The hardest 'What comes next?' (Euler's pentagonal formula)

Looks like I just cannot do short videos anymore. Another long one :) 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 machine (proof part 2)
50:00 Credits

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

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:

A timeline of Euler's discovery of all the maths that I touch upon in this video:

Check out the translation of one of Euler's papers (about the "modified" machine):

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 “perfect” 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 :)

Enjoy!

Burkard

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.

All the best,

burkard

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[Tubluer]

Mathologer: What does a partition have to do with a pentagon (aside from beginning with "p").
Me: *blinding flash of insight* They both end in "n"!!!

[chriscox]

As a 50 year old man, I may have done better in maths if I had teachers like you! Thank you for your simplification of complex maths. :-)

[faastex]

This is literally magic, the video kept getting more and more interesting (and complicated) and I more and more amazed

[numcrun]

"We do real math, which means we prove things" *squashes pentagon into a house shape"

[BillGreenAZ]

I like how this guy laughs at his own presentation. It tells me he is really enjoying himself and I like to see people who are.

[tinkmarshino]

I am so blown away.. I was never a big math guy though I did use a lot of geometry and right angle trig in my construction life.. But now here in my old age (68) I see the amazement of math laid out before me. The wonder that a few of my fiends had talked about but I could not see.. Oh to take this knowledge back 50 years and do it all over again... What fun it would have been.. Thank you my friend for giving me a taste of the fun and joy my old friends had in their day.. They are gone now but I remember.. thank you!

[gardenmenuuu]

3 b 1b and mathologer are the gifts of gods to all the math lovers around the globe

[TheOneSevenNine]

mathologer, coloring numbers green and orange: "and now the pattern should be really obvious to you!"

me, extraordinarily colorblind: oh my god am I bad at math what's going on

[nurdyguy]

Best part of this was realizing how I can use the logic to solve 3 of my yet unsolved ProjectEuler problems! Awesome video!

[theadoenixes]

Ramanujan and Euler is everywhere ... And I love it ... ❤️

[dhoyt]

Dear Mathologer,
Seeing your video this morning has brightened my day so incredibly much. Your videos allow me to transcend my body(have pain) and live in a world of pure mathematics. Please never stop. - Your fan and student.

[drpeyam]

Wow, I’m not even a number theory fan in general, but this was incredible! Thank you so much for this video, really appreciate it!

[nightingale]

What a mathematician!
Whatever problem you approach on math, Euler has done something there.

[joemichelson]

Love the video, Partition numbers are what got me so interested in OEIS. I was hoping you were going to go into A008284 which is kind of a transformation of Pascal's triangle but spits out the partition numbers.

[durian]

"What comes next?"
Me: Almighty Lagrange's interpolation

[Supremebubble]

I just watched the first 5 minutes and have to really compliment the way you present you material. It's inspiring how you structure it in a way that makes it engaging. The "tricking" shows how important it is to really check what's going and that's what math is all about :)

[otakuribo]

Ramanujan may have been The Man Who Knew Infinity, but Mathologer is the Man Who Made Infinity Long Videos About Them :)

[whycantiremainanonymous]

Those "complete the sequence" questions are my pet peave. The thing is, *any* number can continue *any* sequence, and there will be a formula (a polynomial; actually, infinitely many polynomials) to produce the resulting new sequence. That type of question is routinely used in school tests and intelligence tests, but what it really tests for is a kind of learned bias toward small integers.

[lapk]

YES! The guy with the towel hat!! I've always always wondered about this image of Euler, and what he was wearing on top of his head! Lol!!

[SeyseDK]

i always wanted to dig into partitions but never got around to it. Thank you for outlying it and making it so easy to follow! Euler used to be my favourite as well, that dude was amazing.
Good job Mathologer, keep it up