The Amazing Patterns of Modular Exponents

[@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) About modular multiplication: kzbin.info/www/bejne/nnmbaqGLfMlpercsi=d0vqjcJfexWO3bCk About modular division: kzbin.info/www/bejne/h5nChJ2ajNN-psUsi=cJDZaMfSjYj0zcox About "Wilson's theorem": kzbin.info/www/bejne/oKmVonqig9ychpYsi=zi4TAhrDDxJfj1za See the description for more information and links. Love you all :)

[@luiswi]

Maybe pin this comment

[@ComboClass]

I usually pin that type of comment, but must have forgotten this time. Thanks for the reminder :)@@luiswi

[@defenestrated23]

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

[@aperinich]

ahhaha

[@EPMTUNES]

haahah

[@spugnoid]

Took the words right out of my mouth.

[@AntimatterBeam8954]

Definitely 😂

[@TymexComputing]

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

[@delwoodbarker]

I cry out to Domotro when something collapses at home.

[@Intelligenthumour]

Every broken clock empowers him.

[@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.

[@microwave856]

Happy birthday

[@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.

[@colbyforfun8028]

I imagine this happens in any base that is a perfect power.

[@Salsmachev]

@@colbyforfun8028 Yes it must. I am a bit curious, though, why seven, nine, and fifteen are different from the other odds. I wish he had discussed a little more why some numbers have longer repeaters than others.

[@gljames24]

Now I'm curious about Dozenal

[@Salsmachev]

​@@gljames24 Let's see... 0 and 1 repeat a single digit. 2^1 is 2, and 2^2 is 4, but after that it repeats 8 3 repeats three digits in the pattern 3, 9, 5 4 repeats a single digit 5 repeats two digits 5, 1 6^1 is 6, but then it goes to zero (first number to contain all of twelve's distinct prime factors) 7 repeats two digits 7, 1 8 repeats two digits 8, 4 9 repeats a single digit X^1 is X but then it repeats 4 E repeats two digits, E, 1 So that's 4 one digit repeaters, 2 one digit repeaters with a delay, 1 one digit repeater with a double delay, 4 two-digit repeaters, and 1 three digit repeater.

[@ComboClass]

We will discuss modular exponentiation more in the future. There are such a wide variety of patterns that I could fit a small portion of them into this particular episode :)@@Salsmachev

[@LordMarcus]

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

[@muskyoxes]

I only come here to find out what time it is, because i can always get a second opinion

[@Oni_Bread]

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

[@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)

[@GameJam230]

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.

[@pepebriguglio6125]

It would be of an extreme order of surprise, were the result not out there somewhere, yeah, practically everywhere. I could try Wikipedia, and probably find it there just about ten ('one;zero' in base 10) times faster than it's going to have taken me to write this reply. But it's low on my list of priorities ...😅😅😁

[@GameJam230]

@@pepebriguglio6125 I'm not sure what you mean?

[@pepebriguglio6125]

@@GameJam230 I'm just saying that what you worked out, and later lost, is most likely readily available lots of places online. It sounds like something that is already thoroughly established by academia long ago, with formulas and everything. Fascinating, I agree, but nevertheless easily retrieved online, if you don't want to again work it out yourself ... Other than that, I was only trying to be funny 😅😅

[@GameJam230]

@@pepebriguglio6125 I did look it up prior to attempting it the first time actually, but there weren't any direct answers to my specific question. Although, I DID just look it up again, and while it STILL didn't give me the exact question, it DID reveal an answer to the original problem that reminded me vaguely of how I did it in my version though. The method is as follows, at least for primes: Find the number n of the form b^k - 1 such that b is the base and k is the smallest integer that results in n being evenly divisible by your prime p. K is the number of digits in the period of 1/p written in base b. I tested it for non-primes, and it seems like the number of digits in 1/n is related to the GCD. Given f_b(n) returns the number of digits in 1/n for your base b, f_b(n) for a composite n can be described as f_b(p_1) * f_b(p_2)... * f_b(p_k) for all k prime factors of n, all divided by the GCD of all k terms. I'm not sure if this is the actual formula, but it works for at least a few I've tested, and makes sense as to why it works.

