Season 10 Episode 6
This is the last episode of OOMC for 2025, and we're taking a lightning-speed look back at the episodes and further reading notes from seasons 9 and 10 of the show!
Further Reading
Just a short set of further reading notes this week; most of the things we mentioned were recaps of things from previous episodes!
Listen to the Fibonacci sequence
The On-Line Encyclopedia of Integer Sequences is a big database of notable sequences of integers. It has a search feature that lets you enter the first few terms of a sequence and see if that matches a known sequence (which can be helpful for puzzles!) It's founder Neil Sloane often appears in Numberphile videos.
OEIS also has a feature where you can listen to a sequence. To change the sequence, you need to know its reference number (e.g. A000045 is the Fibonacci sequence, A000040 is the prime numbers). Note that clicking the PLAY button downloads a midi file (Gen Z readers may be amused to note that the file extension for this rather basic audio format is “.mid”). Whether or not you can play this downloaded midi file depends on your device.
Catalan Numbers
It’s interesting to me how different resources introduce the Catalan numbers with different applications. See Wolfram MathWorld which introduces the Catalan numbers as the solution to Euler’s polygon division problem, whereas Brilliant’s first example is the number of valid parenthesis expressions and Wikipedia starts with an image of non-crossing partitions. The first example in the book on Catalan numbers that I own starts with polygon division but quickly derives the recursion relation $$ C_n = \sum_{k=0}^n C_kC_{n-k} \quad C_0=1 $$
that relates each Catalan number to a sum involving previous Catalan numbers, and then the book works from that as the fundamental definition of Catalan numbers.
Perhaps I’ll just tell you the relationship to the central binomial coefficients; the Catalan numbers $C_n$ satisfy $$ C_n= \frac{1}{n+1} \binom{2n}{n} $$
where $\displaystyle \binom{n}{r}$ is how I write binomial coefficients (you might prefer $\,^n C_r$).
But by stating just that definition I’m really selling short the sheer range of problems that the Catalan numbers are the answer to!
Dirichlet’s Prime Number Theorem
Dirichlet says that for all positive integers $a$ and $d$, there are infinitely many primes of the form $a+nd$, unless $a$ and $d$ share a common factor, in which case all of the numbers $a+nd$ will share that factor too. I’ve heard the theorem summarised as “any arithmetic progression contains infinitely many primes, unless it obviously doesn’t”. Wikipedia link here.
You might wonder how to prove this. Here are some notes by Anthony Várilly (Professor of Mathematics at Rice University), which I’m linking you to for the most part just so that you can see that the first step seems to be "define the Riemann zeta function”.
Perhaps this is a good place to leave you for this term of OOMC, with a link to one of the largest open problems in modern mathematics.
Thanks for joining us for Oxford Online Maths Club this year. We’ll be back in January 2026, and you’re welcome to join us for our (more admissions-focused) series called the MAT livestream.
If you want to get in touch with us about any of the mathematics in the video or the further reading, feel free to email us on oomc [at] maths.ox.ac.uk.