Primes are like Weeds (PNT) - Numberphile
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- Опубліковано 21 вер 2024
- The Prime Number Theorem shows that primes are like weeds, popping up everywhere! Dr James Grime explains --- Little bit extra cut from this video: • Prime Number Theorem (...
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The internet needs more James Grimes.
hahahah youth channel i was gonna say xd
The world needs more like James Grimes.
James Primes more like, amirite?
James Primes.
@@femioyekan8184 Why not, we shall become like James Grimes?
you should do a video where he explains his Phd thesis to us mortals
Yes.
Original comment is 4 years old which is fitting since I asked him about his PhD when he visited our school 4 years ago.
It was about combinatorics and linking combinatorial math to matrices and linear algebra. I think you can find his PhD on the internet (maybe I havent searched for it)
@@MK-13337 Every PHD thesis is aviable for free
@@shoutz5872 All PhD thesis are in principle available but not all of them are in online archives. But if you go to the department where the PhD was from they have it on hand. As I said I didn't check
Read the title and was like "Smoke primes everyday"
Hey I was going to comment that joke
3 years too late
Anyone else watch Lurd of teh Reings?
That joke is in there a lot.
never gonna prime you up
419 and 421 are twin primes
How about you just change your name to James Prime?
He gets high on maths. He even thinks primes are weed.
*Optimus
Chris Bandy he’s going to marry a sexy prime some day
Prime, James Prime. Agent 00mod7
00bean00 Your comment is underrated
The constant e can be remembered by using the following: Andrew Jackson was president of the USA in 1828; and the angles of an isosceles right angled triangle are 45,90,45.
So remember 2.7; Andrew Jackson; Andrew Jackson; isosceles right angled triangle
That is: 2.7 1828 1828 459045 which is e to 16 decimal places
thats realy cool!
+Andrew Snelson Hi Andrew... Is there a way to determine 1 to more decimal places ? - is there a way to choose how many decimal places you want to, to determine 1 ? ( i know 16 is a lot but im looking to take it further - thanks
+Kris Church Not that I know - e can be calculated - but it will be easier to look it up - I just learnt the quick memory tool to remember it to 16 places. Which should be accurate enough for most real world applications
Ah ok.... i need to go as far as 25 places for a study you see. and the mechanism of calculating would be useful for any adaptations / conversions. will try looking it up. Thanks for the reply
Actually you can calculate it further if you study history, as well phone numbers.
I actually find this site more interesting than 12 years of elementary to highschool education....
the comments are great too. People discussing about this and that. Makes young audiences interested in maths. I hope teachers use this channel
Being a 9th grader who is quite interested in these videos I think it would be very beneficial if other kids my age watched these type of things. This channel made me actually enjoy math.
/r/iamverysmart
I just can't agree more with this comment. It should really make us think about the education system.
2 x 2 x 3 x 5 x 7.
They're like weeds
Smoke 🌿 everyday.
420
3×23
@@hewhomustnotbenamed5912 nice
It is funny that when i started watching numberphile, i didnt understand anything and now i understand EVERYTHING they say!
Look at the video on Godel lol
Wow I actually could follow that . Yay..
If anyone is interested by this video, I highly recommend the book "Prime Obsession" by John Derbyshire. I read through this book as a senior in high-school, and even though I did not fully comprehend the proofs of the theorems presented, it was a great read and really enhanced my problem solving methodology. The author elaborates on Bernhard Riemann and his Hypothesis, and the Hypothesis' intimacy with the PNT. Every other chapter also includes history of the PNT and it's contributors.
I recently found this useful when discussing cryptography. RSA cryptography (simple) creates an asymmetric cypher by providing a very large unfactorisable number (ie the product of two enormous prime numbers) with which you perform a modular exponentiation. Currently a lot of implementations use 1024-bit prime numbers to build the cypher number. So if you were trying to find prime numbers represented by 1024 bits, how many prime numbers is that? Well, base-2 log of 2^1024 is 1024. e is between 2 and 3 (closer to 3) so the natural log of a number is likely to be approximately 2/3 of the base-2 log. But in any case, base-2 log of 2^1024 being 1024, we know that "pi" is going to be no smaller than 1/1000 of 2^1024. Well, if you have a calculator that can handle large exponents (eg MS Calc for Win10 can) you'll find that 2^1024 is about 1.8x10^308. ln(2^1024) is about 710, and so pi(2^1024) is 1.8x10^308 / 710, which is 2.5x10^305. So the PNT tells us that in the realm of 1024-bit numbers, ie 10^308, the number of primes is 10^(308-3) or a still massive 10^305.