[@FirstLast-oe2jm]

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

[@ND62511]

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?

[@asseroy]

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

[@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.

[@mrwess1927]

By raising the exponent to a one higher power it is one higher dimension.

[@darraghmooose]

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

[@eaenki449]

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

[@ComboClass]

A viewer sent that frog instrument to me :) and haha

[@publiconions6313]

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

[@TymexComputing]

Mathematics by examples instead of prolonged syntactic transformations - thats what i like, it easier to find typos and mistakes when graphical WARNING - GRAPHIC CONTENT the clock's hand are being twisted ! :D

[@diversions5693]

serious question, how would a duel between a Pythagorean Druid (ComboClass) vs an Arcane Alchemist (NileRed) go down IRL and who would win???

[@ComboClass]

Probably depends on if/how we are allowed to prepare. If we have time to prepare weapons, I think a chemist would win, but I could win if we were just fighting mma style in the woods or something. Haha

[@diversions5693]

@@ComboClass do you think nile is ever unprepared tho...? you gotta watch what kinda shit he pulls outta that lab coat... even pocket sand !

[@ComboClass]

Honestly I've only seen a couple of his videos (although he seems cool and I will probably watch more sometime) and was just playing along with the joke. However, if he wants to organize a duel, I will certainly agree! haha @@diversions5693

[@TimPeterson]

3:20 ohhhhh "CLass" ok

[@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.

[@ComboClass]

Thanks. And I came up with those visualizations myself :)

[@skoosharama]

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

[@pennywerner9192]

Always interesting

[@ilovecats_og]

0:41 the cats are the best

[@TalsBadKidney]

Heck yeah dude Domotro for POTUS

[@ComboClass]

I could/would never be a politician, but thanks man hahah

[@TalsBadKidney]

@@ComboClass could ÷ would = will be

[@Piocoto123]

awesome!!

[@carterwoodson8818]

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

[@Colour_beaks]

20:24 What happened to Domotro

[@josemarti7487]

Congrats! Now you know more))

[@algorithminc.8850]

Nice catch at the beginning ... hehehe ... Cheers ...

[@MikehMike01]

I love you

[@Dr._Geno]

ok how long did you take to FILM this video?

[@ComboClass]

If you just count filming and filming-related preparations (not research, editing, or other stuff) then probably about 10 hours, spread out over a few different days

[@drh255]

There's an interior to that house? I had no idea.

[@djyotta]

I feel like this guy is trying to suck me in to a conspiracy theory..., but he's just doing math. 😄

[@pepebriguglio6125]

🎯💯😆😂😂😂

[@ComboClass]

Hahaha

[@MiaVilleneuve]

I would do anything to smoke a blunt with domotro

[@willo7734]

I would have liked math a lot more if they gave Combo Class back in school and not the math classes I took.

[@UlfFormynder]

Sudo primes are just prime numbers that you need admin permissions to use in your equations.

[@jan-pi-ala-suli]

this man’s house is falling apart

[@troodon2522]

clock

[@KennHamm]

I think this might be the weirdest math video I've seen on KZbin. I mean, the math isn't weird, that all makes sense like math is supposed to. But... what on earth is going on with everything else in this video?

[@gheffz]

TU^TU

[@hugoperhammer]

I'm a simple man. I see 1729, I press like. 😊😊

[@lloydgush]

WTF do you get all these old clocks.

[@Coffeemancer]

can you mail me a bamboo stick?

[@jonasan3]

how much money did you spend on your props? every video you have many things destroyed one way or another 😂

[@ComboClass]

Not very much money. I find most of my props for cheap or free

[@finnboltz]

First comment.