The Maths-alanche.

1 day ago

5 min read

OpenAI has dropped another MASSIVE bundle of maths proofs. NerdNews asks what this means for knowledge itself, and gives you 4 cool examples you can seriously flex with by explaining to mates.

(First published on my Substack where you can get #NerdNews, marvellous maths and general geekery.)

OpenAI has just dumped 722 maths papers and manuscripts onto the world, organised into 372 families of results. As far as I know, a single release of this magnitude has never happened in the history of mathematical thought.

Some claim to answer longstanding problems. Some can be verified via a painstaking computer process. Others may demand deep human comtemplation.

And it comes at a fascinating time for modern mathematics as it wrestles with the impact of artificial intelligence.

Squaring off.

Let’s be honest, the last couple of months between OpenAI and the maths community have been … um … tense.

ICYMI: Two humans, including gun Aussie Tristan Buckmaster, had been doing significant work towards the landmark Navier–Stokes problem. They had been using OpenAI’s coding platform to test ideas and store drafts. Some suspected OpenAI may have benefited from that unpublished work before unleashing around 10,000 concurrent agents which, over roughly 88 hours, exchanged 2.7 million messages and bulldozed their way to a result.

Buckmaster himself said he did not know whether their data was used and was not accusing OpenAI of doing so. OpenAI says its investigation found that Buckmaster’s Codex prompts from the two months before its 8 September announcement could not have influenced the system, including through training.

Some see it as ‘he said - he said’ stuff. Many people think OpenAI looked a bit grubby.

This time, OpenAI’s cover letter, announcing these results, promises to “continue to act on feedback”. It sounds less like its previous triumphal announcements and more like a chastened C-list celebrity promising to work on being a better person after some awkward old tweets have surfaced.

“We will continue to act on feedback from the community and update our standards for future disclosures of major scientific advancements.” — OpenAI cover letter.

Have they taken their medicine? Maybe. Only time will tell. Some mathematicians still fear a near-term future where they simply verify gigantic, ugly machine-generated proofs.

Drama aside: here are four of the new results I reckon you can actually follow.

1. Five colours ain’t enough.

Imagine colouring every single point on an infinite flat plane, following one simple rule: any two points exactly one unit apart must be different colours.

How many colours do you need to cover an infinite plane obeying that rule?

Seven colours have long been known to work. The mystery was whether five or six could do the job? Or are seven necessary.

OpenAI now claims a proof that five can never work. So the answer is six or seven.

OpenAI claims that if you try to colour the plane using only five colours — say red, yellow, green, blue and purple — somewhere two points exactly one unit apart are forced to share a colour.

“We prove that every coloring of the Euclidean plane with five colors has a monochromatic unit-distance pair.” — OpenAI internal model, The Euclidean plane is not five-colorable, 23 September 2026.

This claim has been formalised in the proof assistant Lean, meaning a computer can mechanically check the logical steps of the proof.

2. You’ve just gotta have (friends of) friends.

Think of a social network where connections have a direction: Alice follows Bob, Bob follows Charlie, and Alice does not follow Charlie. Oh, and in this particular social network you cannot follow someone back (I know weird right!).

Now count the people one step away from each person, and the people exactly two steps away. So here Alice is one step from Bob, but two from Charlie.

A conjecture proposed by mathematician Paul Seymour says that in every such finite network, there must be at least one person with at least as many people two steps away as there are one step away.

Or, loosely speaking: somebody always has at least as many friends-of-friends as friends.

OpenAI claims it has finally proved it. This one is Lean-formalised too.

3. Math like an Egyptian.

Ancient Egyptian mathematicians were awesome at heaps of stuff. They could build pyramids. They could walk like 80s pop queens The Bangles.

And they were absolute bandits for writing fractions as sums of different unit fractions.

Ok, Adam , put it away. WTF are you talking about?

If you’ve got a child in year 7, hopefully they can show you that

Maybe even that

So two-thirds can be written as the sum of two unit fractions (a fraction with a 1 on the top).

Four-fifths requires three unit fractions. For eight-elevenths you need four of the buggers.

The worst-case number of unit fractions required for these sorts of Egyptian sums increases as the bottom number of the fractions — our old mate the denominator — gets bigger.

Turns out 16/17 needs five. By the time you reach 27,538/27,539, no combination of seven unit fractions will do. You need at least eight.

Mathematicians already knew there is no ceiling: somewhere out there are fractions requiring 9, 10, 100 pieces and beyond. At the same time needing five for a denominator of 17 and only hitting eight at 27,539, shows the number of unit fractions required doesn’t exactly race along.

The question posed by the enigmatic mathematical puzzler Paul Erdős was how quickly that required number of unit fractions grows.

OpenAI claims to have finally pinned it down. And the answer is ‘extraordinarily slowly.’

4. Cool for Cat-alan.

Eugène Charles Catalan, a 19th-century Belgian mathematician, gave a lot to mathematics, including having a number named after him - maxx aura.

In his page turner Mémoire sur la transformation des séries et sur quelques intégrales définies to the Royal Academy of Belgium on 1 April 1865, he considered this little beauty:

Turns out it was no April Fool’s Day joke. But Catalan’s constant did make fools out of many mathematicians who for over 150 years could not answer the simple question, is that number a fraction?

In mathematical language, is Catalan’s constant rational (can be written as a fraction) or irrational (like pi, a decimal that just goes on forever, never settling into a repeating pattern)?

OpenAI says … drum roll … Catalan’s constant is irrational.

And the headline result has also been formalised in Lean, so it is looking pretty solid.

Where to from here?

In the short term, 722 papers have to be pored over. Many come with Lean formalisation — see my previous NerdNews for why that matters — but many do not, and OpenAI itself warns that some unformalised results may contain errors.

So while 722 is a gigantic number of papers to drop on the maths community, those geniouses will now have to do some serious heavy lifting.

And hopefully in doing so, not only confirming some new mathematical truths but also learning from the process. Improving our human knowledge base, not just ticking a million tedious and unilluminating boxes.

But the longer-term question is much bigger.

Is this the beginning of a golden age in which AI and mathematicians work together on genuinely great mathematics — humans asking better questions, machines exploring territory we could never search alone, each improving the other?

“It’s a scary time, but it’s also really a deeply exciting time.” — Bryna Kra; Sarah Rebecca Roland Professor of Mathematics at Northwestern, former president of the American Mathematical Society. WIRED Magazine.

Or will we look back and laugh at the naivety we displayed during this brief little window when AI was already doing mathematics at frightening speed, but humans could still, sort of, follow its reasoning?