The Magnificent Patterns of Prime Signatures

Let me show you an underrated way of looking at numbers called “prime signatures”, and some cool connections I discovered with other topics like "partition numbers" and more! Special thanks to my Patreon supporters at / comboclass (top tier supporters are listed in this video's credits and all supporters are listed below)
Special thanks to: Tybie Fitzhugh, Henry Spencer, Mitch Harding, YbabFlow, 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, Max, Joost Doesberg, Adam, Chris Reisenbichler, and Stan Seibert.
more Combo Class links:
Bonus channel: / @domotro
Reddit: / comboclass
Discord: / discord
Disclaimer: Do not copy any physical actions you see in Combo Class episodes, including any use of fire, tools, or other science experiments.
If you're reading this, leave a comment with your favorite prime signature

[@ComboClass]

Thanks for watching! Consider checking out the Combo Class Patreon page I started this month: www.patreon.com/comboclass

[@good.citizen]

one thing I got from the class is counting by primes to infinity seems faster than whole numbers if that holds a bucket of water .

[@liminal3110]

Great video! I'm looking forward on the base 2i numbers still, please don't forget about them!

[@vincehomoki1612]

same

[@brinleyhamer729]

YOU HAVE BEEN NOTICED

[@qwerty11111122]

1:36 the reflection of the back of the whiteboard on the clock feeds suspense! Awesome camera-work!

[@idontwantahandlethough]

I frickin love your enthusiasm bro, you are a great teacher! I really hope you aren't too hard on yourself and are able to see the quality of the work you're putting out (I know I do that sometimes anyway lol). This is some really good stuff man! And judging by the comments, I'm not the only one that thinks so :) Keep it up

[@Fire_Axus]

your feelings are irrational

[@Qreator06]

We need more of that water hose in Comboclass

[@waddupbro]

It's always a good day when there is a new combo class video

[@chester_m]

The OEIS is packed with sequences related to this. Some of them are: 0:46 number of partitions: sequence A000041. 3:22 number of divisors: sequence A000005. 4:12 number of factorizations (like how many divisors can a number with 4 prime factors have): sequence A001055. 4:44 prime signatures: sequence A124832 6:12 smallest number having its prime signature (1, 2, 4, 6, 8, 12, 16, 24, 30, 32): sequence A025487 6:45 primes: sequence A000040 7:12 numbers with two prime factors (aka semiprimes or 2-almost primes): sequence A007774 7:27 squares of primes: sequence A001248 7:36 signature {1,1} (squarefree semiprimes, product of two distinct primes): sequence A006881 9:28 signature {3}, cubes of primes: sequence A030078 10:20 signature {1,1,1}, sphenic number, squarefree 3-almost primes: sequence A007304 10:49 squarefree numbers: A005117 15:10 4-almost primes (numbers in signature add up to 4): sequence A014613 15:16 numbers with 8 divisors: sequence A030626 17:33 5-almost primes: sequence A014614 18:13 numbers with 12 divisors (a dozen divisors): sequence A030630

[@ComboClass]

Yeah i love the OEIS and found a good table of those sequences on there! That was where I was able to find the most info on prime signatures compared to anywhere else online

[@Anonymous-df8it]

@@ComboClass Will it run out? What happens when it reaches A999999?

[@xyz.ijk.]

​@@Anonymous-df8it​ Neil is looking to raise funds for someone to take over after he can no longer run the site. I would think that include the programmer who can alter the numbering of all of the sequences. But he's got quite a ways to go so I wouldn't worry about it just yet

[@Fire_Axus]

your feelings are irrational

[@darreljones8645]

My favorite prime signature is {4,4,4,1,1,1}. Numbers of this form have exactly one thousand factors. The smallest number with this signature is exactly 810,810,000.

[@servvo]

you know it's a good video when it starts with a clarification for what "number" means

[@thedirtykitchenpysic]

Next video: negative prime signatures, complex prime signatures, Zeta-function roots of prime signatures

[@CMDRunematti]

Oh... Damn, i hope those are real things cuz they sound super interesting

[@karlwaugh30]

Negatives being in the prime signatures, at least in theory, would allow you to be classifying not just Natural Numbers but also rationale numbers. But then you could just as easily describe the Rationals by an ordered pair of prime signatures.

