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- เผยแพร่เมื่อ 1 ก.ค. 2024
- Number theory is a field of mathematics that dates back to the earliest civilizations, but why is it so notoriously difficult? In essence, it boils down to addition and multiplication each cannibalizing the nice properties the other brings to the table.
Notes:
Scaling a vector by 0 always produces the 1-vector, corresponding to the fact that n^0 = 1 for all integers n.
I apologize for the audio discrepancies.
#math #numbers #addition
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good vid but the inception sound blew my ears out
0:43 true
Thank you so much for the warning.
and 3:42 watch out
that's the point😂
Did we need the day of judgment?
The real problem is that we have way too many numbers.
Infinitely many
Well it's not a real problem, i'd say it's more of an integer problem ;)
it feels like there's just no end to them
@@cosimobaldi03 indeed, the real problem is that we can't even count them.
@@__christopher__ What is it with these puns 😂
put a normaliser on your audio the loud sounds are rather jarring
Thanks for the feedback.
@@artkalbphd don't listen to him that shit was awesome
@@Benevezzioficiallol. Maybe normalize the volume, and people like @Benevezzioficial can turn up their volume.
@@Benevezzioficial lmfaoooo
dam OP figured out the whole point
Factor space can additionally represent all positive rational numbers, if you allow negative components in the vectors
In music theory (specifically tuning theory in just intonation), doing this creates a monzo. The monzo has a dual structure called a val that interacts with monzos to create a robust reason as to what intervals map to what steps in equal temperament in such a way that you can build chords in a consistent manner.
As such, monzos are typically written as a column vector/ket, while vals are written as a row vector/bra.
And all computable numbers if you turn it from a vector space to a cauchy space (all limits are in the space)
Dude, I spent years thinking the factor space idea was an original thought of mine. Damn!!
Ideas are discovered... Not created
@@Benevezzioficial Ideas are created each and every time anyone considers them. We THEN discover that the idea has a real meaning.
Me too with other ideas 😓 It's common in mathematics
There's a bit on Wikipedia somewhere that talks about being able to encode square-free integers in binary, using the ones and zeros (and that's all there can be and remain square-free) of the prime exponents, which I think is where I first came across it. It extends to being able to encode all power-n-free numbers in base n, and so on to infinity.
This then leads to noting that there are other enumerations of vectors of fixed length (or infinite but the tail is all zeros after some point, which amounts to the same thing). Examining these may help derive insight. "May" being important here. I don't have the necessary brain to work such wonders.
@@DeJay7 I see ideas as 'possible states' of the elements that construct them. Even bad ideas are possible states based on incomplete/false/conflated elements.
Like picking a fruit from Plato's plane of form
This is *fascinating*, awesome video! I’ve taken a handful of higher math classes (and watched a lot of math YT), but I’ve never seen number theory expressed as a semi module before! That makes so much of the random tidbits make sense, and is just so cool.
This is my favourite way to view the primes. I always want to see how many traditional properties of primes simply emerge from ordering these vectors without knowing anything about the underlying arithmetic.
Fundamentally, one could do the same with polynomials. A really nice case are F2 polynomials since they track neatly with binary numbers as well
Indeed. This way of representing natural numbers or polynomials as vectors is knows as "valuation theory". More precisely, the coordinate of the vector corresponding to a prime p is called the "p-adic valuation". As someone commented below, one can also allow for negative entries in the vector to allow for elements in the field of fractions (rational numbers or rational functions). This allows one to work with modules rather than semimodules, and it works for every field of fractions of a factorial ring.
The fact that gcd(a+b,a)=gcd(a+b,b)=gcd(a,b) means that for the sum of two arbitrary numbers, for those entries where they differ, the sum has the lower of the two as entry, while for those where they equal, the sum may have an equal or higher entry. The fact that a and a+1 are orthogonal is a special case of that.
Indeed it seems that if you impose the natural partial order that (1) increasing any component gives a larger vector, and (2) increasing by the same amount farther right gives a larger number than increasing farther left, it seems that the simple rule that every new number gets assigned the smallest available vector compatible with the previous assignments under above-mentioned addition rule already fixes all assignments. I tested it up to 10 at least.
