Gaps between Primes (extra footage) - Numberphile
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- เผยแพร่เมื่อ 26 พ.ค. 2013
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Main video at: • Gaps between Primes - ...
Brown papers available: bit.ly/brownpapers
Prime number playlist: bit.ly/11kSUmF
Featuring Ed Copeland and Tony Padilla (with a very non-expert intro by Brady).
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Prof. Zhang was my basic proofs teacher. I ended up missing a lot of classes due to mono. I regret not working with him more. Very unsurprising, though, that his paper is "crystal clear."
What is mono?
@@solderbuff it's a disease people often get in high school and college. It's usually unpleasant but not threatening, though it can be quite a bit more serious if you're unlucky.
Infectious mononucleosis
@@solderbuff in most of the English speaking world it is known as glandular fever. Americans, of course, have their own way of doing things.
Damn, Brady. I'm translating the subtitles into Portuguese, but the word for 'prime' is the same word for 'cousin', so I get to this 'cousin prime' thing and I'm like "os primos primos"... LOL
Prime cousin translates to primo nobre. Cousin prime translates to primo primo. I don't know which would be grammatically correct.
6 years later
In Spanish it’s the same 🤣
Really appreciate this extra footage. It's astounding to me that even on the edge of infinity the largest gap between primes can be bounded. 70 million seemd such a small gap (in an infinite number system). But as others have reported here, and this quote from inyen1 "Terence Tao and later joined by James Maynard who had found additional new methods, they got the bound down from 70,000,000 to 246 (and if the Elliott-Halberstam conjecture is true down to 16)." For me, 246 is just such an unexpected and insanely small gap between massive primes on the edge of infinity. Incredible work by Dr Yitang "Tom" Zhang and others who have lowered the gap and followed on from his work.
The gaps between primes don't have an upper bound. It's just that it's finally been proven that there exist an infinite number of primes that differ by a certain amount (in this case 70 mil). This sort of thing had never been proven before. There still will be primes that differ by more than that, but now we have a proof that for a gap of specifically 70 mil, there are infinitely many.
@@nicolageorgiev4350 Isn't the proof that there *cannot* be an infinite number of primes with a certain gap that is *over* 70000000?
And that it is only *possible* for there to be an infinite number of primes a certain distance apart if that distance is less than 70 million.
@@ralphy1054 technically, they proved that the minimum (thats what the "inf" in the paper accounts for) gap size of 2 consecutive primes that repeats to infinity is less than 7*10^7 , its believed to be equal to 2 (twin numbers)....if by any chance, someone probe the minimum gap size is over 3 for example, that will mean that there arent infinite twin primes
there is also a theorem about the maximum gap size, it involves a bunch of log functions, but its not a specific number like 7*10^7 (you will see a "sup" instead of a "inf")
@@ralphy1054 I don't think that's what the proof is saying. From what I know, there are an infinite number of prime pairs who are any arbitrarily large distance apart, so larger than 70 million too. It's just that there's an infinite number of pairs with a low distance apart as well.
Massive props to the editor for realising it was a paper worth reviewing quickly
Apparently the limit is now at 246 (down from 70 million).
+tgwnn source, plz?
+Tyko Brian I wrote this a while ago but I see Wikipedia lists 246, with possibly 6 or 12! Sorry I can't look into the exact citations now.
i thought numberphile said 16 is the current limit
siekensou77
in this video?
+siekensou77 16 was conjecture as well.
Eratosthenes not Aristophanes, he was a playwright. The Sieve of Eratosthenes is a miracle of elegance.
This and the Goldbach paper coming out so close together leads me to propose the twin papers conjecture
glad to see someone else had this thought too :D
At UNH, I was taught by Zhang, he is a funny dude. No one pronounces his name correctly so he said at the very first day of class that is name is Tom (affectionately). I took multi-d calc with him and it was easy:P I mean its unh ya know?
Nate Cordova so there are no chinese people there?
@@l.z.7320 , no, there aren't.