Is it 1.8 x 10^308 x 107 or / 107 ?
@@youssefdirani You read all that? The number of prime numbers less than 1.8x10³⁰⁸ is not going to be *bigger* than 1.8x10³⁰⁸, is it? π(n)=n/log(n), log(1.8x10³⁰⁸)=710. 10³⁰⁸ divided by approx. 1000 = 10³⁰⁵. In other words there's still a gigantic number of prime numbers to choose from.
When I first saw the title, I thought it said, "Primes are like Weed"... lol
Math hasn't been the same since I had a chalkboard moved into the bedroom. My math has been longer lasting, more energetic, and better over all. That's funny, because it's probably going to ensure I never have sex.
Why dont u do videos with JAMES GRIME anymore Brady?
His videos are great.. So simple explanations
He's a very likable guy and he's ok. But he's not really just ok.
Dr.Grime kind of looks like an ostrich in the thumbnail.
I’m smiling from ear to ear because I’m in the edge of my seat
These videos just blow my mind every time. Thanks Brady and Dr.Grime
Only understand about half of all your vids, Numberphile... but enjoy each and everyone.
this title is so right, every time I see prime numbers i get so high. there is no multiple to explain this euphoric feeling
Most important thing I like in you is the amount of enthusiasm you have to know about the properties of these numbers. Great explanation of the PNT.
So you are very good with numbers. My favorite number is 3. I've been taught how to find phi by using prime quadruplets. 1st take your 3rd (you could use any of them) 101, 103, 107, 109 and the 4th 191, 193, 197, 199. Then assign a number in the middle: 105 and 195 (101,103, {105}, 107, 109) and (191, 193, {195}, 197, 199). the assign the { } number a prime sequence number. 101 being the 26th prime and 103 being 27th, 107(28th), 109(29th) ... 191(44th prime) 193(45th), 197(46th) 199(47th). Since 105 and 195 ARE NOT primes we have to assign a sequence number so 105 being 27.5th and 195 being 45.5th. Then take 44.5/27.5=1.618. Magic? My question to you is we are a extremely intelligent race of animals(humans). But yet our technology is merely rediscovery something that was already there. Numbers of mathematical fundamental, constant anywhere, and this cyclical nature of number and science. Is it just random chance? Or was it created? Just like your thoughts.
I love mathematicians.
As an engineer I always thought I had a handle on math but honestly thats barely scratching the surface and these guys and gals on this channel are the people that really get math.
That means there is at least one prime between Graham's number and 2x Graham's number. So all you have to do is search that limited interval, and you'll find the biggest prime so far! So get to work!
No.
Ah yes, the limited interval that is a Graham's number large
Euler's constant is absolutely extraordinary.
But can you roll a joint of primes?
Iorveth look up ulam spirals 😂
well if n doesn't have to be a prime then you have the whole set of negative numbers and 0 to work with, assuming it is a real number. In which case there are more examples where there is not a prime in between than there are examples containing a prime between. The actual postulate states that n>3 and it is n
What he says though has to be balenced by the fact that you can have a gap between primes as large as you want. To see this, consider the factorial function n! = 1*2*3*4*5*6*..*n If I want a gap between primes to be, say 100, take 101! Obviously 101! + 2 is going to be divisible by 2, 101! + 3 is going to be divisible by 3 ... 101! + 7 is going to be divisible by 7. So we have all of 101! + 2, ... 101! + 101 all being composite and thus we have a gap between primes of 100.
That may be a true bound, at that magnitude, but there are smaller primes separated by the same bound. You can divide n!s by 2,3,5..p to get n# ("n primorial"), and those are your smallest numbers to start from.
In other words, it is sufficient but not necessary.
