Season 12 Episode 6

On this episode of the Oxford Online Maths Club, we look at some number theory, including a tree-like way to represent all numbers.

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Further Reading

PROMYS problem set

You can get the 2026 PROMYS Europe problem set here (PDF). We had a quick look at the last question, which is about directed graphs (but not all directed graphs!).

Visualisations

We saw Jeffrey Ventrella's Number Tree visualisation.

We also saw Animated Factorization Diagrams made by Stephen Von Worley based on diagrams by Brent Yorgey. 

Higher Dimensions

Last week’s Further Reading had a reference to the Cantor set. Imagine making a Cantor set along the $x$-axis between 0 and 1, then making a Cantor set along the $y$-axis between 0 and 1. The Cantor Dust fractal is what you get if you take the points in the unit square whose $x$-coordinate and $y$-coordinate both like in your Cantor sets. This is a little bit like making a tartan pattern where you have lines going one way crossing lines going the other way, with the intersection points showing up clearly.

It’s considered to have dimension 1.261859... in the following sense; you need four copies of the Cantor dust fractal to make a version of the fractal that is three times as large along the side. By comparison with a square, for which you’d need $3^2$ copies, or a cube, for which you’d need $3^3$ copies, the Hausdorff dimension roughly says that if you need $3^d$ copies to make something $3$ times as large along the side, then it’s got dimension $d$. Here, $4$ is $3^{1.261859...}$ so we say the Cantor dust has dimension 1.261859... . If you know about logarithms, you’ll know where that number has come from. You don't have to do anything with this information, I'm just telling you in case the idea of irrational dimensions between the familiar integer ones is appealing to you!

Complex numbers

If you’re trying to make your own version of the factor dance animation in something like Desmos then... firstly, wow! that sounds like hard work... and secondly here’s some advice on plotting equally spaced points around a unit circle. Set Desmos to complex mode in the settings, and you can use $\exp(2i \pi [0,...,n]/n)$ to get $n$ points equally spaced on the unit circle. These numbers are called the roots of unity, and they have the property that (as complex numbers) they satisfy the equation $z^n=1$.

If you'd like to know some more facts about roots of unity, and you'd like to see some tricky questions (including International Mathematics Olympiad questions), here's a handout that I found on Ray Li's website at Stanford.

I’ve decided that the number theory that the animation reminded me of was the Hardy-Ramanujan-Littlewood circle method. See the book Closing the Gap by Vicky Neale for a bit more on this, and the story of the search for the truth about twin primes.

Prime Number Theorem

Luci mentioned the prime number theorem: the number of primes less than or equal to $N$ is about $N/ln N$, when $N$ is large. This was conjectured by Gauss and proved in 1896 by (independently) Hadamard and de la Vallée Poussin. The  is genuinely quite a good read (if you ignore all the logarithms, perhaps). It’s got it all; an unproved claim by Gauss, a surprising result, the number $10^{10^{10^{34}}}$, the Riemann hypothesis, and the phrase "this proof is not easy".