Should've added 2 seconds to the video to get 19:01 1901 is a prime or added 12 seconds to get 1151 seconds total which is a twin prime with 1153.
Can we get a follow-up? I would *love* to hear how this proof has been evolved by mathematicians!
The polymath project with Terence Tao and later joined by James Maynard who had found additional new methods, they got the bound down from 70,000,000 to 246 (and if the Elliott-Halberstam conjecture is true down to 16).
This was back in 2014. According to Terence Tao they could not get any further with those methods.
I love how this video is a prime number of minutes long.
+NoriMori Twin prime number of minutes (17 and 19)
Walter Kingstone Oooh!
No it's 18:59. Which is 18.9833333333333333 minutes. Not even a whole number.
18m59s is 18*60+59=1139 seconds and 1139=17*67...so not really prime...
or 1859, which is divisble by 11 to get my favorite number 169...
At 8:04 the subtitle says "Aristophanes" but it should be "Eratosthenes".
indeed, would expect to have edited the video by now to correct that (even the subtitles got it wrong). Aristophanes was a master of ancient comedy
As soon as you said "You know that's what I would have done," I looked down to see the length of the video. I love your attention to detail!
I find it hard enough to wrap my head around any number over 10000 being a prime number. This is just mind-blowing.
Thanks for taking the extra time to explain this further. I had to watch the first video a few times before I finally grasped what it was trying to say. And now that I do get it, I agree it's pretty cool.
Really awesome video. Enjoyed this extended talk quite a bit.
Thanks for the extra footage Brady
loving these longer videos!
I love these conjecture discussion videos.
Brady is really nerding out in the beginning. Usually he lets the talent do the talking, but you can tell he is a worthy candidate for the title of Numberphile.
i absolutely loved what he said at the end of the video.
thanks Brady, this is a really exciting video about primes which I didn't know before.
Brady, do you always have this much extra footage? This stuff is fascinating...you should publish extra footage more often. It's great stuff!
good to hear
The spark can also go into it. It also goes into the first one. Genius!
That was funny, professor Copland part of listing all numbers and using a computer would take a long time. A very very very longish time.
This is pretty amazing. When I first read the comment you replied to, I thougth "this can't possibly be true", then I checked the first couple of primes by hand, then the primes between 5 and 10000000000 with a Python script. Pretty amazing.
I haven't read this thread carefully, but from what I see my impression is that I completely agree with you. An apology does not even require the word "sorry." The form of an apology is: (1) I recognize that what I did was wrong; (2) I recognize you were hurt; (3) I feel bad about it; and (4) I will try never to do it again. Saying "sorry" is, as you say, the opposite of saying sorry.
The ancient Greek mathematician which had the sieve idea was Eratoshenis not Aristophanis.
Aristophanis was an ancient Greek comic playwright.
+Michail Panagopoulos Actually it's Eratosthenes.
Ali Lahijani Actually Greek people pronounce i as e :)
+Ali Lahijani That's the right spelling.
What appeals to me about a subject like this, like Fermat's Last Theorem etc...is that it is so easy to understand the problem, and even picture how difficult it is to prove it--yet I KNOW I can't keep up with the math, but I imagine I can with proper explanation. It's a lot like watching a chess analysis of a a Carlsen in the World Championships...I imagine I too would make that move. Dreamers...I am one.
Excellent video, very informative.
Happy 2019 year. Prime # year. Excitinggg
Brady, I think you have the most awesome job in the world.
8:59 and 857 are twin primes, well done Brady.
extra footage is the best part IMHO
Very Inspirational!!!
Oh boy.
I'm really in for it now.
Props to you, that was kind of hilarious.
Numberphile is COOOOOOL
Brady thanks for this big extra footage, I really enjoyed watching it!
The video is just 1141 seconds long which is a prime number. Nice work Brady!
I love the last bit
"I can easily say without a doubt..." Made my damn day.
Hahaha. Best response ever. Showing clearly that you're not only a reasonable person, capable of acknowledging your errors/mistakes, but also honest and sincere not to get your undies all in a bunch about trivial things.
