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- เผยแพร่เมื่อ 24 ก.ย. 2024
- In this video, we explore the "pattern" to prime numbers. I go over the Euler product formula, the prime number theorem and the connection between the Riemann zeta function and primes.
Here's a video on a similar topic by Numberphile if you're interested: • A Prime Surprise (Mert...
There are a few mistakes in this video, so I clarified them in a pinned comment. Sorry about that!
This video adapted several concepts and ideas from this article: / the-riemann-hypothesis...
More sources: docs.google.co...
Riemann Hypothesis by 3b1b: • But what is the Rieman...
Complex Exponents by 3b1b: • e^(iπ) in 3.14 minutes...
Harmonic Series Divergence: web.williams.e...
P-Series Convergence: www.khanacadem...
This video was animated using manim: github.com/3b1...
Source code for the animations: github.com/viv...
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Music by ChillHop
#primes #zeta #math
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What's a vcubingx video without errors?
At 1:54 it should be "Q - P = 1" instead of "P - Q = 1"
At 3:04 it should be "Converges" instead of "Coverges"
We're humans, don't worry :). I loved your video and animation, I didn't the primes were so amazing! Keep it up!
5:45 Gauß don‘t show he supposed and Dirchlet to.
Fun fact he supposed that with the Age of 15 with no Computer .
P - Q = 1 is also fine 😂
vcubingx how do you animate the text like in your video???
Same as 3blue1brown
As a statistician, I twitched when I heard ‘when the p-value is greater than 1’.
f1f1s uppercase P or lowercase p?
well he is doing real math here so it's ok.
(just kidding ;) )
@@tofu8676 dumbass lol
@@tofu8676 stfu you look like a girl
Then 0 and 1 are prime or composite?
You’re almost like a spiritual successor to 3Blue1Brown. Keep going, your videos are beautiful.
I honestly thought that beginning is fragmet of 3b1b video :0
He uses his Animation script
Besides using the same drawing tool, he does not explain nearly as fine as 3B1B.
A very good teacher who is spreading knowledge for free --- a noble deed!
It is intuitive to feel that primes have structure. Using Euler and Euclid, Reimann subjected this intuition to rigorous analysis.. He got further than anyone else and left a great legacy. This is a fantastic video, unless you are a prime number, hiding out there in integer space somewhere. In which case you should be worried, because soon your number will be up!
Incredibly high quality video. In those 16 minutes you went on such a structured clear and deep route into a topic in a way that most other popular mathematics channels never will.
Thanks!
I'm only 5 minutes in but already have to comment! I love your explanation of the Euler Product formula, it seems like it would be intimidating to derive given its connection to the Zeta function but you did it beautifully
thank you so much!
确实，这是个很直观的推导，虽然并不严格
Your channel is hidden goldmine. Underrated!!!
I dont know, but your style is like 3B1B's
I use the same animation engine as him (which he made)
@@herrmarx973 manim
@@herrmarx973 Xd
@@vcubingxwhat engine he uses
@alien3200 He uses manim
i think euclid's theorem works like this (noting this corrects the slight mistake in the video where it suggests that P - Q = 1 at 2:02):
- assume there is a finite number of primes
- then there exists a number P which is the product of this finite set of primes
- consider a number Q = P + 1
- by definition, Q is either prime or non-prime
- CASE 1: if Q is prime, then P is NOT the product of all primes (because Q = P + 1 implies that Q > P and no number greater than P can be a factor of P)
- hence, Q being prime leads to a contradiction
- CASE 2: if Q is NOT prime, then we should be able to factor Q as a product of primes (in the manner demonstrated for 30 earlier in the video)
- let one of Q's prime factors be the prime number p
- recalling that P is the product of ALL primes, p must also be a prime factor of P
- therefore p divides both P and Q
- i THINK there's a theorem which says that it follows that p must also divide Q - P (e.g. think of 3 as a prime factor of both 9 and 15 which leads us to know that 3 is also a prime factor of 15 - 9 = 6).
- by rearranging the original equation, we find that Q - P = 1. hence p should divide 1 by this logic.
