The Most Underrated Concept in Number Theory

This is probably my favorite video I've made yet. It's about an underrated mathematical concept known as "integer complexity" and my personal journey to discover it.
0:00 - Introduction
1:20 - A Mathematical Question I Stumbled Into
3:23 - Discoveries Among the First Dozen Numbers
6:49 - What is the Largest Number We Can Build?
11:19 - Number Webs With Mysterious Gaps
13:54 - Incorporating Subtraction and Division
17:23 - How I Found the Name of This Concept
21:00 - Further Directions We Could Take This
24:40 - A Philosophical Question I Stumbled Into
27:27 - Outroduction

Clarifications and corrections:
- When I draw the web of which numbers you can build with eight ones I forgot to draw a dot for 14, which should also be on that web.
- During one of the whiteboard b-roll shots, I showed 4 as being built by "1+1+1+1+1" which is one too many ones.
- Lots of comments on this video mention that exponentiation would make things have lower costs. That's true, but I wasn't including exponentiation here, because we have to define the exact mix of operations we are using at each point, not take any operations for granted (like, if we automatically allow exponentiation, why not automatically allow tetration?) so this video shows how I just began with addition/multiplication, then included subtraction, then included division. You could include exponentiation (which would change costs of course) if you wanted to look for other further patterns, and I'm sure there's interesting patterns to find there too (there are some OEIS entries related to that on Glenn Whitney's table I mentioned). However, you could also include any number of other operations if you chose (tetration, roots, logarithms, etc.).
- For some reason, various comments on this video claimed that the integer complexity was always just equal to a number's prime factorization, which is not true.

Special thanks to all of my supporters on Patreon! (Supporting the show not only helps me keep improving my content, but also lets me avoid needing to incorporate any product placements from brands in episodes)
Evan Clark, Max, George Carozzi, Peter Offut, Tybie Fitzhugh, Henry Spencer, Mitch Harding, YbabFlow, Joseph Rissler, Plenty W, Quinn Moyer, Julius 420, Philip Rogers, Ilmori Fajt, Brandon, August Taub, Ira Sanborn, Matthew Chudleigh, Cornelis Van Der Bent, Craig Butz, Mark S, Thorbjorn M H, Mathias Ermatinger, Edward Clarke, and Christopher Masto, Joshua S, Joost Doesberg, Adam, Chris Reisenbichler, Stan Seibert, Izeck, Beugul, OmegaRogue, Florian, William Hawkes, Michael Friemann, Claudio Fanelli, The Green Way, Julian Zassenhaus, Bailey Douglass, Jan Bosenberg, Brooks Boutwell, David Irvine, qe, George Sharabidze, Jack Dwyer, Fredrik, Dave Brondsema, and Chandler Smith!

If you want to try to help with Combo Class in some way, or collaborate in some form, reach out at combouniversity(at)gmail(dot)com

In case anybody searches any of these terms to learn about them, some topics in this video include: number sequences, the OEIS (or "Online Encyclopedia of Integer Sequences"), Mahler-Popken "integer complexity" and extensions to it, my fun personal mathematical journeys, and more.

This episode was directed/edited/soundtracked by me (Domotro) and was filmed by Rishi Amutas and Carlo Trappenberg.

Disclaimer: Do NOT copy any dangerous-seeming actions you may see in this video, such as any actions related to fire.

[ComboClass]

This is probably my favorite video I've made yet. It's about an underrated mathematical concept known as "integer complexity" and my personal journey to discover it.
0:00 - Introduction
1:20 - A Mathematical Question I Stumbled Into
3:23 - Discoveries Among the First Dozen Numbers
6:49 - What is the Largest Number We Can Build?
11:19 - Number Webs With Mysterious Gaps
13:54 - Incorporating Subtraction and Division
17:23 - How I Found the Name of this Concept
21:00 - Further Directions We Could Take This
24:40 - A Philosophical Question I Stumbled Into
27:27 - Outroduction

(see video description for more links and info!)

[infinitesimalperinfinitum]

I'm glad you're free-range, I'd be terrified to see what would happen if you were contained

[anoNEMOs]

I like to imagine that he's just going through his day and suddenly he thinks up a next sentence to say in his video, so he records it in any place where he is at that moment.

[M-Orion-Nebula]

1. You don't have enough clocks.
2. The "e" reveal was beautiful.

[soingpeirce]

This is the most flat earth theory vibe I've ever seen on a video that has actual substance

[undrs]

"Dad, why is there smoke in our neighbor's house?" "Damn it, it's that guy again"

[EionBlue]

I fully believe that one day Domotro will discover some sort of eldritch mathematical concept that opens his mind to some horrific elder god which will drive him absolutely mad.

