Le problème n°397, posé il y a plusieurs décennies par le célèbre #mathématicien #PaulErdős, qui porte sur l’existence d’une infinité de #solutions pour une équation impliquant des #coefficients #binomiaux #centraux, vient d’être résolu… par une #IA d' #OpenAI

https://www.journaldugeek.com/2026/03/09/une-ia-dopenai-resout-un-probleme-qui-cassait-la-tete-des-matheux-depuis-des-decennies/

“Juggling is sometimes called the art of controlling patterns, controlling patterns in time and space”*…

A skill for our times…

The Library of Juggling is an attempt to list all of the popular (and perhaps not so popular) juggling tricks in one organized place. Despite the growing popularity of juggling, few websites are dedicated to collecting and archiving the various patterns that are being performed. Most jugglers are familiar with iconic tricks such as the Cascade and Shower, but what about Romeo’s Revenge or the 531 Mills Mess? The goal of this website is to guarantee that the tricks currently circulating around the internet and at juggling conventions are found, animated, and catalogued for the world to see. It is a daunting task, but for the sake of jugglers everywhere it must be done.

For every trick found in the Library, there will be an animated representation of the pattern created via JugglingLab, in addition to general information about the trick (siteswap, difficulty level, prerequisite tricks, etc.). If I am able to run the pattern, then I will provide a text-based tutorial for the trick with the help of animations. I will also include links to other tutorials for the trick that can be found online, ranging from YouTube videos to private sites like this one. If I am unable to provide my own tutorial, there will still be a short description of the trick in addition to outside tutorials and demonstrations…

… if you have come to the Library looking to find out how to start juggling, than it would be best to begin with the Three Ball Cascade pattern. If you are a juggler who is already familiar with the basics, then the various tricks included in the Library can be accessed via the navigation tree on the left, or you can click here to view all of the tricks by difficulty…

Enjoy “The Library of Juggling.”

And see also: “The Museum of Juggling History,” the resources at the International Jugglers’ Association, and “The world cannot be governed without juggling.”

* mathematician (and juggler) Ronald Graham

###

As we toss ’em up, we might send carefully-calculated birthday greetings to G. H. Hardy; he was born on this date in 1877. A mathematician who made fundamental contributions to number theory and mathematical analysis, Hardy juggled other interests as well– for example his  Hardy–Weinberg principle (“allele and genotype frequencies in a population will remain constant from generation to generation in the absence of other evolutionary influences”) is now a basic principle of population genetics.

In Hardy’s own estimation, his greatest contribution was something else altogether: from 1917, Hardy was the mentor of the Indian mathematician Srinivasa Ramanujan, a relationship that has become celebrated.  Hardy almost immediately recognised Ramanujan’s extraordinary (albeit untutored brilliance), and the two became close collaborators. When asked by a young Paul Erdős what his greatest contribution to mathematics was, Hardy unhesitatingly replied that it was the discovery of Ramanujan, remarking that on a scale of mathematical ability, his own ability would be 25, Littlewood would be 30, Hilbert would be 80, and Ramanujan would be 100.

source

What’s your Epstein Number?

The release of the latest batch of information relating to disgraced financier and sex offender Jeffrey Epstein got me thinking about the number of physicists on friendly terms with that individual and that in turn got me thinking about the Erdős Number, which I blogged about here, and about constructing some sort of metric relating to a person’s connecttion to Epstein.

The Erdős Number? It’s actually quite simple to define. First, Erdős himself is assigned an Erdős number of zero. Anyone who co-authored a paper with Erdős then has an Erdős number of 1. Then anyone who wrote a paper with someone who wrote a paper with Erdős has an Erdős number of 2, and so on. The Erdős number is thus a measure of “collaborative distance”, with lower numbers representing closer connections. A list of individuals with very low Erdős numbers (1, 2 or 3) can be found here. As it happens, mine is three.

The main difference between an Erdős Number and a putative Epstein Number is that most people think’s a nice thing to have a low Erdős Number whereas the opposite is probably the case for evidence of close collaboration with Jeffrey Epstein…

It is also difficult to define an equivalent to the Erdős Number for Epstein as the form of “colloboration” is less easily catergorised than publishing a paper. I think it is probably fairer to base a number simply on the number of people you know who met Epstein personally (assuming you didn’t know him yourself). Anyone who did know Epstein personally therefore gets an automatic red card. It would also be very difficult for a typical person to work out how many people they have met who have met someone who has met Epstein, etc.

I was intrigued by this because it is known that Epstein liked hanging out with scientists and, being a scientist myself, I wondered if anyone I knew had been drawn into the Epstein circle. It’s unreasonable to count anyone who appears in the Epstein files as having “known” Epstein because many of the names simply appear on emails sent by Epstein to which no reply was apparently ever received or which were not indicative of a working relationship or personal friendship, sometimes quite the opposite.

Anyway, based on a not very thorough bit of research I came across the following people who I have met in person who met and knew Jeffrey Epstein to a greater or lesser extent.

First, there’s Lawrence Krauss who left his position at Arizona State University as a consequence of a sexual misconduct case. He features prominently in the Epstein correspondence, including many messages about the disciplinary case brought against him at ASU. I met Lawrence Krauss in the 1990s at an Aspen Summer School for Physics, where I shared an office with him for about two weeks. I wouldn’t say that we got on well.