I am the original 'idiot' and I now accept your apology further. You have become one of my favorite TH-cam commenters. LOL. Cheers.
Wish you would put this in the prime number playlist. Thanks!
exciting stuff! :) thanks guys!
GPY was mis-stated in the video. What Goldston, Pintz and Yildirim actually proved is that, for all ε>0, there are primes P>Q with P-Q < ε*log(P).
Choosing ε to be very small doesn't guarantee that the difference between P and Q is small in absolute terms; it's just small compared to log(P). So, for example, if you choose ε=0.0001, it might be that the value of P you end up with is something like 10^1000000 and the only guarantee you get then is that P-Q < 100.
I love that the video length (if read as 1901) is prime
Harald Andrés Helfgott was born in Peru, he finished high school in Lima, at the Alexander von Humbolt College and is now workigh at the Ecole Normale Supérieure, in France.
Now it's down to 6!!!!! Exciting times!!!!
Yes it is. Consider that there are 5 numbers between two multiples of six (for instance, 7 8 9 10 11 are the five numbers between 6 and 12). Of those, two are divisible by 2 and one other is divisible by 3. The only possible remaining numbers that have a chance to be prime are the ones bordering the multiples of 6.
Thank you for your answer! Now I understand better, how this system works.
Something interesting I have noticed noticed: Base six behaves pretty excellently with primes. It can really quickly be shown (and I imagine proved) that all primes end with a 5 or a 1 in base six.
And don't even get me started on 'sexy primes'. It's just 'sexy cousin'. I mean, really.
ive done this "takin' it back/Im sry thing" twice between december 2009 and may 2010... it is not that unheard of.
The extra footage is always my favorite
Seeing people being excited about math is equally great as the matter itself.
They do! It doesn't show up in subboxes but you can see it via annotation at the end of the video it's usually paired with.
This video is 18 minutes and 59 seconds, so you got a "59" in there, Brady!
The length of this video is 1901
Yay! I managed to find a working video on youtube!
honly cow, who would have thought of such a thing. I would never have thought myself into that corner
thanks for the leak into the paper. I want the original text! even though I will probably not understand it, lol
You should do a video explaining the relationship between number of episodes and bradys %camera time and extrapolate to find the date from which all numberphile videos will just be brady being a boss
Would be really nice to hear about the Goldbach conjecture and Riemann hypothesis, and maybe even the abc conjecture, as well!
Sorry for the confusion -- typo. Both of them should have been "Primes P>Q with (P-Q) < ε*log(Q). Thanks for pointing that out.
"So this has applications beyond number theory" as a Physics major those were the magic words I was waiting for.
Thanks !
Bradys comment at the end ☺ Best part of the video.
one guy took a right way.. figuring exceptions, not only acseptions.. worked on primes 2 weeks alredy, and I feel like gone further than any man before..
Wow, 19 minutes. Think I might grab lunch before starting this.
144 videos! my favorite number!
Those last words explains it all: original vid is lenght 859(twin prime with 857). This vid is also prime and also special: 1901 is a Sophie Germain prime(2p + 1), since 2 x 1901 + 1 = 3803 is also prime.
Nice touch.
thank you and the other ppl.
now i got it:D
Lovely video. Would be nice for a link to the paper. And what about a video on the Riemann hypothesis?
1901 minute,seconds video, a prime. Very clever Brady.
What progress we have here. We must live in the era of golden age of math again.
Brady , you should definitely check the Collatz Conjecture ! Easy to present and very interesting !
