The Fundamental Families of the Most Divisible Numbers

[@ComboClass]

Let me tell you about an interesting web of number families that contain the most divisible numbers ever.... 0:00 - two mysterious sequences 2:45 - two super-divisible sequences 5:00 - an assortment of related number names 6:37 - explaining the first layer of sequences 7:17 - explaining the second layer of sequences 9:50 - explaining the third layer of sequences 12:47 - ways that these numbers could be important 15:40 - outro/credits Previous episode about fractals: kzbin.info/www/bejne/jHaQnX6gbtl5as0 Thanks to my Patreon supporters (and supporters on the KZbin memberships on my @Domotro channel). See the credits at the end of this episode for the names of my supporters. Consider supporting Combo Class here: www.patreon.com/comboclass

[@deltalima6703]

15:00 thats because you are an american. Just switch to metric already you troglodyte! 🤣

[@thewatcher7579]

I think we can all agree this man is a wizard

[@Salsmachev]

Larger highly divisible numbers are great for units (personally, I think that a mile should be 5,040 feet) but they kinda suck as bases. The only way to make them work is by using a mixed radix system, and I think people severely overestimate how useable mixed radix is. Even the relatively friendly case of base 60 (split into a seximal and a decimal component) requires you to memorise your base 10 times tables twice (once in base 10 and once in base 6). For a base like 5,040 you would likely need a triple-radix like 21:15:16, meaning you would have to learn your base 21 times table three times over. The effort required simply isn't worth it when you can just use a binary compression base (I like hex), learn how to use binary divisibility rules, and have the best of both worlds.

[@bjorntorlarsson]

7×11×13=1001 is nice. There's also 2×3×17=102. Every multiple of that, call it XY, will have 2X=Y. As in 2×1=02, so 102. Like 2×43=86 so 4386 is divisible by 17 seen instantly. Works simply up to 5000 when Y gets a third digit that overlaps with X. So it's 5100 as in 2×50=100. And 10098 as in 2×99=198. 17 isn't quite as pesky a divisor as it looks.

[@ibrahimali3192]

7

[@publiconions6313]

Cool stuff. I love how Domotro took those couple of seconds to point out the 1001 thing. Even though it's base dependent, still cool... And I bet there are other small sets of prime factors in other bases that function in similar ways

[@PhilFreedom]

Yeah man! Great vid. Love your work. I, too, feel that Super Abundance and High compositability should be celebrated in number. Factors are like building blocks for measuring, making time, space, rectangular arrays in packing and shape (even 3D space), the stars or astrological houses, and more, easier to work with.

[@soninhodev7851]

720720 is probably my favourite out of all these larger numbers... though my favourite overall is 2!

[@yumm_6]

when 2 factorial is equal to 2

[@SirSX3]

Can you explain why 720720 is your favourite, or where I can find out more about it? Thanks.

[@soninhodev7851]

@@SirSX3 well... idk, it's because i like the "normal" 720, for a variety of reasons one of them being that it is 6! (pronouced "six factorial") among other things, and that a number who is a mutiple of 720 is cool! (720720 is obviously 720*1001 with 1001 itself being interesting as it is 7*11*13 meaning that 720720 is actually 7!*143 = 5040*143 = 720720)

[@SirSX3]

​@@soninhodev7851 Thanks for replying! I've also learned that 720,720 is divisible by the first 16 numbers as it is their lowest common multiple!

[@SirSX3]

​@@soninhodev7851 Thank you for replying! I've also learned that 720,720 is the smallest number divisible by the first 16 numbers as it is their lowest common multiple!

[@alexdamman6805]

Up my alley, baby, and highly entertaining as always. I particularly appreciated the discussion of real world application. It is real and also important. Everybody, please do keep these numbers in mind when you need a large number to work with.

[@publiconions6313]

Almooooost 50k guys!.. love your work so much!

[@bjorntorlarsson]

On the format of this show, I must say that I am also crazy, and I too get away with it (mostly) by just pretending that I am crazy. That's confusing enough. But a good advise is to marry someone who doesn't care enough about you to figure it out. Or someone who also pretends to be crazy in order to hide the true craziness in plain sight. I got a job where the boss hired me, as a specialist advisor so we met alot, by at the end of the interview saying that he likes my sense of humor. (That sounds good!) When he years later laid me off because of the general economic downturn, he said that he always liked my sense of humor. I looked him in the eyes and in the most serious way said: "- I never made a joke."

[@sharpshooter-ot2ew]

i see a sequence within the smaller terms that reminded me of resolutions sizes ie 360, 720, 1440 or close enough ie 2160 to 2162, 120 to 160 that I still considered them for a second lol. Anyways it is probably used for graphics optimization to varying degree; great content & vid. Update: After a quick reference 160px V 120px H was used before at 4:3(1.3) aspt ratio Quarter-QVGA allowing for 80x30 character text and graphics.

[@Nixitur]

Even just the existence of that first sequence means that a superior highly composite number is always a divisor of the next superior highly composite number. I think that's really interesting, and not at all obvious! I know this isn't true for highly composite numbers in general, after all, so where is this property coming from? Google has been spectacularly useless in this, so I'm curious about the proof, though it's probably far beyond me.

