Gaps between Primes (extra footage) - Numberphile 

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

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."

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.

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).

This and the Goldbach paper coming out so close together leads me to propose the twin papers conjecture

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!

Eratosthenes not Aristophanes, he was a playwright. The Sieve of Eratosthenes is a miracle of elegance.

I find it hard enough to wrap my head around any number over 10000 being a prime number. This is just mind-blowing.

I love how this video is a prime number of minutes long.

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?

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!

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.

good to hear

i absolutely loved what he said at the end of the video.

At 8:04 the subtitle says "Aristophanes" but it should be "Eratosthenes".

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.

Brady, do you always have this much extra footage? This stuff is fascinating...you should publish extra footage more often. It's great stuff!

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.

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.

The spark can also go into it. It also goes into the first one. Genius!

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.

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.

Thanks for the extra footage Brady

The ancient Greek mathematician which had the sieve idea was Eratoshenis not Aristophanis. Aristophanis was an ancient Greek comic playwright.

Really awesome video. Enjoyed this extended talk quite a bit.

thanks Brady, this is a really exciting video about primes which I didn't know before.

8:59 and 857 are twin primes, well done Brady.

The extra footage is always my favorite

loving these longer videos!

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.

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..

Brady thanks for this big extra footage, I really enjoyed watching it!

I love these conjecture discussion videos.

This video is 18 minutes and 59 seconds, so you got a "59" in there, Brady!

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.

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.

ive done this "takin' it back/Im sry thing" twice between december 2009 and may 2010... it is not that unheard of.

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.

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.

And don't even get me started on 'sexy primes'. It's just 'sexy cousin'. I mean, really.

Brady, I think you have the most awesome job in the world.

I love that the video length (if read as 1901) is prime

Seeing people being excited about math is equally great as the matter itself.

The statement of the Goldston-Pintz-Yildirim theorem given in the video at 6:00 (for any ε>0, there are primes separated by less than ε*log(N) for large enough N) is incorrect. That statement is trivially true: for any ε, just take N=e^(1+1/ε). Now, 2 and 3 are primes separated by less than ε*log(N), since ε*log(N)>1. Goldston, Piltz and Yildirim actually showed that, for all eps, there are primes P>Q with (P-Q) < ε*log(Q).

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.

honly cow, who would have thought of such a thing. I would never have thought myself into that corner

This is true for all primes greater than 3, and it is easy to see why. All numbers are either a multiple of six (in which case they are not prime), a multiple of six plus or minus three (in which case they are a multiple of three), a multiple of six plus or minus two (in which case they are a multiple of two), or a multiple of six plus or minus one (in which case they may or may not be prime). 2 and 3 are exceptions because they are the only multiples of 2 and 3, respectively, which are prime.

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.

"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 RU-vid commenters. LOL. Cheers.

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

extra footage is the best part IMHO

I love the last bit

All I can think is what's the point. To be honest, thanks to Asperger's I'm technically a genius, Particularly if I care about something, because then I become obsessed and being able to notice patterns like crazy only helps further it. One of the first things I became obsessed with and still am to this day was writing, be it numbers or letters. Math appropriately became my first whiz subjects. I not only learned what was taught but went further, even creating my own rules to do even the...

The property of being prime or not is universal for all bases. You simply factorize a number and than translate those factors into other bases. Example: 15= 3*5 in base 10 F=3*5 in base 16 17 = 3*5 in base 8 1111 = 11*101 in base 2 So if you want to work on prime numbers, you can simply pick your favourite base ;)

"So this has applications beyond number theory" as a Physics major those were the magic words I was waiting for.

Would be really nice to hear about the Goldbach conjecture and Riemann hypothesis, and maybe even the abc conjecture, as well!

Numberphile is COOOOOOL

At around 7:40, Brady expresses surprise that the bound, 70,000,000 is a nice round number. The video explains where the number came from but not why it's so round. The reason is that, once you know that the actual number isn't very interesting, you tend to just prove that it's smaller than some round number. He presumably knew that the exact bound from his proof was much, much bigger than 2 and did a back-of-the-envelope calculation to say it was less than 70 million.