[@Domotro]

@@karlwaugh30 Yeah incorporating negative exponents into prime factorizations will be in a future episode :)

[@JirenSlr]

This sound very interesting, do you have resources on this?

[@thedirtykitchenpysic]

@@JirenSlr sorry, was just roleplaying a neural network. With a faint hope that CC could surprise us more

[@EPMTUNES]

I always appreciate your number theory videos. It’s such a fundamental part of math, present and past, yet we learn nothing of it in school. As such I still find there is so much to learn about it.

[@karlwaugh30]

Excellent stuff as always! I love how it's elementary stuff, but also kinda deep and beautiful. Also in this vein, if you extend to negatives in the prime factorisation you could extend this to the rationals. I'm not sure how usefully but in some sense for sure. It all reminds me a lot of Ring Theory, not that I've done tons of that since degree level.

[@ComboClass]

Yeah, I actually have an episode planned (which will come out in not too long) that involves negative exponents in prime factorizations and how that allows you to include rationals! :)

[@MuffinsAPlenty]

This video contains a decent amount of elementary number theory (if it were taught at an undergrad level, it would contain most of this information, but possibly in a different order). But number theory has a strong connection to ring theory! So it makes sense you feel a connection there!

[@sobertillnoon]

I'm impressed how well that whiteboard does near fire. I expected it to discolor.

[@Anonymous-df8it]

It did discolor

[@peppermann]

Brilliant. My wife isn’t a mathematician but she loved it too! She says you have neat handwriting but you need to wash your coat 😂❤

[@torydavis10]

Edit: I am dumb. That was excellent! For anyone interested, the deeper reason behind all of this and the reason why the 'add one and multiply' method works is really a matter of convention. As we define the number of prime factors, only distinct primes count. When we count the number of divisors, each identical prime is counted.

[@ComboClass]

Thanks, but that doesn't really explain the pattern as to why divisor numbers are an "add one and multiply" version of the prime signature. When you count the number of prime factors, you do include non-distinct primes within the multiplicity. And with the divisors, you count more than just identical primes, you count combinations of them. I didn't really explain the whole pattern (just the example with hypereleven-like ones) in the video, but to figure out why the "add one and multiply" method works, you'd have to look at how different sorts of picking items works in combinatorics, with how many combinations of a set of items you could pick if some of them were duplicates

[@coopergates9680]

@@ComboClass I suppose one could imagine each prime (or prime power) as a dimension, so each time a new distinct prime pops up, the already existing list of divisors can all be multiplied by the new prime, as many times as it's in the factorization, and still yield distinct divisors. For instance, a number like 18, which is 2*3^2, has divisors 1, 2, 1*3, 2*3, 1*9, and 2*9. Each divisor is formed from a product of the prime factors where the exponents range from 0 to the max found in the factorization, and all integer exponents in between are allowed.

[@angelmendez-rivera351]

Saying that it "is a convention" is false, and it betrays a misunderstanding of the theory. In number theory, we work with something called the p-adic valuation of a positive integer n. This is defined, for prime integers p, via the formula v[p](n) = max({m in Z : p^m divides n}). The canonical representation of n in terms of prime factorization is this written as Π[p(m)^v[p(m)](n)]. The prime signature S(n) of n is defined by the formula S(n) = {v[p](n) in Z : p divides n}. This is not a matter of convention: it is a conceptual distinction, which meaningfully separates powers of prime numbers from prime numbers, for example, and squarefree 2-almost primes numbers from squares of prime numbers. The Ω function is defined by the formula that Ω(n) = Σ(S(n)). The pattern in question has nothing to do with convention. The pattern in question is actually a theorem. The divisor function, d, which counts the positive divisors of n as d(n), is defined by the formula d(n) = |{ν in Z : m > 0 & ν divides n}|. It turns out that for all coprime m, n, d(m•n) = d(m)•d(n). Now, consider the ω function, defined by ω(n) = |{p in Z : p is prime & p divides n}|. Since n = Π[1 -> ω(n); p(i)^v[p(i)](n)], it follows that d(n) = Π[1 -> ω(n); d(p(i)^v[p(i)](n))] = Π[1 -> ω(n); v[p(i)](n) + 1], demonstrating the formula exactly. It turns out that the formula incorporates both the multiplicities, as well as the number of distinct prime factors. Both functions are equally as necessary in number theory. Actually, it is easy to see why: while Ω(n) = Sum(S(n)), ω(n) = |S(n)|. Both are defined in terms of the multiplicities anyway, the p-adic valuations.