- as the video-maker then explains, no number divides 1, so p cannot divide 1 either
- hence, assuming Q being non-prime led to a contradiction
- therefore, the original assumption that there is a finite set of primes must be false
- therefore, the set of primes is infinite
Thanks for this!
@@vcubingx thank you for making the amazing videos!
@@mohsenardalan8934 ah, great - thank you!
You can arrive at a contradiction in the second part directly from the ring axioms. By the definition of divides, p|Q implies there is an integer a, such that Q=pa, and similarly an integer b, such that P=pb. Then Q-P=pa-pb=p(a-b) (by distributivity), and p|Q-P=1 by the definition of divides (a-b is an integer by the existence of an inverse and closure under addition). Integers are rings, and this works under it.
This the best Riemann hypothesis video till date...it take from first basic prime theorem to non-trivial zeroes of zeta function, and this video is not to complicated , I loved it.
Your video editing skills are really good!
The pattern to prime numbers is that they are prime
The primes here are indeed made out of primes
please head to collect your 1M
hmmm yes, the floor is made out of floor...
The internet remains undefeated.
*BIGBRAIN*
Maybe the best video on the topic I've seen yet. Nicely done!
“Prime numbers are solitary numbers that can only be divided by 1 and itself. It gives me strength” - Someone who achieves heaven
Good work, expanding on 3B1B while giving credit. You defiantly add significantly to 3B1B's phenomenal presentation.
I appreciate this video, I’ve always been confused as to how the zeta function relates to primes but you laid it out pretty solidly. I feel like that section would benefit from more clearly explained math but I understand it.
100%, this video is an old work of mine and I really wanna re make it in the future
I love the way that you perform on Manim, subscribed! I hope you talk about many other interesting topics and stuff
I'm on shock, I didn't know the primes were so amazing!!
Wait until you get in to the spirals in prime numbers, the Fibonacci sequence, the fabric of reality......
Thanks! I was specifically looking for a video that directly explained the relationship between the prime counting function and Reimann zeta function zeros. This video did exactly that!
best video that explains the background but also covers different aspects of Riemann function and primes. but have you or anyone found a pattern yet ?
I... This looks exactly like 3blue1brown...
Huh Neat his engine
I... why do you write like this...
@@Tulanir1 ... I... Don't know...
@@huhneat1076 Ok... fine...
Irony is I clicked this video because I thought it 3B1B
Thanks man, I didn’t know about the approximation of the prime counting function and I loved the way you explained it, it’s my first time in your channel and I’ll proceed to watch your other videos, great work
Brilliant video- thank you. You've given an explanation for a number of facts that I was aware if, but had not seen any justification for.
Big fan of the channel, came from 3B1B. Just some constructive criticism: I've watched quite a few of your videos btw. Whenever you're going through the steps of some proof or result, the sudden animation that replaces the previous expression is very confusing. It's hard for the brain and the eyes to follow along with so many changes happening simultaneously, so if you animate the steps one at a time with continuous frames rather than discrete frames, I think it would be a lot easier to follow along. Maybe you could try presenting videos to a friend and have them follow along; they could point out the points of their confusion so you can fix them before posting the vids. It's just hard for visual learners (at least me) to follow along sometimes. Thanks, and I love your work otherwise!
Euclids theorem makes no sense to me, what am I missing? How is P - Q = 1 and why should 'p' divide it?
p should divide P and Q as Q is made up of primes like p
And P is product of such primes
Therefore p should divide P-Q, means it should divide 1, which is not possible for any prime p, hence Q is divisible by some prime not in the product of P, hence it would bigger than all the primes present in P, hence number of primes cannot be finite.
Hope you understood 😁
@@chirayu_jain thanks, I guess what confused me is that P - Q should be -1 since Q is defined to be P + 1
Sorry yeah it should be Q-P and I think @Chirayu Jain's explanation covers it.
@@chirayu_jain But what does "made up of primes" mean? Composite number or also numbers that are primes added or subtracted with potential exponents?
Coprimes of course won't divide one another without remainders or fractions but what's the one there for if primes are at least 2 apart except 2 and 3?