@@__christopher__ That seems interesting! Would you mind posting some examples you've worked out?
@@ChaseRoycroft wel, we start at 1 (obviously 0 can't be represented this way). Since we haven't yet used up any representation, we use the smallest one that exists; that's unambiguously (0,0,0,0,....).
Next comes 2, which is 1+1. The addition rule gives no restriction, so we look for the smallest representation not yet taken, which is (1,0,0,0,...).
Next comes 3=2+1, where the addition rule forces a 0 on the first coordinate, which makes the smallest allowable unambiguous: (0,1,0,0,....)
Now it gets interesting: 4 can be written as sum in two ways: 4=3+1=2+2. The first sum enforces the second coordinate to be 0, and the second sum forces the first coordinate to be at least 1. Again the smallest allowable non-taken vector is unambiguous: (2,0,0,0,....).
5=4+1=3+2. The second sum enforces the first two coordinates to be 0, the first doesn't add another restriction. The smallest allowable vector is (0,0,1,0,...).
since S is not linear on the vector space, I would avoid dropping the parentheses around the argument.
you write at a few points stuff like S²p, which is notation usually reserved for linear operations.
a quick proof that S isn't linear:
a²+1 ≠ (a+1)²
=> S(2A) ≠ 2S(A)
where A is the vector corresponding to a
love reading the comments and seeing how many other people independently thought to represent the naturals like this
there are only 48 comments so far (+1) But having only basic knowledge in fundamental math, I asked me the same question, but now I have a lot to follow ...
Factor space is great! It intuitively feels like there is so much potential mathematics to discover in relationships between numbers and patterns/functions in factor space. I haven't seen anyone else using it at all.
I started experimenting with it in relation to the Collatz conjecture and made a couple of interesting "discoveries." A really simple and fun one is that each factor has its own predictable fractal pattern as you contually apply the successor function. If you take the component of 2 of numbers 1,2,3,... etc. you get the sequence 0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,... etc. Every time you double the length of the sequence, you duplicate the existing sequence and then add one to the final element. This self-similarity in the sequence becomes very interesting when you start to think about integers with a potentially infinite number of digits (similar to p-adics) as they will still have a finite component of 2 (so long as they are a p-adic infinite power of 2.) It gives you a way to think about all possible numbers at the same time when attempting to prove theorems about repeated application of multiplication, such as with Collatz. When you keep dividing by 2, that component of 2 will hit zero, but your local space of integers on either side will share have a similar overall shape (hard to explain but if you picture the numbers as "peaks", the local neighbourhood will have the same peaks as your older neighbourhood did, just on a smaller scale with smaller numbers. In particular, the side on which the taller peaks are will not fundamentally change.)
Note that the sequence works in a similar way for higher factors, the sequence is just repeated self-similarly k times where k is the factor e.g. for 3 it's 0,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,0,3,... etc...
Another interesting note on Collatz is you can actually change the (3n+1)/2 rule when n is odd into a new rule where only the multiplication is repeated:
1. add 1 to n
2. multiply n by 3 and divide by 2
3. repeated step 2 until n is odd
3. subtract 1 from n (and go back to dividing by 2 until n is odd again.)
After you add 1 to n, you can look at the component for 2 and it will tell you the number of times you are going to need to repeat step 2. This is because each repetition of step 2 is dividing by 2, so the component of 2 will fall until it hits zero and the number becomes odd. Interestingly, this means that the highest number of possible repeated (3n+1)/2 in a row is always the exponent of the next highest power of 2 greater than n, and the number with the most will always be the one that is 1 less than that power. Note that each repetition of step 2 is adding to the component of 3, which relatively moves the number within its range between powers of 2, unlike dividing by 2. (All combined, this means your local neighbourhood stays similar in the sense that the shape of the valley you are in will stay the same, but which side the taller peaks can be found on will change predictably.)