[@torydavis10]

@@ComboClass Wow, it was downright rude of me to be so dumb...clearly I wasn't paying attention. Watched it again, I'm hip now. Take the signature {3,4}. The divisors will have signature {a,b} where a is from [0,3] and b is from [0,4], for two independent choices from 4 and 5 options respectively. At least, I think I'm hip?

[@torydavis10]

@@angelmendez-rivera351 You are right, but I think it would have been more accurate to say my blatant misunderstanding belied a misunderstanding. Sadly I am not fully fluent in this notation so your comment is less illuminating to me than it should be. It's not entirely greek but I can't quite parse it all either. Does my reply to combo indicate that I pulled my head out of my ass?

[@HipNerd]

Who knew a video about prime signatures would have such a brutal ending. 🙂

[@shreyjain3197]

very interesting highly underrated channel

[@ahobimo732]

The crazy person vibe could not be stronger. Love it.

[@RandomAmbles]

I assure you, it can. It most certainly can.

[@ahobimo732]

@@RandomAmbles I should have be more specific: The crazy person vibe could not be stronger - while still maintaining a coherent, informative explanation of mathematical concepts. Setting aside this minor misunderstanding, I agree with you. Crazy person vibes can ALWAYS get stronger.

[@WhattheHectogon]

Never heard of prime signatures...let's go!

[@xjoke6981]

you were pretty chill this video

[@seedmole]

This is great, reminds me of a lot of higher level geometry things like making patterns by reusing midpoints as vertices and such. Endless ways to generate new, related data from preexisting things.

[@nathanvandevyver]

let's make this channel grow

[@monoman4083]

you're on fire with these videos ...

[@GameJam230]

Wow! I didn't know this was a real thing people of studied. I discovered parts of this myself in Grade 10

[@braznartn5176]

The mad scientist is burning up... OMGsh... I better pay my dues... don't wanna miss out on the fun.

[@Guil118]

He has the same voice as Kevin from Vsauce2.

[@calebclark9114]

Spectacular video!

[@good.citizen]

thank you for the number video

[@kokulanselvakumaran]

Great video!

[@MayorVideo]

i saw it and clicked without even reading the title lol

[@braznartn5176]

Hey folks... another EINSTEIN is among us!!!

[@hkayakh]

0:29 oh no don’t put the leaves in the fire! They create tons of smoke!

[@HomieSeal]

Absolutely loved how everything came together at the end! (Although at the very end it must have been quite cold for you) Also, when I saw Squarefree on the board, all I could think about was Stone Free lmao

[@balbarard4041]

alien level knowledge 👽

[@absxn]

nice

[@isobarkley]

bart, say the line! "hyper elevens..."

[@echoawoo7195]

I can see the script in the new clock xD

[@Idekaks]

Have you considered making a video on the Collatz conjecture?

[@rogerkearns8094]

Interesting.

[@evenaxin3628]

Going with your idea for hyper elevens you can actually do the same for all others, but if there is one that is a power of something it adds that many options but you can only choose one of them at any given time. EX (2^2)*3 where 2^2 is a¹ and a² while 3 is b: 1:{ } 2:{a¹} 3:{a²} 4:{b} 5:{a¹, b} 6:{a², b}

[@Anonymous-df8it]

Prime signature= {2,1}. For the first factor, there are three states: not present, contains one, contains two. For the second, there are two states: not present and contains one. In general, if you add one to each number in the prime signature then multiply the new numbers together, you get the number of factors of a number!