@@chirayu_jain "p divides P, and p divides Q, therefore p divides Q-P"
I think there's a missing step there. It's not self-evidently obvious why that would be.
Honestly, this is the best video on the riemann hypothesis I have ever seen
You should make the quote at the beginning last like 3 seconds longer.
good point, I will next time
Wow, so honored! "THE FORMULAS OF NONPRIMES REVEALING ALL THE PRIME NUMBERS" was named one of the best new Arithmetic books by BookAuthority!
Dude, this made me understand stuff, like I don't even care about all this.. but this made me learn new stuff and you made it easy for casual viewers like me. Thanks.
That's awesome! It's exactly the point of me making the video!
Most of the math shown here, i learn it on my first semester of the first year of college(computer science). What i find interesting in math is that if you want to be good at it, you need to be good at every part of it: ecuations, trigonometry, integrals, etc.
that's what i absolutely love about it. its all interconnected in such an interesting way.. sadly you also have to be smart to fully comprehend everything in it :(
This video with lofi music is perfect * - *
Vừa vào đã nổi cả da gà 藍giọng a Phúc hayyy quá, mong sẽ tiếp tục cover ạ ❤️
Just amazing, I liked the video before watching. BTW how do you get such ideas for making videos?
Very cool! Sometimes tho, the things you've shown were too complex for me to follow along, so I had to grab a pencil and paper and really think about it, but in the end I think that is a good thing! Thanks for forcing me to actually do something :D
Thanks for the EXPLICIT definition (extension) of the Riemann ZETA function for numbers less than 1. It is surprisingly hard to find.
Thank you
Now I'm not just relaxed but know how to distress in difficult situations
By using the Riemann Zeta Function?
So in other words, the prime counting function can use the Reimann-zeta function to predict the values of prime numbers, but only as the number of zeroes tends to infinity. Problem is, it's not proven that all these zeros are at Re(x)=0.5.
Would the person who finds the pattern of primes be legally allowed to reveal it because of the encryption/cryptography implications
yes
lets say decryption is n times more difficult than encryption, if decryption becomes easier our abilty to encrypt data becomes easier too, thus decryption will be around n times harder than decryption again. Unless we have the ability to predict the future, which seems imposible.
@@juliansoto2651 if decrypting became easier wouldnt encrypting become harder?
Yes
Loved the video, i am currently reading the music of the primes and this video put it all together beuatifully! Thanks a lot for the content!
Thanks bro it was more clear for me than previous videos about zeta function
"Give calculus a chance" -YT ad -- Finally, a positive message.
Finally a worthy heir to 3B1B. Similar calming voice, and technically strong explanations.
AYYYY 10/10 would watch again
đón chờ những ca khúc tiếp theo của Phúc, càng nghe càng thích giọng ca của Phúc ❤
Video is excellent. Warbling music can induce nausea, unless you are tone-deaf perhaps. It makes listening to the video physically painful. What a shame considering how amazing the rest of the video is.
Thank you for linking to *manim* in the description! It's crazy I haven't find about it through 3b1b!!
about primes and the zeta function: consider the x funciton f(x)=1/n*n^(1/2+n*ni), the prime numbers when considered n = prime will give alternated sings for the sin(f(x). and every integer z number will lead to sin(x)=x , a special class of numbers that i called misiec´s zeta complex numbers, as i have not found no reference about the numbers that respect the squeeze theorem. do a wolphram alpha for the plot you will see how interesting the behavior of the graph.consider sqrt (-1) instead of i in ni.
Is there any other possible visual other than the 3B1B style?
The pattern is the removaval of patterns (divisible by irreducibles 2,3,5,7, 11, 13, 17, etc is in itself, a pattern....)
Thanks to you and 3b1b , so i understand what makes this hypothesis be very important. Let me go home and prove it.
Glad you liked it!!
Let me know when you finish 😅
This channel is like the child of 3Blue1Brown. Not because the software used is the same, but because the explanation is good.