This all comes together to give you new ways to reason about the Collatz conjecture and what is really going on under the hood. It opens up the potential to define better understand which patterns are actually possible in the hailstone sequence and how patterns in the sequence in smaller numbers can relate to similar-looking patterns in larger numbers. In particular, you can see how there is ain interesting pattern to where numbers in lower intervals of powers of two will end up positioned on higher intervals.
Anyway, this is where it's really out of my depth, I don't have the mathematical skills or patience to take this any further and find exactly how the neighbourhoods change when the sequence is applied.
@@isogash that modified Collatz rule is a very interesting finding. In tems of the vector, the repeated multiplication with 3/2 means that you add the first component to the second and then set the first to 0. Repeated division by 2 of course just means setting the first component to 0. All other components are affected only In the increment and decrement step.
I think 'addition ruins factorization' explains better, which I think is true in all nontrivial cases. And number theory is just very about factorization.
A lot of number theory is not about factorization at all. Or even about divisibility.
@@rosiefay7283 what is it about?
*@[**0:11**]:* They are all divisible by 30.
This video was recommended to me out of the blue. Not my regular piece of feed but I was intriguided by the title and also have some interest in number theory. But I was a bit disappointed by this because this was like an extremely high level overview of a lot of math jargon. Like vector, linear algebra, vector space basis, mobius function, totient function. Felt like a half-hearted attempt. Would have loved some details and intuition behind the theorems, application etc. rather than just taking the name of some random math concepts. You earned a subscriber and a like, hoping for better content.
good video content, but i'm kind of confused as to how it explains why number theory is hard (the description helped me understand, but i don't feel i got it from the video).
Because most theorems and things we know how to solve talk about multiplication. Statements involving addition don't connect nicely with things we already know so we are left in the dark for most of such questions
Wow its crazy that I got recommended this video just after having the idea of factor space myself. Very cool to see how its used :)
I stopped at 3:41 when the music blew out my speakers. This is hostile to your audience.
Awesome way to look at primes!
So highly informative ^_^ And the music that accompanied the introduction of the successor function is very fitting. Thanks for the explanations.
I think there a, while trivial, interesting philosophical observation to note - Raising the number below the arrow to a power calls for the multiplication operation; Multiplying the number below the arrow with another number calls for the addition operation; Both call for their 'primal' origin. Adding a number to the number below the arrow should call for whatever operation it is that is 'primal' to addition.
And this is where is gets beautiful and complicated.
i immediately disagree cuz addition is kind of the base for everything but yt shoved this vid infront of me so many times that it cant be that bad
There's a beautiful interpretation of the Lifting the Exponent lemma in this space
3n+1. Whoever develops the theoreticap framework that solves this will be laying the groundwork for understanding addition and succession in this space.
The fundamental theorem of arithmetics is said to be corollary of the book VII proposition 30 of Elemeta, which rests on the proposition 29 which proves and defines coprimes. Translations from Heath's edition:
PROPOSITION 29.
Any prime number is prime to any number which it does not measure.
Let A be a prime number, and let it not measure B; I say that B, A are prime to one another.
For, if B, A are not prime to one another, some number will measure them.
Let C measure them.
Since C measures B, and A does not measure B, therefore C is not the same with A.
Now, since C measures B, A, therefore it also measures A which is prime, though it is not the same with it: which is impossible.
Therefore no number will measure B, A.
Therefore A, B are prime to one another. Q. E. D.
PROPOSITION 30.
If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers.
For let the two numbers A, B by multiplying one another make C, and let any prime number D measure C; I say that D measures one of the numbers A, B.
For let it not measure A.
Now D is prime; therefore A, D are prime to one another. [VII. 29]
And, as many times as D measures C, so many units let there be in E.
Since then D measures C according to the units in E, therefore D by multiplying E has made C. [VII. Def. 15]
Further, A by multiplying B has also made C; therefore the product of D, E is equal to the product of A, B.
Therefore, as D is to A, so is B to E. [VII. 19]
But D, A are prime to one another, primes are also least, [VII. 21] and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore D measures B.
Similarly we can also show that, if D do not measure B, it will measure A.
Therefore D measures one of the numbers A, B. Q. E. D.
