mastodon 4.7.2I am a science fiction writer and computer programmer.
Latest novel: MORPHOTROPHIC.
Latest collection: SLEEP AND THE SOUL.
Web site: gregegan.net
“A study that claimed OpenAI’s ChatGPT can positively impact student learning has been retracted nearly one year after publication. The journal publisher, Springer Nature, cited “discrepancies” in the analysis and a lack of confidence in the conclusions—but not before the paper racked up hundreds of citations and made the rounds on social media.”
AI-assisted answers to Erdős conjectures get all the buzz, but good old-fashioned human mathematicians [*] recently settled a 50-year-old conjecture:
Is there a positive constant c such that every binomial coefficient Bin(n,k) has a divisor in the interval (cn,n]?
The answer turns out to be: YES when n is large and k is large compared to n, but strictly NO.
It's a long proof, but I managed to learn something cool from the first few pages.
Erdős originally guessed that maybe Bin(n,k) would always have a divisor in the interval [n-k+1,n], which would be a tighter result. After all:
Bin(n,k) = n (n-1) (n-2 ) ... (n-k+1) / k!
and it's easy to imagine k! failing to “spoil” *all* of the factors in the numerator here.
But in 1958, Schinzel found the counterexample Bin(99215, 15), which is NOT divisible by any of 99201 ... 99215. In fact, Schinzel found an infinite number of counterexamples of the form Bin(n, 15), where the values of n form an arithmetic series, and for n = 13085213870159810495 this is also a counterexample to the conjecture that [cn,n] might always contain a divisor for the choice c=3/4.
So this was an early hint that maybe even the conjecture allowing for any choice of c would fail ... but it took until 2026 to settle this!
https://arxiv.org/abs/2605.21221v2
* Edit: the authors did make some use of AI tools, but it seems to have been predominantly a collaboration between between several authors combining their own results on aspects of the proof.
Dear Benjamin Franklin
I have travelled back in time from 2026 to tell you that we’re still not certain exactly how lightning works.
There are relativistic electrons and gamma rays involved, maybe positrons, and (possibly extragalactic) cosmic rays.
But it’s complicated, and we’re still thrashing out the details.
A new knot invariant that “uniquely identifies more than 97% of the knots with 18 crossings. By comparison, the Jones polynomial, one of the most widely used invariants for cataloging knots, identifies about 42%, and the Alexander polynomial only about 11%.”
A great write-up in Quanta:
https://www.quantamagazine.org/a-powerful-new-qr-code-untangles-maths-knottiest-knots-20260422/
The paper itself is fun and fairly accessible. It’s also full of questions for the future!
“A Fast, Strong, Topologically Meaningful and Fun Knot Invariant” by Dror Bar-Natan, Roland van der Veen
We have observed electron neutrinos oscillating into mu and tau neutrinos.
But could we ever observe electrons themselves oscillating into muons or tauons?
The short answer is “probably not”, for reasons mostly to do with the much greater mass differences between the charged leptons compared to those of the neutrinos. But it’s worth reading this paper in full, as it gives a clear, detailed exposition of the creation, propagation and detection of the relevant states.
H/T Martin Bauer
Breaking: Rubio, Araghchi announce negotiations to divide up parallel worlds between those in which US, Iranian conditions are met to each party’s complete satisfaction.
I’m two episodes into “For the Record: An Incomplete History of Music” from the Cosmic Shambles. It’s a sprawling, joyful, smart and generous tour, from the baryonic acoustic oscillations of the early universe to the Pythagorean comma, the Indian Tala system, Arabic quarter-notes, and much more to come (9 episodes in total, not all of them yet released).
https://www.youtube.com/playlist?list=PLoVTI-iJf1y2EfoTD-fOQLfQ3hQunroWy
“In the 80 years since George Romero's Night Of The Living Dead ambled into public consciousness … ”
So Romero managed to insinuate his movie into the zeitgeist when he was six years old, 22 years before it was released. That’s one hell of a slow-burn viral marketing campaign.
Quote is from the preamble to an interview with Yeon Sang-ho, director of “Train to Busan” [which I loved] and the new film “Colony” [not yet seen].
https://www.abc.net.au/news/2026-06-20/yeon-sang-ho-colony-the-interview/106813376
Breaking: Google AI summary of leaked Australian budget papers confirms that $2B will be allocated to phasing out all bicycle gears that rotate in the wrong direction by 2035.