"Pi is a step function" - shows a smoothed version of that function that is not even monotonously growing... -.-
I’ve found a function which the line is vaguely close to the line of prime numbers (like you go up on the y axis every time x= a prime number)
12(square root(x+30))^0.7-38 It’s very vaguely resembling
Next prime is near P+ln(P) and always will exist a new prime betwen P and P+2ln(P)
The best explanation I've seen of this
Excellent video. Very well explained.
Congrats!
Hey, great video, just found you on my feed!
Thank you so much!
Damn your manim animations look clean!
Consider this: instead of looking only at the distribution of the prime counting function, EXPAND the prime counting function to look at numbers with N prime factors (where 1 has 0, primes have 1, and numbers like 4, 6, 9, and 10 have 2). What you will notice is that each line that can be formed by this extension will be "random" like the prime counting function, but getting the sum of the lines will be equal to the input of the functions, X. This means the sum of a set of random sequences is a predictable sequence, and so we can't REALLY say that they're random, can we?
Nice and brief retelling of “Prime Obsession”, John Derbyshire’s book.
for challenge 2 i think its more fun to derive the gamma function:
consider:
∫exp(-at)dt
where a is positive and the bounds of integration are from 0 to infinity. its easy to evaluate this integral to get that it equals 1/a. so:
∫exp(-at)dt = 1/a
differentiate both sides wrt a:
∫-t exp(-at)dt = -1/a^2
∫t exp(-at)dt = 1/a^2
differentiate both sides again:
∫t^2 exp(-at)dt = 1*2/a^3
in general, after differentiating n times:
∫t^n exp(-at)dt = (1*2*3*4*...*n)/a^(n+1) = n!/a^(n+1)
just setting a=1 we get:
∫t^n exp(-t)dt = n!
Theorem: There are no primes between any two consecutive primes.
How does multiplying by a fraction subtract just one partial sum?
Why did 1/4^s disappear along with 1/3^s?
There is a very interesting recent research book that have miraculously answered almost all the questions concerning Prime numbers, it is available on Amazon by the name of: THE FORMULAS OF NONPRIMES REVEALING ALL THE PRIME NUMBERS
i like how u didnt cut the clips where u stumbled. thanks.
My recollection is that the Riemann formula for the error term does _not_ depend on the truth of the Riemann hypothesis. The formula is true regardless; the hypothesis in merely an observation of where the zeros needed for the formula will be found.
You're Euclid proof is back to front. Q - P = 1 not P - Q =1
Its funny. I learned math via Euler and Ramanujan so when encountering the sequence definition of series in an analysis text I was shook. A few years later and I primarily think about series in terms of their sequence definition. Computational utility eclipsed by generality, I blame my study of functional analysis lol. Going to study Euler after learning the basics of Algebraic Topology from Munkres. It will be nice to go back to the Eulerian view of function.
As for Euclid's proof, it can be paraphrased more simply: the list of primes is endless because the lowest factor greater than 1 of p!+1 must be a prime number and must be greater than p.
(Remember that a prime is a factor of itself.)
This is not a 'proof by contradiction'. It is a simple direct proof.
Cống hiến hơn nữa trong sự nghiệp của mình. Một lần nữa cảm ơn Đức Phúc và chương trình đã mang đến những giây phút thư giãn này. Mãi yêu!!!❤❤❤
16:27 I’ll tell you when I find a solution.
The error gets lower and the counting function improves due to we have to enter prime numbers in the formula. So that's reminds me the same problem that we have with prime representing constants.
Very nice video. I have a question. The curve @15:30 looks like steps, can there be a smooth curve going through the primes? Such that one can ask what is the 2.5th prime.
Yes, that is the Riemann R function.
RiemannR(4.18142) = 2.50000
Some examples:
Pi(10) = 4, RiemannR (10) = 4.56458
Pi(100) = 25, RiemannR (100) = 25.662
Pi(1000) = 169, RiemannR (1000) = 168.36
Pi(10000) = 1229, RiemannR (10000) = 1226.93
Pi(100000) = 9592, RiemannR (100000) = 9587.43
the error is on the order of √x/ln(x)
@@ckq Thanks for sharing this. So as I understand, xth prime would be RiemannRInverse(x).