Greek pure geometry did not consider that numbers really exist. Numbers are just measurement partitions of continuous geometry. The measurements are purely intuitive deductions, drawings in the sand are just a tool to aid intuition and intuitive coherence. What is the origin of coherence for Greek mathematics? The ontology of Elementa is holistic mereology, which human mathematicians perceive as projective decompositions.
What does this mean? Euclid's definitions and proofs lose their meaning, if they are detached from the ontology in which they were constructed. Modern approaches through Peano axioms are not corolaries of Euclid, because the axiomatic language games starts from the reductioistic direction of claiming that the idea of numerical object has independent existence merely be arbitrarily declaring so.
Over the years I've tried to get a very basic understanding of number theory...this has opened the door for me to start again.
It's a trivial proof, but my favorite rendition of the irrationality of root 2 is what introduced me to p-adic valuation.
Suppose for contradiction that root 2 is rational -> there exist n,m in the naturals s.t root(2)=n/m
2 * m^2 = n^2
v_2(2 * m^2) = v_2(n^2)
2 * v_2(m) + 1 = 2 * v_2(n), which are Odd and Even respectively.
The log-like behavior of v_p is because it's working in "factor space" - ie, the scaling operation at 2 minutes.
Darn, just a couple of days ago I had an idea for which I had to come up with a similar vector construction
Fascinating!! I've only taken basic number theory through my discrete math class last semester but nonetheless enjoyed this video! Not a pure math major by any means (just physics) but I'd be heavily interested if I could find an intersection between it and number theory.
Great video!
Alternatively, maybe multiplication ruins everything.
So you watch loads of numberphile and think you have a fair handle on number theory and then you see the prime factors as vectors for the first time
What is the name of the vector with prime inside i want to know more,does this prime vector have a magnitude?
What will you see, if you look to the factor space from direction (1, 1, 1, 1, 1, 1, 1....)?
Your factor space and its notation exactly matches my discovery/invention of the same thing in 2014. I am happy that I am not alone!
Yeah, repeatedly re-discovering known math makes me want to try and read current math journals-see if I'm just reaching the obvious conclusions that others already immediately discovered when the known facts lined up, or if I can actually independently discover something new and interesting.
One of my favorite re-discoveries is that you can generate the Stern Brocot tree by multiplying successively by one of two 2×2 binary matrices.
What, if the dimension of the vector itself is prime? Then we can add /subtract the projection of the units vectors and they end up at the unit circle? just be accident?
At 3:23 is there an error in the typing? Should it be "x" in the brackets not "+" ?
I'm in the math Olympiad world and number theory is the hardest one for me because of how technical it is, you can't even understand the problems or even have an idea of what to do if you don't have experience nor any solid basics but when you get used to it's pretty manageable, you can solve problems and keep learning more without as much difficulty as as when you started. Also I love how mysterious it is sometimes but anyways good video ❤.
I really enjoyed hearing Verdi's requiem out of nowhere. You don't here that everywhere. However it almost woke up my brother xD
This factor space framework, it's not new is it? I'm not familiar with the area of study. But having linear algebra experience that'd be my thought proces for representing integers as infinite dimensional vectors, whose span of the primes is the whole space.
I've been thinking about this "factorspace" for quite some time! I've been representing them not as vectors but as a number system that isn't quite positional. For example, six is 11 (3^1 * 2^1).
Have you ever noticed that twelve is 12 (3^1 * 2^2)? I haven't found any numbers like this up to 100,000,000,000. What do you think?
The squarefree numbers are the numbers with no repeated prime factors, in other words, the semimodule in question over the field with two elements. This is a "lattice" which corresponds to a n-dimensional hypercube. The squarefree numbers therefore live in a square.
As someone tackling 2^a +1 (which ones are prime) yes addition in fact does ruin everything
Wait, if 19×…×54 = 23 ×…× 57, can't you just remove the explicitly common factors ([23,54]) to get two ranges that… (re-watches the beginning of the video). Ah, that's actually one of the known examples. I'm thinking now that must have been implied.
Has anyone found a pattern for which all numbers are constructed?
What was that random Quidditch World Cup theme at 3:41 about?