Second, there’s Harvard theoretical physicist Lisa Randall, whom I met at a meeting in South Africa about 25 years ago. The disturbing thing about her case is that she carried on interacting with Epstein even after his conviction for sex offences, visiting Epstein’s island home and travelling on his private jet.

Another name that comes up frequently in the Epstein files is John Brockman, a well-known literary agent. I met him at the Experiment Marathon in Reykjavik in 2008. In fact we were placed next to each other alphabetically speaking in the list of contributors:

Our conversations at that meeting were limited to small talk. As a matter of fact I didn’t really know who he was! He certainly didn’t offer me a lucrative book deal like he did with certain other physicists. The topic never arose.

The files also contain references to Stephen Hawking (who died in 2018), including allegations about him made by Virginia Giuffre. Hawking was never charged with any crime but it is the case that he met Epstein at least once, at a meeting organized by Lawrence Krauss on St Thomas, close to Epstein Island. I met Stephen Hawking on a number of occasions.

So according to this my Epstein Number is four. I have had no contact with any people who knew Epstein since 2008 and very little before that. Although it is perhaps indicative of a lack of eminence, I can’t say I’m sorry this number is low. I may have missed some, of course.

P.S. It is worth reading Peter Woit’s blog post on this topic and Scott Aaronson’s here.

Paul Erdős, one of the most prolific mathematicians of the 20th century,
left behind hundreds of puzzles when he died.
To help keep track of which ones have been solved, Thomas Bloom,
a mathematician at the University of Manchester, UK, set up erdosproblems.com,
which lists more than 1,100 problems and notes that around 430 of them come with solutions. 

When Sebastian Bubeck celebrated GPT-5’s erroneous breakthrough, Bloom was quick to call him out.

“This is a dramatic misrepresentation,”
he wrote on X.

Bloom explained that a problem isn’t necessarily unsolved if this website does not list a solution.
That simply means Bloom wasn’t aware of one. There are millions of mathematics papers out there, and nobody has read all of them.
-- But GPT-5 probably has.

It turned out that instead of coming up with new solutions to 10 unsolved problems,
GPT-5 had scoured the internet for 10 existing solutions that Bloom hadn’t seen before. Oops!

There are two takeaways here.
One is that breathless claims about big breakthroughs shouldn’t be made via social media:
Less knee jerk and more gut check.

The second is that GPT-5’s ability to find references to previous work that Bloom wasn’t aware of is also amazing.
The hype overshadowed something that should have been pretty cool in itself.

Mathematicians are very interested in using LLMs to trawl through vast numbers of existing results,
François Charton, a research scientist who studies the application of LLMs to mathematics at the AI startup Axiom Math, told me when I talked to him about this Erdős gotcha.

But literature search is dull compared with genuine discovery,
especially to AI’s fervent boosters on social media. Bubeck’s blunder isn’t the only example.

In August, a pair of mathematicians showed that no LLM at the time was able to solve a math puzzle known as Yu Tsumura’s 554th Problem.
Two months later, social media erupted with evidence that GPT-5 now could.
“Lee Sedol moment is coming for many,” one observer commented,
referring to the Go master who lost to DeepMind’s AI AlphaGo in 2016.

But Charton pointed out that solving Yu Tsumura’s 554th Problem isn’t a big deal to mathematicians.
“It’s a question you would give an undergrad,” he said. “There is this tendency to overdo everything.”

Meanwhile, more sober assessments of what LLMs may or may not be good at are coming in.
At the same time that mathematicians were fighting on the internet about GPT-5,
two new studies came out that looked in depth at the use of LLMs in medicine and law (two fields that model makers have claimed their tech excels at). 
Researchers found that LLMs could make certain medical diagnoses,
but they were flawed at recommending treatments.
When it comes to law, researchers found that LLMs often give inconsistent and incorrect advice.
“Evidence thus far spectacularly fails to meet the burden of proof,” the authors concluded.

But that’s not the kind of message that goes down well on X.
“You’ve got that excitement because everybody is communicating like crazy
—nobody wants to be left behind,” Charton said.
X is where a lot of AI news drops first, it’s where new results are trumpeted,
and it’s where key players like Sam Altman, Yann LeCun, and Gary Marcus slug it out in public.
It’s hard to keep up—and harder to look away.

Bubeck’s post was only embarrassing because his mistake was caught.
Not all errors are.
Unless something changes researchers, investors, and non-specific boosters will keep teeing each other up.
“Some of them are scientists, many are not, but they are all nerds,” Charton told me.
“Huge claims work very well on these networks.”
https://www.technologyreview.com/2025/12/23/1130393/how-social-media-encourages-the-worst-of-ai-boosterism/

#paulerdős
https://www.erdosproblems.com/

#LB Aliás, aproveitando o aniversário de #PaulErdös para recomendar a biografia dele: "O homem que só gostava de números", do Paul Hoffmann. Ele era muito excêntrico, porém de um jeito muito adorável.

https://mastodon.social/@rrmutt/114228897663298035

Two announcements: AI for Math resources, and erdosproblems.com | What's new

Nice. This week's #BBCInOurTime is about #PaulErdos. Looking forward to listening to the discussion.

#Mathematics #maths