Yes, here I agree. There is also constructive argument - one can easily check that n!+2 is divisible by 2, n!+3 is divisible by 3... up to n!+n, so all n-1 numbers between n!+2 and n!+n are composite and create prime gap.
If he had been my teacher when I was a teen, I would have been so in love.
Correction. PNT is NOT an acronym. It is just an abbreviation. An acronym is a special abbreviation that spells a word, or is pronounced as a word. So, NATO is an acronym, because we say it like a word. So is PIN. TLA is not. It actually stands for "Three letter abbreviation".
Why would you multiply [the average gap up to N] by [N] to get the N'th prime? Doesn't he mean the average gap up to the N'th prime?
The average gap between primes up to 135,221,143,753 * 5.500.000.000 = 140.965.975.573, which is a lot closer. Could someone please explain?
+T Geijtenbeek This confused me too. It doesn't make sense to multiply the average gap up to 5.5 billion BY 5.5 billion. That means you are saying the first 5.5 billion primes are separated by an average of ln(5.5 billion). But according to the PNT the first 5.5 billion primes are separated by an average of ln(135 billion).
If I had to guess it's because you will have two unknowns in the equation if you don't know the prime numbers. Therefore you can substitue the prime number itself with the number of primes (by using n for both). As you approach very large numbers the difference becomes less and less significant because you are taking the natural log. Maybe a mathy person can testify to that.
So I just asked my maths teacher what the PNT is...
He had no clue whatsoever :)
What a great teacher I have!
Smoke primes everyday.
Lel
421 every day
Your question is important. And yes, in an infinite way, there is a bijection from N (the positive non zero integers) to Pn. This is easy to see by setting a function such that f:N->Pn, with formula f(n)=pn, and it is easy to prove that the function is both injective and surjective. So it is a countable set. Unlike R (real numbers) the set of prime numbers is the same infinite size as N.
TIL tilde's have more of a meaning than simply approximately.
Just thought Id mention something. Log and Natural Log are different things, I know it says base e on the picture but it still might be confusing to people who enter log(1000000000) in to google or a calculator and get 9. (Its because its in base 10 so instead of e^9 you need to do 10^9.) You can have logs in any base, the base ten is most common in calculators and such and is appropriately called the common log. (Denoted by lg rather than the ln used in the video, maths syntax for ya.)
As a musician, it's nice to have opportunities to engage in STEM disciplines in fun ways, such as this channel!
Also, if Dr. Grime is single....I call DIBS!
Yes, and for precisely the reason you're suggesting. There seems to be some confusion in the replies to your question, so let me clear up that two infinite sets have the same cardinality ("size") if there's a 1-to-1 mapping from the elements of one set to the elements of another.
Interestingly, primes, positive intergers, intergers, and fractions are all equally sized sets.
Woohoo HTC One!
I really enjoy all your videos. Im about to get really motivated to envestigate.
Thanks - and please don't stop.
Why aren't you The Doctor?
The nth prime will always be greater than n, and the estimate for the nth prime is n*ln(n). If you check, n*ln(n) grows bigger as n gets bigger. And ln(n) > ln(m) provided that n > m. No contradictions. If you disagree, please elaborate.
I think we should call them Grime Numbers from now on.
Or call him James Prime.
I'm sure I've said this before, but I love how genuinely excited this guy gets when talking about math!
well the title was primes are like weeds so i read math as meth
If someone managed to predict the actual prime would it affect Rieehman hypothesis in any way? For instance if we know the precise 500,000,000th number and not just approximation
That other theorem you're thinking of doesn't say that the largest gap between primes is 70 000. It says that however high you go on the numberline, there will always be a couple of primes that are separated by less than 70 000. Most primes at that level will still be separated by more than that.
One question related to number primes:
I think that with the only number that you can form prime numbers by repeating it n times is number one : 1 and 11.
There is any other combinations of the number one that get a prime number ?
Thks.
The first part of your statement I can tell you is true. If I was a gambler, I would guess that the answer to the question is no. I don’t know though...