I just properly learned linear algebra this week and I'm going crazy realizing that everything is representable via a vector space. It's good fuel for my thesis, but not so much for my sanity.
Fun fact. Microtonal/xenharmonic music theorists affectionately call these vectors "monzos" and do all kinds linear and geometric algebra with them, slightly tweaking how large the prime numbers are. This results in tempering e.g. we can make it so that (3/2)^4 ~ 5/1 or equivalently [-4 4 -1> ~ [0 0 0> which is the Regular Temperament Theory way of specifying the meantone tuning system prevalent in Western music (meantone temperament predates the now ubiquitous 12-tone equal temperament which is itself a special case of meantone).
Oh wow! By Euler ... the video starts AMAZING!!!
I legit read the thumbnail as addiction ruins everything
I haven't seen this kind of vector representation of natural numbers ,
Is there any book that explains more about this topic?
It’s just the coefficients of log(p) in log(n) .
So, e.g. log(6) = 1 log(2) + 1 log(3) + 0 log(5) + 0 log(7) + …
@@drdca8263 Raised by e
I raised this question "What will you see, if you look to the factor space from direction (1, 1, 1, 1, 1, 1, 1....)?" for purpose. If you look to a 3-d space from direction (1, 1, 1) you will see an area divided in 3 segments. The three lines you see are the projection of the base vectors to a plane normal to (1 1, 1) If you look to a higher dimensional space along the inner diagonal (1, 1, 1,....) you also should see the projection of the base vectors, which have to be equally distributed and show up with a projected length of 1/sqrt(n) . And if you look here: th-cam.com/video/OS2V6FLFmxU/w-d-xo.html you will see, if n is prime, you can add up the "spokes" in a certain way and end up at sqrt(P). But if the spokes are seen as the projection of base vectors, you will end up always at the unit circle (0,1) or (1,0). And there is a much more elegant way to add up the spokes, related to the "quadratic residue," whatever it means. But now, with this "factor space" brought up here, I ask: what does it mean if the spokes itself represent primes? And what, if you go on with this, as primes of primes of primes....
in 4:08 fortunately even this is still a "veruni form?" The transcript says: "fortunately even this is still a little useful we know.."
Dies Irae jumpscare
3:14 why perpendicular ? shouldn't it be just orthogonal ? is the vector space you define in this video euclidean ?
Or is it that you have the visual/physical interpretation of vectors which makes you think of the two as the same thing ? I know some people make this confusion. It's my first intuition and also you bio says applied mathematician lol
I'm confused on the difference between "perpendicular" and "orthogonal" except that perpendicular is just a specific example of orthogonal .. but for practical purposes aren't they essentially the same thing?
Synonyms
I opened this video to find out why number theory is hard.
I must have missed something.
At 0:22 the text says in two ways, but the voice over ignores this...
ive been trying to find this, it didnt make sense to me... specifically, "why is the 'factor space' treated as finite dimensional, when it seems to follow that there are infinitely many primes?"
I'm guessing, because any given integer is finite. So after a certain point, all the remaining "coordinates" are 0.
Where is it treated as finite dimensional?
this video have the best intro ever.
But what if you took them numbers inside those semimodule vectors and then represented them as semimodule vectors? like, why go half way with it if you gonna be changing up the way you rep the ints anyway bruh?
@@IIAOPSW this method does not represent the non positive integers, but requires 0.
Good vid 👍
Thanks
(w+x)!/w! = (y+z)!/y!
Gamma(w+x+1)/Gamma(w+1) = Gamma(y+z+1)/Gamma(y+1)
This question is equivalent to asking for the integer points of the level sets of f(w,x):=Gamma(w+x+1)/Gamma(w+1) .
Not to say that framing it in terms of such level sets is helpful for addressing it.
Suppose it could make searching for candidate solutions easier? But presumably every solution where this approach might be helpful has already been found.
I suppose this “viewing it as a semi-module”, is kinda basically taking the logarithm of the integer. Or perhaps a “formal logarithm”, as the actual numeric values of log(p) aren’t used.
log(a*(b+1))= log(a)+log(b+1)
But also, log(ab+1+1+1+…+1) for a 1s.
