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919444¹⁰⁴⁸⁵⁷⁶ + 1 is prime

Distributed internet prime number search PrimeGrid has found a new largest generalised Fermat prime.

The discovery was made on 29th August, and was double-checked before being announced on 2nd September. PrimeGrid uses a distributed computing approach and uses spare computer time donated by volunteer computers connected to their network.

A generalised Fermat prime is a prime number of the form $a^{2^n} + 1$, with $a \gt 0$. It’s called ‘generalised’ because ‘Fermat prime’ is the name for the particular case $a=2$.

Much like Mersenne primes, there are special tests which make it much easier to check if a number of this form is prime than for a general number. For this reason, they’re a good place to look for new large primes.

Until now only 392 generalised Fermat primes had been found: this new discovery makes 393. At 6,253,210 digits long, it’s now the 12th largest of all known primes, and the second-largest known non-Mersenne prime.

PrimeGrid have put out an announcement in PDF format giving some more details about the search, and credits for the many people involved writing algorithms and providing computers to run them on.

The PrimeGrid homepage has more information about the many different prime number searches they run, and how to join in the search with your own PC.

Maths at the British Science Festival

The British Science Festival is organised annually by the British Science Association, and this year it’s hosted by the University of Brighton and the University of Sussex from Tuesday 5 to Saturday 9 September. For more details and full listings, see the main British Science Festival website.

We’ve pulled out some of the mathematics-related events in the main programme – from theatre reproductions to puzzle workshops and plenty of talks and lectures, there’s something for everyone!

London Mathematical Society launches Mathematical Sciences Directory

LMS MSDirectory entry

The London Mathematical Society yesterday launched its Mathematical Sciences Directory (LMS MSDirectory), a directory of mathematical scientists in the UK. Entries include some personal information, academic networks and social media, current employment and information on education/qualifications. Yes, it’s yet another place to list all this information.

The LMS website suggests a set of benefits for being on the list, including networking with others in UK mathematical sciences and the opportunity to contribute data anonymously to projects such as the Mathematical Sciences People Pipeline, which are used “to make representation to national policy-makers regarding the mathematical sciences”.

Those eligible to be listed include people with a maths degree from a UK institution, those currently working in mathematical sciences in the UK with or without a maths degree, and current students. You don’t have to be an LMS member to be on the list. The FAQ suggests the list was initially populated with data from “over 5,000 mathematicians” (though some may have opted-out before launch – they first emailed me in March asking me to check my data or opt-out) and people can opt to join.

More information

Further details and information on how to join the list from the LMS.

P might not be NP, reckons Norbert Blum

Norbert Blum of Universität Bonn has uploaded to the arXiv a preprint of a paper claiming to resolve the problem of whether $\mathrm{P} = \mathrm{NP}$, in the negative.

“Proofs” one way or the other turn up on the arXiv pretty much every day, but this one might actually be correct. At least, it’s not immediately obvious it isn’t.

Here’s the abstract:

Berg and Ulfberg and Amano and Maruoka have used CNF-DNF-approximators to prove exponential lower bounds for the monotone network complexity of the clique function and of Andreev’s function. We show that these approximators can be used to prove the same lower bound for their non-monotone network complexity. This implies $\mathrm{P} \neq \mathrm{NP}$.

John Baez has very quickly put together a post explaining the very basics of Blum’s argument.  Even more briefly, Blum claims to have shown that the best-case complexity of a function solving the clique decision problem is exponential, not polynomial.

Colin Wright reckons that the proof passes all of Scott Aaronson’s immediate ‘sniff tests’ for a claimed proof of a big problem, and his supplementary list for proofs to do with P versus NP. Those help you spot charlatans and Walter Mitty types, rather than looking at the actual mathematical content.

Obviously, none of us are qualified to even offer a hot take on this, so we’ll all have to wait until more experienced sorts have had a good look.

So, watch this space.

(Personally, my money is on this not quite working, purely based on my natural pessimism)

Landon Clay, founder of the Clay Mathematics Institute, has died

The Clay Mathematics Institute, home of the Clay Millennium Maths Prizes, has announced the sad death of its founder, Landon Clay. “Driven by a deep appreciation of the beauty and importance of mathematical ideas”, Clay donated generously to many organisations and projects, including the Institute which he founded in 1998.

Statement on the CMI website, including an addendum from Andrew Wiles

via @LondMathSoc

News of some significance

You may have heard of the replicability crisis in science, and practices like outcome switching and p-hacking. BuzzFeed reports on a paper that proposes a solution: rather than assign statistical significance at $p=0.05$, change this value to $0.005$.

Sir David Spiegelhalter is quoted saying: “Dealing with the size of the p-value fixes some things. But it’s not dealing with the most important issues.” There’s a lot more in the BuzzFeed article and the full paper.

More information

PsyArXiv preprint: Redefine statistical significance.

BuzzFeed: These People Are Trying To Fix A Huge Problem In Science.