The Amazing Patterns of Modular Exponents

In this episode, I'll show you some clock-like number tricks that reveal patterns about exponents, prime numbers, "pseudoprimes", other bases, and more! See below for timestamps of different parts (and related episodes I've made)
0:00 - Introduction
0:24 - An Interesting Trait About Last Digits
2:47 - Connecting the Mathematics to Clocks
6:09 - Why 0, 1, 5, and 6 Did a Special Thing
9:30 - Interesting Symmetries Within Our Base
11:15 - Prime Mods/Bases and Fermat's Little Theorem
16:01 - Pseudoprimes and Carmichael Numbers
19:27 - A Further Generalization / A Chaotic Outro

Here are some previous episodes I've made related to modular arithmetic: (which can be watched either before or after this video)
(And many of my other episodes are also related to modular arithmetic in more subtle ways!)

Make sure you're also subscribed to my @Domotro channel for bonus content!

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 having any product placements from brands in episodes)
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Domotro
1442 A Walnut Street, Box # 401
Berkeley, CA 94709

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: exponents, modular arithmetic, modular multiplication, modular exponentiation, base ten and other numeral bases like base two (binary) and various others, mathematics related to clocks, prime numbers, pseudoprime numbers such as Fermat pseudoprimes, Carmichael numbers, Fermat's Little Theorem, Euler's (Totient) Theorem, and more!

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

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

[ComboClass]

In this episode, I'll show you some clock-like number tricks that reveal patterns within exponents, prime numbers, "pseudoprimes", other bases, and more! See below for timestamps of different parts (and related episodes I've made)
0:00 - Introduction
0:24 - An Interesting Trait About Last Digits
2:47 - Connecting the Mathematics to Clocks
6:09 - Why 0, 1, 5, and 6 Did a Special Thing
9:30 - Interesting Symmetries Within Our Bases
11:15 - Prime Mods/Bases and Fermat's Little Theorem
16:01 - Pseudoprimes and Carmichael Numbers
19:27 - A Further Generalization / A Chaotic Outro

Here are some previous episodes I've made related to modular arithmetic: (which can be watched either before or after this video)

See the description for more information and links. Love you all :)

[defenestrated]

This channel is great for learning because of the constant low-grade anxiety of chaos it induces, which boosts salience due to norepinephrine activation.

[TymexComputing]

Pseudo primes, pseudo perfect numbers - are my.... pseudo-favourites :)

[delwoodbarker]

I cry out to Domotro when something collapses at home.

[Nachiebree]

The clock thing makes me think of a dividing head (I'm a CNC machinist). It's a device used to rotate a part to a precise angle- there's a handle that you rotate, usually at a 40:1 ratio to turn the part. There's plates with a certain number of holes as well. For example, for 6 divisions you'd do 6 full rotations plus 10 more holes on a 15-hole plate.

[Salsmachev]

Had some fun playing with this in hex. A bunch of numbers, rather than repeating, will spit out a few distinct ones digits, then go to 0, where they stay forever. Four, eight, and twelve all do it immediately, since they have four as a factor, and four squared is hex. Numbers like two, six, ten, and fourteen all hit zeroes on the fourth power when their single twos finally multiply out to hex. Odd numbers don't ever hit zero like that. Seven, nine, and fifteen each repeat two digits, and three, five, eleven, and thirteen all repeat four digits.

[EPMTUNES]

Very nice video. I love how you make a discovery come about organically, then show the fact a little more abstractly and formally (like with the modular equations)

[Oni_Bread]

So glad i stumbled upon this channel. Unique and interesting. Keep it up.

[FirstLast-oejm]

nice touch with the intro edits, your videomaking has done nothing but improve since you started, keep up the good work, great video!

[darraghmooose]

Your videos keep getting better. Keep up the great work big man!

[LordMarcus]

The clocks may all be broken, but there's always time for Combo Class.

[ND]

Modular arithmetic is such a cool sect of mathematics; there's a lot of things you can use it for in ways you wouldn't think make sense to apply it to. For example, you can use it to show how many real number answers arise from exponentiating a root of unity by a certain power.

Let's say we have the equation z^N = 1, and we wanted to figure out how many real number answers we get when we raise each root of unity by power M. Well, we can take the numbers 1 up to N, multiply them by M, and take the results mod(N). If any result is equal to 0 (and N/2 if N is even), we've found a real number answer, corresponding to the root of unity 1 + 0i (and -1 + 0i for the even number case). This is mainly possible due to the symmetry involved in the roots of unity, as well as the fact that e^ix = cos(x) + isin(x), but you don't actually need to use those formula to do this calculation; just some simple multiplication and modular arithmetic.

It's really cool, wouldn't you say?

[GameJam]

Honestly, the one thing that always bothered me was when people looked at these kinds of patterns specifically in base 10 and went "WoooOOOOAAAAHHHH! I wonder why that happens!", when it would become obvious or at least far more clear if you looked at it in other bases. This is something I love about this channel compared to some other math ones, as it makes it clear that these patterns aren't magical, they're just fun byproducts of the number system we use by pure coincidence.

This isn't specifically a criticism of the video in question, as they DID vaguely touch on a subject related to it, but the video The Reciprocals of Primes - Numberphile discusses the quantity of digits in the period of a prime number reciprocal that occur before repeating. I decided after watching the video to attempt the same problem in different bases, and while I don't remember EXACTLY what my conclusion was (I lost the pages I noted them on in high school), there WAS a predictable sequence using the base B and an integer N to determine the number of digits in 1/N. And yes, all integers N, not just primes. I believe I was able to use the Totient function's multiplicative property in extension with this to show how even the number of digits in the period of composite numbers could be predicted using their prime factors, and by filling in the gaps betwrem the primes, it made it easy to see the pattern in different bases.

If this is enough information for someone else to give it another try, let me know what you come up with! I'm curious to see what the pattern with it was again. I might try it again myself, in which case I'll return here to give an update, but it's low on my priorities.

[publiconions]

Excellent work as usual Domotro and Carlo!.. love the vid

[FlockAndField]

This reminds me of a printout that I made of fractions in binary. Each number has a symmetry, kind of a Nyquist barrier, where when you cross the middle (i.e. looking from the 3 to 4th result of sevenths) the binary digits invert their pattern. It's like looking in the midst of a state change or something. I might not be using some of those words right.

[asseroy]

18:42
I kind of had goosebumps when I realized that the number 1729 is on the list

[sophiophile]

Did you come up with the grid/spreadsheet presentation style shown at 9:39 yourself, or did it come from a text somewhere? If so, what text- I'd love to read it.

Great presentation.

[carterwoodson]

alright, i love number theory, you finally earn my sub haha

[skoosharama]

I hope you've got a really good insurance policy. Thank you for risking life and limb for our edification and entertainment!

[eaenki]

I have one of those frog instruments as well, they go well with frog-based drugs, which are indeed very inspirational for doing math