Carnival of Mathematics 248

Welcome to Carnival of Mathematics issue 248! Before we dive into our assortment of monthly maths, let’s talk about the number 248.

Its digits are $2^1, 2^2$ and $2^3$. There doesn’t seem to be a name for numbers whose digits are all powers of a single integer -even when those powers ascend from 1 to 3(!) – but I think there should. In the spirit of the Valentine’s month, I shall call it a Lovely number (not to be confused with Love numbers which measure the rigidity of a black hole).

Now onto properties of 248 that do have names. It is refactorable, meaning that is divisible by its number of divisors. In this case 248 has 8 divisors, one of which is 8. It is an untouchable number, which is to say that it cannot be expressed as the sum of all the proper divisors (divisors excluding itself) of a positive integer. It is palindromic in bases 13 ($161_{13}$), 30 ($88_{30}$), 61 ($44_{61}$) and 123 ($22_{123}$). I can’t help but sprinkle in a bit of mathematical physics; the Lie group $E_8$ has dimension 248, and $E_8\times E_8$ happens to be the gauge group of one of the two heterotic string theories.

On the less mathematical but just as fun side of things, the 248 Bee Network bus in Manchester goes from Manchester airport to Eccles, passing my childhood home on the way. I’ll have to take that route next time I’m there. And finally, Ross 248 is a red dwarf star that will overtake Alpha Centauri as the closest star to the Sun sometime in the next 80,000 years. Any day now…

Now, to this month’s roundup. There are eight courses of maths for you to enjoy!

Course 1: An amuse bouche

I can hear the bells… by myself at Letters and Words

I did a very quick delve into the mathematics of bell ringing! I hope to go into more detail in the future, but in the meantime, here is an appetiser on the art of bell ringing and its undeniably mathematical nature. 

Course 2: A surprising starter

The centre of the Earth is not straight down by John at John D. Cook Consulting

John gives an insight into why approximating everything to a sphere (including our own planet) might not always be technically accurate. A concise proof with interesting real world consequences – even if they are very small!

Course 3: A champagne toast

JLMS centenary by Peter at Cameroncounts

The Journal of the London Mathematical Society is celebrating their 100^th anniversary with a special issue of ten papers. Peter has contributed a paper on Philip Hall’s Marriage Theorem. There’s something for everyone in the issue, so it is well worth checking out. 

Course 4: A hearty main course

The Axiom of Choice is Okay, Almost Surely by Brian at Skewray

A nourishing discussion on the axiom of choice with lots to chew on here. The question is: is the axiom of choice consistent with probability theory? The answer is no. But the journey to getting there is lots of fun. 

Course 5: A cheese board to share

The integrated explicit analytic number theory network by Terence at TerryTao

Explicit analytic number theory avoids asymptotic notation such as $\ll$ in favour of more precise statements with numerical constants; however, estimates of these constants are often years out of date with potential numerical accuracy. Enter ‘The integrated explicit analytic number theory network’, which aims to bring together current results from the field! A great way to foster collaboration. 

Course 6: Celebratory cake!

SoME4 Prizes by Grant at 3blue1brown

As part of 2025 Summer of Math Exposition, prizes are given out for entries helpful to teachers. Here are the five selections! They include content on the history of expectation values, visualisations of Lorentz transformation, a secret ratio property of cubic functions, the Euler algorithm, and the science of complexity. All five are wonderfully visual and break down challenging concepts into easily digestible chunks. Student, teacher, or neither, you are guaranteed to enjoy these learning resources. 

Course 7: A (birthday) espresso shot

A birthday discovery by Peter at Cameroncounts

A quick, invigorating proof. The theorem involves equivalent statements about graphs on groups and the payoff is instant. Whether it’s your birthday or not, this is a great way to spend five minutes of your day.   

Course 8: Chocolates for the road

Online Talks on Accessible Theorems! by Bill at Computational Complexity

Bill signposts us to a series of talks that break down recent theorems to be understood by a wider audience. Unfortunately, we have missed the first one, but there are still four more to catch between now and late May!

That’s all for this month. Look out for carnival 249 on Tony’s Maths Blog next month! You can submit entries here.