And nobody will notice the difference.

[lincolnuland]

How does this guy only have 40k subs!? He lit stuff on fire and then started talking about math.

[curtiswfranks]

The OEIS is one of the most-important websites ever.

[TheDJRiffin]

Never heard of "throdd" and "threeven" before, love it.

[jansustar]

I thought that the whole "i couldnt find the numbers in the millions since i wasnt a programmer" segment would lead to a Brillinat sponsor.

[jvcmarc]

I was bizarrely discussing about this same thing with my cat the other day while high. However, I wasn't able to find anything about it on the internet and ended up forgetting it. It's such a mystique coincidence for you to have posted this video so close to those thoughts, thank you

[SirNobleIZH]

That revelation with e reminds me of when i realized a^b will always be greater than b^a so long as a is closer to e than b

[auztenz]

Watching the first minute; pls dont burn ur self.

[mmorizes]

On that last philosophical point: there's also this feeling that if the new thing you discovered hasn't been researched before, that it's perhaps too contrived or useless or uninteresting. It's not too difficult to discover a sequence that isn't on the OEIS, but you have to ask yourself if you were motivated purely by the math, or if you were motivated by finding the lowest hanging fruit of undiscovered (uninteresting) math.

If something you discovered has been researched before, at least you know it's important enough to have been worthy of the time and effort of serious mathematicians. If not, you're left wondering how important your discovery really is, even if it is new.

To analogize this to exploring islands, if you discover a new island, and it's extremely rich in natural resources, nature, hospitable for humans, spacious, etc., chances are, someone has already discovered the island, and people already live on it. If, however, you discover an island that truly no one has discovered before, chances are, it's probably just a small rock full of bird poop. Should you be excited in the first case or in the second case? The second case is your original discovery, but it's also less meaningful and useful than the first case, but the first case isn't your original discovery.

Rather than be disappointed in either case, may as well be excited in both cases. Like you said, if it hasn't been discovered before, that's an awesome feeling. Even if the reason for it not having been discovered before is that it is somewhat contrived, it was clearly interesting enough for you to stumble across naturally. You should feel proud in that case. If the thing you discovered has been discovered before, and researched before, you should feel proud that your mathematical intuition is well-honed enough to tread the same paths that the giants before you have. You should also feel excited that you can skip to the front of the line. The foundations and path has been built for you, so you have no excuse to not rush to the frontier as fast as you can (by learning and studying and catching up on research) so you can start making new discoveries from a different (more developed, new) starting point.

[joshuazelinsky]

Hi! I'm one of the mathematicians who has done some work on Integer Complexity, especially work with Harry Altman.

A few quick notes: A related fun open problem: do powers of 2 have the obvious complexity? That is for any n>1, is the complexity of 2^n just 2n?

Also, I gave a version of the Gaussian problem as a research problem to a student group a while ago. There work is I believe still under review.

Edit: Your thoughts about rationals are interesting. I think you are correct that that problem has not had much work (or at least if there is work on it, I don't know of it). To some extent, your 5/6 example seems to be taken advantage of "Egyptian fractions" which are ways of writing a number as the sum of fractions of the form 1/a for various distinct a. In particular, for 5/6, you are using that 1/2 + 1/3=5/6. Frequently it seems that using an efficient Egyptian fraction representation for a number will give rise to a low cost way of writing that fraction.

[arystkrat]

This feels tailor-made for people with ADHD. There's never a moment to get bored because there's always something new on screen to pay attention to while listening.

[nissantzvitovey]

This just showed up in my feed, first time watching one of your videos. As a math major myself, I truly admire your mad mathematician vibes, with your calculations done in the wold with no clear uses, the true calling of all mathematicians.

[JRAN]

Incredible coincidence that this video found its way into my recommendations.
A few weeks ago I encountered exactly this problem while working on an idea about bypassing some filter in a program that blocked digits, but allowed string like ‘true’. By using things like ‘true+true+true’ which equals 3, I could create numbers that I needed. But it would take a lot of text to create, say, the number 100.
I realized that through multiplication and parentheses I could create larger numbers more quickly but had to figure out an algorithm to generate these. It ended up being a pretty messy brute-force algorithm but I could generate the equations for the numbers I needed with exactly this idea!

[rfjihesrhydxdc]

Very entertaining. You're the Explosions&Fire of math