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The curious mathmo talks to David Roberts

Way back at the end of last year I put out a call to mathematicians I know: hop on Skype and chat to me for a while about the work you’re doing at the moment. The first person to answer was David Roberts, a pure mathematician from Adelaide. 

We had a fascinating talk about one thread of David’s current work, which involves all sorts of objects I know no more about than their names. I had intended to release this as a podcast, but the quality of my recording was very poor and it turns out I’m terrible at audio editing, so instead here’s a transcription. Assume all mistakes are mine, not David’s.

If you’ve ever wanted to know what it’s like to work in the far reaches of really abstract maths, this is an excellent glimpse of it.

DR: I’m David Roberts, I’m a pure mathematician, currently between jobs. I work – as far as research goes – generally on geometry and category theory, and the interplay between those two. And also a little bit of logic stuff, which I thought I’d talk about.

13532385396179 doesn’t climb to a prime

Someone called James Davis has found a counterexample to John H. Conway’s “Climb to a Prime” conjecture, for which Conway was offering \$1,000 for a solution.

The conjecture goes like this, as stated in Conway’s list of \$1,000 problems:

Let $n$ be a positive integer. Write the prime factorization in the usual way, e.g. $60 = 2^2 \cdot 3 \cdot 5$, in which the primes are written in increasing order, and exponents of $1$ are omitted. Then bring exponents down to the line and omit all multiplication signs, obtaining a number $f(n)$. Now repeat.

So, for example, $f(60) = f(2^2 \cdot 3 \cdot 5) = 2235$. Next, because $2235 = 3 \cdot 5 \cdot 149$, it maps, under $f$, to $35149$, and since $35149$ is prime, we stop there forever.

The conjecture, in which I seem to be the only believer, is that every number eventually climbs to a prime. The number 20 has not been verified to do so. Observe that $20 \to 225 \to 3252 \to 223271 \to \ldots$, eventually getting to more than one hundred digits without reaching a prime!

Well, James, who says he is “not a mathematician by any stretch”, had a hunch that a counterexample would be of the form $n = x \cdot p = f(x) \cdot 10^y+p$, where $p$ is the largest prime factor of $n$, which in turn motivates looking for $x$ of the form $x=m \cdot 10^y + 1$, and $m=1407$, $y=5$, $p=96179$ “fell out immediately”. It’s not at all obvious to me where that hunch came from, or why it worked.

The number James found was $13\,532\,385\,396\,179 = 13 \cdot 53^2 \cdot 3853 \cdot 96179$, which maps onto itself under Conway’s function $f$ – it’s a fixed point of the function. So, $f$ will never map this composite number onto a prime, disproving the conjecture. Finding such a simple counterexample against such stratospherically poor odds is like deciding to look for Lord Lucan and bumping into him on your doorstep as you leave the house.

A lovely bit of speculative maths spelunking!

via Hans Havermann, whom James originally contacted with his discovery.

Right answer for the wrong reason: cellular automaton on the new Cambridge North station

Cambridge North is a brand new train station, and the building’s got a fab bit of cladding with a design ‘derived from John Horton Conway’s “Game of Life” theories which he established while at Gonville and Caius College, Cambridge in 1970.’

One problem: that’s Wolfram’s Rule 135, not the Game of Life. You can tell because of the pixels.

Rule 135 is a 1-dimensional automaton: you start with a row of black or white pixels, and the rule tells you how the colour of each pixel changes based on the colours of the neighbouring pixels. The Cambridge North design shows the evolution of a rule 135 pattern as a distinct row of pixels for each time step. Conway’s Game of Life follows the same idea but in two dimensions – a pixel’s colour changes depending on the nearby pixels  in every compass direction.

Either way, it’s a lovely pattern. I suspect the designers went with Rule 135 instead of the Game of Life so that they’d get a roughly even mix of white and black pixels, which is hard to achieve under Conway’s rules.

Just in case gawping at train stations is your cup of tea, here’s a promotional video with lots of lovely panning shots of the design:

EDIT: James Grime has now also done a video, which can be seen here:

More information

Delayed £50m Cambridge North railway station opens on BBC News.

Cambridge North Station information from Atkins Group, the design consultancy responsible for the station building.

Press release from Greater Anglia trains.

The Game of Life: a beginner’s guide by Alex Bellos in the Guardian.

Brought to our attention by @Quendus on Twitter.

We want your best #proofinatoot on mathstodon.xyz

Mastodon is a new social network, heavily inspired by Twitter but with a few differences: tweets are called toots, it’s populated by tusksome mammals instead of little birds, and it’s designed to run in a decentralised manner – anyone can set up their own ‘instance’ and connect to everyone else using the GNU Social protocol.

Colin Wright and I both jumped on the bandwagon fairly early on, and realised it might be just the thing for mathematicians who want to be social: the 500 character limit leaves plenty of room for good thinkin’, and the open-source software means you can finally achieve the ultimate dream of maths on the web: LaTeX rendering!

Timetabling choreography with maths

Earlier this week my sister-in-law (“SIL” from now on) sent me an email asking for help. She’s a dance teacher, and her class need to rehearse their group pieces before their exam. She’d been trying to work out how to timetable the groups’ rehearsals, and couldn’t make it all fit together. So of course, she asked her friendly neighbourhood mathmo for help.

My initial reply was cheery and optimistic. It’s always good to let people think you know what you’re doing: much like one of Evel Knievel’s stunts, it makes you look even better on the occasions you succeed.

I’d half-remembered Katie’s friend’s Dad’s golf tournament problem and made a guess about the root of the difficulty she was having, but on closer inspection it wasn’t quite the same. I’m going to try to recount the process of coming up with an answer as it happened, with wrong turns and half-baked ideas included.