[@stickfiftyfive]

One can take the pi-product of p^ɛ, for any prime p, with p raised to some epsilon ɛ = (floor(1/p^(1/x)-1)), for any real x, where that equality is a monotonic mapping from the reals, in order to generate the set of SHCN. You can see from this that each element will have the last as a divisor, and hence the sequence Domotro showed on top exists, and is cool.

[@stickfiftyfive]

Simply put, similarly to factorials and primorials, the prime factors of any member multiply into every greater member.

[@Saol.Alainn]

How do you have so few subs compared to your second channel?! Either way, thanks for sharing so much interesting knowledge. I love hearing about deeper(?) math like this. I need to get back into school, start doing something with my mind again. Construction just isnt it lol.

[@ComboClass]

The subscriber difference is because I post "shorts" on the bonus channel and some of those have gotten tons of views from random people on the shorts page

[@jasonhildebrand1574]

@@ComboClass I thought that I enjoyed Number Theory, but you constantly remind me that I truly love the subject !

[@BhbtheRock]

This one is for my exact tism

[@caspianmaclean8122]

Before watching the video, I noticed the sequences had the prime factors for 2, then 3, then for 4,5,6,7,8 but not 9. Okay I'll see if that gets explained by the end.

[@adambarlev8992]

Is this a sign for me to get into combinatorics? I've always been curious.

[@X3MgamePlays]

Aliens will have everything different. Their second will be different. Their light year will be different. Their "kilogram" will be different. Their "kilo" will be different. etc.

[@Gamr-bc6kp]

Play at double speed for mathematician ben shapiro simulator

[@deltalima6703]

BS is dead to me, I will pass. Lol

[@ckq]

I watch everything as 2.5x this is pretty slow

[@HipsterShiningArmor]

In a better world, we would round to the nearest highly abundant/colossally abundant number instead of to the nearest number ending with a bunch of zeroes. Fwiw personally i think base ten, though not perfect, does the job well enough. I think the real problems are decimal notation and our cultural obsession with numbers that look like 100000… (etc)

[@IvoKintobor]

"base 6 would be the best base for humans to use" 2 things 1, are you saying you're not human? 2, base 12 gets my vote (i see you said that'd be good too) simply because you can use your thumb and count the segments of each finger... makes it easy to count with your fingers still, the same way base 10 is

[@thesharpestknife]

You can easily use your fingers to count in base 6 as well. Each hand is a digit, which just needs to be a number 0 to 5. If you want to know why base 6 is a popular choice of base, you should watch "a better way to count" by jan Misali. Those who want to explore this rabbit hole even deeper should watch "the best way to count" as well. It's essentially a response video to "a better way to count" arguing why the best base is 2.

[@felipevasconcelos6736]

Seeing someone saying "base 12 is actually the best" fills me with rage. Say DOZENAL. Or even duodecimal. Base twelve. Or even base DEC12. But if you're talking about a base different than decimal being better, don't use implicit decimal. Either use a different base or explicit decimal. In a discussion about different bases "12" is absolutely meaningless.

[@Saol.Alainn]

​@@felipevasconcelos6736 I was confused for a minute, that's actually hilarious xD

[@samuraisecretary]

​@@thesharpestknife American highschool basketball jerseys are in base 6. This allows the referee use simple hand signals to reference each player.

[@cameronbigley7483]

@@thesharpestknife Both are great, but I always had problems with the finger counting to visually show information. Using the thumb as the most/least significant digit *will* cause the same issues as "thirsy two" v. "dozen three", if you forget which is which, or how the system is.

[@bpark10001]

You did not explain how to derive the sequences!

[@publiconions6313]

I know Standard measurement system gets a lot of flack from the Metric people... Buuuut, there are some reasons. I could be convinced either way

[@leeprice133]

What does 'standard' mean here. Because if there is a standard, it's the metric system.

[@publiconions6313]

@@leeprice133 it's capitalized because it wasn't meant as a adjective. Heh.. but I think you knew that and were just being snarky. : )

[@JR13751]

I think primorials are most important.

[@stickfiftyfive]

What constitutes the property of importance, in what context?

[@TymexComputing]

Like

[@pennywerner9192]

1st😊

[@evandrofilipe1526]

2nd

[@charlesgaskell5899]

4th

[@Toenailslikkepind]

8th

[@soninhodev7851]

16th

[@deltalima6703]

32nd

[@bjorntorlarsson]

Depending on seemingly random political events in the Middle East (and even more so in the West), this whole base 6 and base 12 Babylonian stuff might suddenly be classified as terrrorism and banned.

[@mienzillaz]

Hmm.. so much definitions and so little practical examples.. worst episode through all combo classes..

[@landsgevaer]

Squirel, cat, geese, and whole lists of numbers; what more do you want...? 😉 This is class -3: it gets more negative all along. So does your comment, apparently.

[@issholland]

Idk man, I never made any friends learning base 6 but it does lead to unique things in game design. I think all of these videos ->you

[@bjorntorlarsson]

What do you mean "practical"? The only practical with number theory that I can imagine is memorable patterns for mental arithmetics when watching a table or a diagram or when shopping based on cents per protein content per ounce.

[@bjorntorlarsson]

Take a look at 𝕌, the undefinable set of numbers! Not much definitions there, as you wish for. But on the other hand, not many examples either. (𝕌 is btw an undefined letter in the blackboard bold font of math, so I thought it fits here).

[@stickfiftyfive]

Disagree. Practical examples were given, if learning about the mathematics of the sequences is practical, and in the field of math, it is.