This doesn’t seem well expressed in the semimodule thing….
at first it felt like just a normal method and we wouldn't get much far.....but damn i was wrong
So one is a prime number?
A 0 at the end?
This answers (no, shows me a way to look for an answer) one of my favorite questions: What happens here? th-cam.com/video/OS2V6FLFmxU/w-d-xo.html And why Brian Convey doesn't show a more simple way to get the result?
It doesn't beg the question, it raises the question. You're a mathematician, you shold know better.
That being said, this is a fantastic video. I've been thinking about this exact concept (infrequently) for two years as applied to the Collatz Conjecture. This video has given me some great things to think about.
Good video
Man I'm an english major to be, I got lost at the number 3
210 = 2*3*5*7 and that caught my eye. Unfortunately none of the other examples of a!-b! = c!-d! match this.
@@Qermaq do you mean a!÷b! = c!÷d!?
@@artkalbphd I was mirroring what was presented at 0:30 - 210 = 7! - 4! = 14! - 12! and so on.
@@Qermaq You do mean division, then, not subtraction, as you wrote?
@@iang0th Yes, interpret the - as a division sign. I had a BIT OF sCOTCH IN THE INTERIM. Yeah, see.
Agh, now I really want to try encoding this into Idris or maybe Lean (surely someone already did it)
But I can't. I have to do other things today : (
Extending the allowed components, the vector [1,1,1,1,1,1,1,1,1,1,1,...]=4π² (the product of all the primes). So [-1,1,1,1,1,1,1,1,1,1,1,...]=π² and [-½,½,½,½,½,½,½,½,½,½,½,...]=π.
How do you make the vector a number, and how you normalize the product of all primes?
The topic was interesting. The video/audio creation/editing was not good. The video got confusing once you introduced "S" or The Succesor function. Maybe you should've better explained how this was the same or different from standard addition of 1. You were right that the Successor function is problematic but mostly because I got confused from your video's explanation and not necessarily the function itself.
0:17 - The narration leaves out the key phrase "two different ways." The point the narrator is trying to make doesn't make sense unless the viewer is reading the text on the screen rather than only listening.
0:24 - The phrase "multiplying 720 by 7 to both factorizations" doesn't make sense for at least two reasons: (1) Since the wording "two different ways" was left out (see previous comment), the idea that there are two factorizations of each number has not yet been introduced by the narrator. (2) I don't know what it means to multiply a number by another number "to" a factorization.
Bottom line: The ideas in the video are good, but the presentation needs some work.
College lecture in TH-cam form.
Holy shit the music interludes are FUCKING loud! Absolutely painful with earphones. The worst part is your speaking volume is low so naturally you turn the volume up.
Hello lol
1:54 हाँ-३ ठीक है; powers से deal कर रहे हैं, तो सामान्यतः multiplication addition बन गई; कुछ नया नहीं है। आगे बढ़ो ...!?
This all seems so trivial to me that I wouldn't think of spending time with the obvious: focusing on the true problems, like why is 1+1=2.
I was JUST getting greatly frustrated over addition. Namely, how to define addition of two numbers with different, not absolute, signs. Like,
0:11 - They all end in zero. Done! Next video.
3:40 WHY???? Are you purposefully talk very silently for the whole video, so that you can later rupture people's eardrums with super loud music??
Definitely interesting but YOU FAILED HORRIBLY when it came to proofing your audio mix levels. ⚠️
Misusing your creativity that way should not be an option.
I'm already with hearing impairment so please think about it.
It's not fun.
⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️
A bit dickish to have those subitaneous loud sounds.
Jesus christ the audio is so bad, mumbling voice at 20% volume and unnecessary sound effects at 110%
apllied mathematician booooo booooooo booooooooooooo 👎👎 NORMALIZE NON-PURE MATHEMATICAN HATE
NORMALIZE NON-PURE MATHEMATICAN LOVE 👍👍
ACTUALLY FUCK IT, ALL MATHEMATICIANS ARE AMAZING!!!!!!!