# The hardest “What comes next?” (Euler’s pentagonal formula)

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From Mathologer.

Looks like I just cannot do short videos anymore. Another long one đ In fact, a new record in terms of the slideshow: 547 slides!

This video is about one or my all-time favourite theorems in math(s): Euler’s amazing pentagonal number theorem, it’s unexpected connection to a prime number detector, the crazy infinite refinement of the Fibonacci growth rule into a growth rule for the partition numbers, etc. All math(s) mega star material, featuring guest appearances by Ramanujan, Hardy and Rademacher, and the "first substantial" American theorem by Fabian Franklin.

00:00 Intro
02:39 Chapter 1: Warmup
05:29 Chapter 2: Partition numbers can be deceiving
16:19 Chapter 3: Euler’s twisted machine
20:19 Chapter 4: Triangular, square and pentagonal numbers
24:35 Chapter 5: The Ramanujan-Hardy-Rademacher formula
29:27 Chapter 6: Euler’s pentagonal number theorem (proof part 1)
42:00 Chapter 7: Euler’s maching (proof part 2)
50:00 Credits

Here are some links and other references if you interested in digging deeper.

This is the paper by Bjorn Poonen and Michael Rubenstein about the 1 2 4 8 16 30 sequence: http://www-math.mit.edu/~poonen/papers/ngon.pdf

The nicest introduction to integer partitions I know of is this book by George E. Andrews and Kimmo Eriksson – Integer Partitions (2004, Cambridge University Press) The generating function free visual proofs in the last two chapters of this vides were inspired by the chapter on the pentagonal number theorem in this book and the set of exercises following it.

Some very nice online write-ups featuring the usual generating function magic:
Dick Koch (uni Oregon) https://tinyurl.com/yxe3nch3

James Tanton (MAA) https://tinyurl.com/y5xj2dmb

A timeline of Euler’s discovery of all the maths that I touch upon in this video:
https://imgur.com/a/Ko3mnDi

Check out the translation of one of Euler’s papers (about the "modified" machine):
https://tinyurl.com/y5wlmtgb

Euler’s paper talks about the "modified machine" as does Tanton in the last part of his write-up.

Another nice insight about the tweaked machine: a positive integer is called âperfectâ if all its factors sum except for the largest factor sum to the number (6, 28, 496, …). This means that we can also use the tweaked machine as a perfect number detector đ

Today’s bug report:
I got the formula for the number of regions slightly wrong in the video. It needs to be adjusted by +n. In their paper Poonen and Rubenstein count the number of regions that a regular n-gon is divided into by their diagonals. So this formula misses out on the n regions that have a circle segment as one of their boundaries.

The two pieces of music that I’ve used in this video are ‘Tis the season and First time experience by Nate Blaze, both from the free YouTube audio library.

As I said in the video, today’s t-shirt is brand new. I put it in the t-shirt shop. Also happy for you to print your own if that works out cheaper for you: https://imgur.com/a/ry6dwJy

All the best,

burkard