Very clear, thanks :-)
thumbnail is priceless
use quadratic equation x^2-dx -n=0 as sieve can comb any gap d, have infinity solution for every gap d by induction, add up all of d prove goldbach conjecture, twin prime conjecture is it's special case at d=2, for example : a*b=a*(a-d)=n=5*3=15, (2^2+4*15)^0.5=8, (8+2)/2=5, (8-2)/2=3 two solution, if a or b is composed number have more than two solution and gap d not equal to 2, 7*5=35, (12+2)/2=7, (12-2)/2=5, 11*13=143, ((11+13)+2)/2=13, ((11+13)-2)/2=11, 7*3=21, ((7+3)+4)/2=7, ((7+3)-4)/2=3 for d=4, 13*7=91, ((13+7)+6)/2=13, ((13+7)-6)/2=7 , d=6, for prove Riemann Hypothesis use realization of sieve of Eratosthenes ,mean keep remainder, for example : pi(2^2)=4*(2-1)/2+0/2+1-1=2, pi(3^2)=9*(2-1)*(3-1)/(2*3)+1/2-3/6+0/3+2-1=4, pi(5^2)= 25*(1*2*4/2*3*5)+1/2-1/6-5/10+25/30+1/3-10/15+0/5+3-1=9.
They're all a big happy family of little primes and odd numbers.
59 is also a very particular kind of prime number, one that sees a fellow prime two units above, and that sees 57 two units below which is a honorary prime number
Thank you for the video. Can you provide a link to the paper of Mr. Zhang, please.
I love the way he says epsilon
It depends on what your definition of prime is.
In many abstract algebra classes, an element p of a ring is prime if whenever p divides ab, then p divides a or p divides b.
Under this definition, there are infinitely many negative prime integers. In fact, if p is a positive integer prime, then -p is a prime.
But again, it depends on the definition, what branch of math you're working in, and if it even matters concerning the problem you're working on.
Most people would say no, though.
Working in other bases has proven useful for CHECKING primes, on the other hand, because the divisibility rules for numbers are in fact different in different bases. For example, numbers that end in 1 in base 6 tend to be prime because those numbers can't be divisible by 2 or 3. Obviously not all numbers that end in 1 in base 6 are prime, but since most composite numbers are divisible by 2 or 3, this particular check makes it easier to find many primes in quick succession.
"An order of magnitude in the way people think", or maybe it's all over long ago and we need a refresher. Thinking about Time as a substance, something like a drumhead for example, means that the numbers can be imagined as wave phenomena in Principle and actual perspective.
So the trick is to think of primes as dominant probability integration positioning, ie wave-packages, and twin primes or any other, are nodal dominance/interference of symmetrical reflection, e-Pi-i numberness in time duration timing modulation.
I started out with this idea from a text book on Chemical Bonding explained through QM, Phys-Chem, and accumulated observation of excellent presentations like these of Brady's.
Sieves of varying grid sizes are somewhat similar to "Ring down" interference patterns. (Some like to infer Entropy, which is a lot like Temperature, and other vaguely applied words, but the real number properties are manifestations of e-Pi-i resonance, so every word description is another aspect of infinity.., loosly, or dualisticly connected, reminiscent of the solid-fluid dualism of all QM temporal substance)
*Sees papers
*Sees camera
*Sees Brady
Analizing is entertaining
One of the most interesting books I've read on this is the Mystery of the Primes series by Matthew Watkins. It is a very visual book, and I went through it slowly and it was a very intriguing read. I'm still waiting for the last in the series....
Seriously, sexy primes? Hmmmm....
Basically, the refined sieves catch the ores, but you still have to separate the gold nuggets from there 🤔.
Woud love a video on the "P vs Np"-problem. It would suit computerphile as well!
Yaaay Tony Padilla is a Liverpool fan
+1 for a video on the Riemann hypothesis!
According to the video, GPY states that: for any epsilon(e), u can get two numbers smaller than N with a gap in between is smaller than eN, provided that N is large enough.
so even if e is very small, u may need N to be very large for GPY to work, but then the gap eN may not be small
I'm interested. Did you indeed brute force trial divisions / sieves up to sqrt(n)? If so, how did you optimize it so well?
2:44 Hello Brady! :D
A lot of domains use these, especially cryptography. Every secure data exchange uses prime numbers in the encryption process (for instance, logging in a website, money transfers with ATMs or between banks, VPN protocols, etc...)
so cool! any other numbers like 6 with this property or very similar?
Cool story!
The length of this and the original video are prime numbers.