest. You can see the presentation here (1.7MB pdf; it's in French but I bet your browser's AI offers translation and "this looks long and boring, want a summary?" options).

 

And to conclude this discovery section, a mathstodon post by Oscar Cunningham that asks "Can you colour the faces of a dodecahedron black and white so that if you stand on one of the faces you can determine your position and orientation by looking at your face and the adjacent faces?" It's a great question: it suggests all kinds of rabbit holes! What about other Platonics? What about three colours? Is there a link to Hamilton's icosian game?

Or you could ask an LLM. Maybe it would say No and give a page of TLDR stuff. A destination, but no journey, no discovery.

Well, on we go. I should add a heading of my own:

Good Old-fashioned Breakthroughs
Which is where exciting non-LLM things live!

"A sensational Ramsey breakthrough by Domagoj Bradac" posted by Sam Mattheus on his Points and Lines blog (strictly speaking in late May but it's a June update that I link to, where the breakthrough is even more sensational). 'Ramsey' means how big a party before you're certain that if there aren't \(M\) mutual friends then there must be \(N\) mutual strangers, or vice versa. Off-diagonal Ramsey means how big if you fix \(M\) and ask for more and more mutual strangers. And now, after a 70-year wait, we know (up to the usual logs-of-logs-of-logs small adjustments).

Gil Kalai reports that Matching is in NC, "a holy grail of complexity theory". Given an \(n\times n\) matrix of 0s and 1s, can you find \(n\) 1s with one in each row and in each column? Now we can answer in subpolynomial time, given a polynomial number of computers: in other words the problem is not inherently sequential. NC, by the way, stands for 'Nick's Class' because it was Nick Pippenger who formulated this accepted way of defining 'not inherently sequential': how cool is that, to have something named after you by your first name!?

And Christian Lawson-Perfect alerted mathstodon to the fact that a certain \(k\)-hyperperfect number of interest had been discovered: "Alex Violette writes on the SeqFan mailing list," he writes (referring to the SeqFAN Google Group—a mailing list for Integer SEQuence FANs). And he reproduces the reveal: there's a 2-hyperperfect number that doesn't look like the Euclid–Euler Theorem for ordinary perfect numbers. This is not a breakthrough in the Ramsey theory class or in the Nick's Class class, but it's a fifty-year-old question nonetheless.

Now, I have a gloomy feeling that an LMM could swat like bluebottles a lot of \(k\)-hyperperfect questions, including this one. But we (Tim Gowers, Terence Tao, people who matter) are coming to realise that what matters and needs to be nutured is that mathematics is a community activity. I didn't know about \(k\)-hyperperfection. Now I do because mathstodon relayed a mailing list post from someone who knows a lot about it. If, instead of being Alex Violette's discovery, this was something buried in a OpenAI brag-list, I wouldn't have given it the time of day. Because I felt reached-out-to I cared enough about it to make a presentation (300KB pdf), so I could share it with some colleagues. So there, this belongs under another Fractal Kitty heading "Community, Celebration & Resources".

This Carnival has got me ranting rather, which has never happened before (this is my 10th Carnival; 10, a poor number, yay!). So I think I should stop there, although I had a few personal finds saved up...

But before I go, here's one last submission, so there's at least one video to watch: How to draw square roots in Google Slides! by Zye, Sunny, and Arglin. There are 167 comments (at time of writing) including ""skip if you have a math-phobia" they say, nearly an hour into a math video". Indeed, this is a feature-length production (well, Toy Story 5, not Odyssey) but it covers a huge amount of constructive geometry, going as far as field theory (whence the above comment) and there's some piano playing and some jokes. Watch as much as you want, I'm pretty sure you'll learn something new.

Thanks for reading, clicking through ... and submitting to future Carnivals! Indeed, submit right here to Carnival 254 to be hosted by Karen Campe at Reflections and Tangents.