[@evenaxin3628]

@@Anonymous-df8it I'm sorry, I can't understand your reply as it's worded.

[@Anonymous-df8it]

@@evenaxin3628 What parts don't you understand?

[@evenaxin3628]

@@Anonymous-df8it I was just waking up at the time, that is what he already explained in his video, but he was explaining squarefree prime signatures through lists so I was doing the same for other signatures.

[@karlwaugh30]

I'd be interested in functions that act on them. Eg. A function F that took a set Signature S to { s +2 : s in S } Or G that took S to something like itself twice so { 3, 1, 1} -> { 3, 3, 1, 1, 1, 1 } What does these representations of these functions tell us about them? Compared to more traditional representations of the same functions.

[@Anonymous-df8it]

That just becomes functions between sets of integers though

[@karlwaugh30]

@@Anonymous-df8it it does in principle yes. Although I think it's relatively "easy" to convert to N-> N by preserving the primes referenced by the exponents. Functions that are extremely easy to define in terms of Signatures (S + 1) have more interesting behaviour when viewed over the natural numbers - all the sphenic family of numbers get squared but other numbers rise by a lower amount - essentially each number is multiplied by it's "sphenic root" (aka just it's prime divisors multiplied together). Functions that take signatures to 2s result in simply raising the number to the power of 2 Functions that square the signatures have a more complicated (almost factorial like) behaviour that motivates the definition of other functions acting on the natural numbers. These are merely a few observations I made in an hr or two after watching this video. I think there's some interesting numerical behaviour going on.

[@Anonymous-df8it]

@@karlwaugh30 What about the other direction? What would adding one (in positive rational number space) do to a prime signature?

[@karlwaugh30]

@@Anonymous-df8it well I think that would be really hard to know... Like a full answer would probably be akin to knowing where the primes are going to occur etc...

[@Anonymous-df8it]

@@karlwaugh30 Couldn't the same be said for your suggestion?

[@toanpham4110]

✅

[@Klaevin]

"back when KZbin was just a guy with a camera"

[@cabobsstopmotion4983]

6:40 me too

[@AlexBaklanov]

Nothing have fallen at the beginning))) but all happened at the end. Nice patterns! Did you discovered them?

[@ComboClass]

I read about prime signatures myself. I figured out the relationships with the partition function and other combinatorics property myself. I couldn't find a huge amount of easily available information online about prime signatures, they should be more well known!

[@AlexBaklanov]

@@ComboClass Wow! That's impressive! Looking forward to all negative Grades and Grade i =) GL to CC

[@iloveyouuu3]

i love you

[@kabalofthebloodyspoon]

Yes, 24, the highest number. Like you got ten. You got ten more and four after that. Forget about it 😉

[@Illure]

Great video! Although, I can't help but get distracted by the fire... Man fire looks different in a video than in person. It's more disjointed and less lively... Wait what were you talking about again... Sigh...

[@KnuxMaster368]

Combo class hasn’t covered Combinatorics? 🧐

[@evenaxin3628]

Is there a hyper-eleven super-eleven?

[@ComboClass]

Great question! Yep a good amount of them (at least base ten hyperelevens) have prime signatures like (1,1) or (1,1,1) or etc

[@evenaxin3628]

@@ComboClass I didn't know how I was asking, I meant numbers like 11111 that have prime signatures of only the digit 1. (ignoring of course primes.) are there any primes with only the digit 1 besides 11?

[@deltalima6703]

😜 Head spinning

[@datbubby]

:)

[@Jkauppa]

1^inf, has only itself as the prime one (1)

[@Jkauppa]

in the case of one, separated from zero, you need to include/show the one

[@Jkauppa]

if you dont show the one, what number that is

[@Jkauppa]

one must be included in the signature, contradicted/shown by the case of number one (1)

[@Jkauppa]

one must have its own composition signature, it cannot be empty

[@Jkauppa]

be consistent. 1 = 1^inf or just one (1)

[@mackwebster7704]

😭 Promo>SM

[@ChaverYadim]

are you homeless?