And still this won't be exact right? Since RiemannR itself doesnt exactly match the prime counting function.
Very Inserting. Thank you very much!
The last minute is the crescendo!
thank you so much for this ♥
If we discover that spacetime is quantised, what bearing does that have on the foundations of mathematics ?
Se toman todos los números primos conocidos uno atrás del otro y se los junta para tener una serie, si esta serie corresponde a las propiedades de las series aleatorias de números, entonces no hay patrón para calcular los números primos, pero si hay una pequeña discrepancia mas allá de lo aceptablemente probable, entonces los números primos deben aparecer en un patrón determinado
La cuestión quedaría en lo "aceptablemente" probable
Amazing video! I love math :D
Part 1: There are an infinite amount of primes. Because take this: 2 and 3 are primes, right? so you take the square of 3, 9, and calculate the process. Repeat. Also, if there are no primes between one square and the next (which i doubt is possible) until you get to the 'last' primes square, just get 2*3*5*7*11*... until the last prime plus 1. That will be a prime.
At 1:25, shouldn't the text read "all prime numbers" rather than "every single prime number"? Otherwise, how would you define the product of a single number?
At 4:22 you say you subtract it from the "original series" but you don't. You subtract it from the PREVIOUS series.
I figured something out the other day. All primes are odd numbers. Hope that helps 😁
Except for 2
Question: Is it possible that there is a different function that will approximate primes more accurately than Zeta function?
I'd argue, if you could disprove that, then you've already 'solved' the million dollar question.
@@drew-id Andrew, to clarify: You mean, we assume there is only 1 function to approximate primes and it is zeta function? Thanks :)
@@kam1470 in reference to your Question 'is it possible'...
If you could definitively say it's not possible, then you'd be proving the zeta function is the best we can ever hope for... Right?
Very nice video. I can't wait to see how you improve your videos and explanations. Good job, but there's a lot of work to do yet.
the first quote is pretty deep
In your proof for Euclid’s theorem 2:01, you said P - Q = 1 but I think you meant Q - P = 1. Just a heads up.
I believe the Prime Counting Function Pu(x) uses capital Pi, as opposed to lowercase pi used in circles. Otherwise, nicely explained.
Thanks! I'm not too sure either now that your mention it. I used lowercase pi because Wikipedia uses it - en.m.wikipedia.org/wiki/Prime-counting_function
@@vcubingx If Wikipedia uses lowercase _pi,_ that's probably correct, and I retract. Uppercase Pi is used as the multiplication analog of Sigma, that is, the product of a series (where Sigma is the sum of a series). That symbol is also used in the development of Riemann's formula, as you showed, hence my mistake.
One of my favorite math shit posts goes as follows:
Sum n = - 1/12
Sum x^i = 1/(1-x)
So. Sum n = -1/12 = 1/(1-x). Therefore x = 13.
All integers are powers of 13
I think I discovered a pattern to the Primes. None of them are divisible. I checked this up to 101 and it seems to be true, still working on a proof.
2:17 definition of series - which i always forget and equate to sequence
Analytic number theory student here, you should contact a professor working in the area with your script next time. You mentioned a couple of things that werent quite true or were quite a bit more delicate than you made them seem. Other than that great video it was fun to watch.
I agree. It's been a while since this video came out, and its what I do nowadays. Thanks for watching!
have a look at \left( and
ight in latex @4:26
ight)
P - Q = 1 should be Q - P = 1 ? Also how can you tell that a prime factor p of Q should divide Q - P ?
15:30 to 15:55 How to calculate the sum of non trivial zeros.
"This question is so difficult!!!111"
me: add 2 and if its obviously a composite then add 2 again until its obviously a prime
There is a guy in TH-cam named Sergio fernandez....saying that he has an equation which can tell us how many primes in a given rage....i didn't understand it's true or not?
Btw your videos is always awesome keep it up.
ni, san, go, nana, ju ichi, ju san, ju nana, ju kyu, niju san...
Even if there is a pattern for primes, which I doubt, what would be the practicality? Or benefit?