@@void9720 it's trivially true as all the other n repeats can be divided by the number itself
the 'log' button on a calculator is base 10. So for example, 10^3 = 1000, thus the LOG of 1000 = 3. [in general 'what power of 10 is required to get a number']
'ln' as stated in the video is to do with 'e' (the exponential) - so that's "what power of 'e' is required to get a number'
Please make a video on 1^infinity
+sam disum hard time with limits?
It is the composite numbers that are the weeds.Primes are a thing of beauty.
The person who made the PNT shouldn't have reused pi.
He should have used CAPITAL PI -> Π
+Aidan Dorgan Π is reserved for the product over a set of terms.
OK that makes sense
Aidan Dorgan He should have used 8=====D
As opposed to π
δ maybe?
Division is the only basic function that converges instead of diverges
Addition and multiplication tend towards infinity while subtraction diverges to negative infinity
Division converges to 0
A twin prime is a prime number that has a prime gap of two
4:03 i like twin primes
example for reminding myself: (5,7); (11,13) and so one.
But 420 isn't a prime number
True
+Morgan Freeman
But 419 and 421 are, so it all evens out
yay for twin primes
...just add 1. presto!
By definition, according to Merriam-Webster, an acronym is "a word (as NATO, radar, or laser) formed from the initial letter or letters of each of the successive parts or major parts of a compound term." The proper term would be "abbreviation." All acronyms are abbreviations, but not all abbreviations are acronyms.
Primes are like weed... Oh.
By the way, it is impossible to pause a video with James and have his face look normal ;).
Definition of a prime number (so people stop arguing about whether 1 is or isn't prime):
An integer greater than one is called a prime number if its only positive divisors (factors) are one and itself.
_I'm proud of California, for legalizing primes 😎_
I just realized the insane amount of abstraction Maths has. At 6:25 he literally said "If you pick any number, let's pick n, ...". Usually, everybody would say "Erm... 'n' of all things you could've picked is not a number, it's a letter." But in Maths, everybody is fine with it. Imagine there was some weird field of science where you'd say "Pick any vowel, let's pick 'k', ..." and everybody silently nods and goes on to listen the rest of the sentence. It's so weird in a sense.
watched it 3 times.. didn't understand anything lol
Nice clarity on the tilde!
Error:410 upper lips not found
The hypothesis about the gap between primes is saying that there are an infinite number of gaps between primes less than 70,000. But there are still an infinite amount of gaps that are larger than 70,000. 70,000 is by no means the largest possible gap between primes
I read this as "primes are like weed" at first, and ironically I'm actually high as fuuuuhhh
#nerdscanbestonerstoo
m4kefile there's different types of irony
+m4kefile it's not an irony, but the situation is ironic.
I didn't mean stare into the camera, just see the camera for what it is, your audience's eyes. You can glance at the camera once in a while to acknowledge that others are watching and not just the guy who's filming.
i got here by shearching 420
I read the title as Prime Weed (DMT)
Pie for the munchies too at the start, How joyful!
and then Constant E.
nf is the original formula's variable, but I was saying if you make ni in your formula 1, you don't quite get back the original. It makes sense that if nf is 1 the fraction of primes between 1 and 1 (an interval of zero) should be undefined, since it's a formula for non zero lists of integers.
this is probably one of the most informative youtube comment i've ever read, lol - English is not my native language, and while i've studied various languages and speak english fluently (and have been most of my life) i didn't actually realize there was a difference between acronyms and initialisms. Virtual highfive to you.
The paper discussed in the video you linked proved that there are an infinite number of paired primes that differ by 70,000. There are paired primes that differ by millions; in fact, the paper proves there are an infinite number of primes that differ by millions.
In the case of infinite sets, the fact that elements of one set pop up "less often" than another doesn't imply that there are fewer. That metaphor of how "frequently" the elements appear rests on the arbitrary assumption that you'd go through the intergers in sequence.
It depends on your definition of division. In abstract mathematics, we define every single operation. Some definitions of divisibility apply only to members of certain algebraic structures, such as a division ring or field, and infinity is not a member of these structures.
The ln(150 million) = 18.83. My program gives 17.76 for n = 150 million (biggest n without crashing on my old PC). So, the error curve at the end of this presentation is much appreciated!
If you get deeper in mathematics (extending number theory into rings) it gets clearer. Units are the numbers with multiplicative inverses, while primes are numbers p where p dividing ab implies that p divides either a or b. The familiar definition of not having a factorization ab with a and b not units is instead called irreducible, though they are the same for integers. The point is that 1 is fundamentally different than the primes. (Note the integers actually have two units: 1 and -1)
1 is a special case. It is neither prime nor composite. This came up in a Numberphile video. Check out the list of prime number videos in the description.
if n=1 then the n
Everytime Dr Grime appears in a video, UA-cam thinks i'd need subtitles... Which i dont, and i'm German...
Ln is log for the base of e, while a log function on a calculator is a log for the base of 10. That's why ln is called natural log
The best way to think about them is as the building blocks of every other number. every even number can be built as the multiples of 2. 3,6,9,12 are the multiples of 3s. 11,22,33,44,55 is the multiples of 11s.
They're useful because the are *mutually exclusive*. That means each prime can't be built with any single other prime. i.e. 5 can't be built with 3. 11 can't be built with 7, or 5, etc...
Hi, Pi(n) is the number of primes less than n. It grows as n grows larger, and there is no contradiction as the number of primes less than n increases as n increases.
What you are thinking of, the rarity, is given by the proportion, Pi(n)/n which tends to 1/ln(n) as n tends to infinity. This becomes smaller as n, and ln n grows larger. Hope to have helped.
The formula is quite simple, and yes I have a mathematical demonstration that it works for whatever number >=0. What I want to do is to publish officially this formula (that can be implemented in whatever programming language), and for doing that I probably need someone who is 'inside' the mathematics environment, to write together an article and send it to the University of New York, which is the center of research about Numbers Theory.
Nearly 45000 views and only 2k likes?? Come on, guys! Show Numberphile some love!
Yes it is, it fits all the criteria for being a prime number. My dad is maths teacher at college and he agrees with me
god! you can see how happy Dr.Grime gets when talking about numbers
i wish he was my combinatorics lecturer, would've made it alot more exciting
LASER is acronym for: Light Amplification by Stimulated Emission of Radiation
This is my favorite.
This stuff is crazy people even thought it up. What sort of practical applications does it have?
(2) For example p_3 is 5 and is estimated 3*ln(3) = 3.296. The absolute error is 1.704, and the relative error is 1.704/5 = 0.3408. p_1000 is 7919 and estimated 1000*ln(1000) = 6907.755. Absolute error is 1011.245, and the relative error is ~0.1277. Notably, the relative error to the magnitude of the number is lower, but of course the number is larger and so the absolute error is also larger. It should be intuitively obvious that it gets unfeasibly hard to approximate primes as they get large.
When you just write "log" it is usually interpreted as log base 10. "ln" is log base e also known as "natural log".
E = 2.7 1828 1828 45 90 45 2 3 5 360
2.7,then 1828 twice,then the angles in an eqilateral triagle,then the first three primes,then the sum of the oofs in a psqare.
I thought it said "Primes are like weed"...it perfectly matches James' face in the thumbnail
I was doing some math and found that (2n)+(n^2)-1 created primes very well if n is even. Example: (2 x 99922222222220)+(99922222222220^2)-1 is prime. I also saw that up to 200 being n (leaving out odd numbers) it spit out a prime 42% of the time.
Hey Brady, you should do a video on Matrices, and how they are used in the real world, and how they are conventionally multiplied and added, etc.
I hate when people ask for thumbs up, but you should thumbs up this if you think it's a good idea, so Brady can see it!
Dr James Grime the King of Prime.
They have done a video on this. James explains the long system and the short system in it.
The theorem n
I Like this one - Bertrand's postulate (Bertrand-Chebyshev theorem) : Chebyshev said it, but I'll say it again; There's always a prime between N & 2N. :)
Grimes explaining Primes. Nice.
There really aren't more in any sense unless you take subsets. Intuitively you'd think there would be "more" integers, but the method of proving two cardinalities are the same kind of infinity requires you to actually match them up perfectly. It just seems like there should be more because it's difficult to imagine otherwise.
I like the number 557 it's a prime number, the sum of it's digits is prime and each digit itself is a prime.