Yes, and they're the same.Remember that the greek (such as Aristotele, who the professor was talking about) didn't use our number system. They did basically all of their math based in geometry.

The video is just 1141 seconds long which is a prime number. Nice work Brady!

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

A better explanation: All numbers 6k, 6k + 2, 6k + 4 are even, if k is an integer. 6k and 6k + 3 are multiplies of 3, so all primes must be of the form: 6k + 1 or 6k + 5 = 6(k+1) - 1 = 6k' - 1.

For one: very large primes are used in the encryption of valuable information. Math is also very useful for programming, especially for video games (very large matrix multiplications) to determine has light reactions with objects. This channel mostly explains different ideas within Math, but the applications of these ideas are enormous. You seem like a smart individual, so figure some more out!

Oh boy. I'm really in for it now.

Happy 2019 year. Prime # year. Excitinggg

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...)

"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)

Because prime numbers have an additional clause: they have to be greater than 1. Semi-sarcastic explanations aside, many of earlier theorems involving prime numbers repeatedly found themselves having to state "X is true for all prime numbers excluding 1", so it was easier for everyone if prime numbers excluded 1. One such theorem is no less than the fundamental theorem of arithmetic.

Yes there are, and they are the same in all bases. For example 25 hex is a prime, and is not divisible by 5 obviously. Primalty is an intrinsic property of a number, it is not dependent on the way you write it. And number bases are just that, diffirent ways to write numbers :)

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

Wish you would put this in the prime number playlist. Thanks!

Thank you for your answer! Now I understand better, how this system works.

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.

Now it's down to 6!!!!! Exciting times!!!!

It's not saying that 70,000,000 is the highest gap between primes, it's only saying that there are an infinite amount of primes where the gap between two primes is

the primes exist independent of bases. while they may look different in different bases, they always represent the same number. for instance, 19 in decimal and 17 in dozenal are both prime, and both represent the same amount of objects. it's just a notational difference.

Edderiofer's Prime Conjecture states that: If n is an element of N (set of natural numbers), then n+1 MAY BE PRIME. Not all such numbers are prime, but we can find primes with this formula as well!

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.

Not 2 and 3, but yes. Say we have a number n>3, and we put it in the form n = 6a + b where 0

Essentially, but there are also the "trivial zeros": zeta(z) = 0 when z is any negative even integer. The Riemann hypothesis states that the non-trivial zeros all have real part 1/2.

The length of this video is 1901

it's just fun, man. it's like making a video about pi 3:14. yeah, that's not really pi because time isn't represented decimally, but it's still neat.

Bradys comment at the end ☺ Best part of the video.

They're all a big happy family of little primes and odd numbers.

*Sees papers *Sees camera *Sees Brady Analizing is entertaining

The fundamental theorem of arithmetic says that every number can be decomposed into prime factors only in one way. For instance 252 = 2 * 2 * 7 * 9. You can shuffle the terms, but they will always be these 4. Now if you define 1 as prime, then you can do 6 = 2 * 3 = 1 * 2 * 3 = 1 * 1 * 2 * 3, so the factorization isn't unique anymore. That's why 1 is not prime. It's just a convention, so that the theorem works.

The twin prime conjecture is to do with the idea that primes which differ by only 2 will always pop up, despite the fact that primes tend to be further apart as their value grows.

I know that was sarcastic, but I'll add mine anyway because I think it's interesting :P Seiterarch's prime conjecture (easily provable): If n cannot be described by n = m(6k +- 1) +- k where m, k are integers, then {6n-1, 6n+1} are twin primes. If you require (6k+-1) to be prime, this gives you exactly the set of all twin primes. Either way, I'm fairly sure it's not very useful.

Brady , you should definitely check the Collatz Conjecture ! Easy to present and very interesting !

Yay! I managed to find a working video on youtube!

And 19 is a twin prime with 17. I think Brady initially intended to make the video 19:00 long, that 01 probably appeared after the upload.

What progress we have here. We must live in the era of golden age of math again.

Wow people seem to really have missed your point. You are actually correct, you can not write 2 as the sum of 2 primes. The Goldbach Conjecture actually states, that every even number GREATER then 2 can be written as the sum of 2 primes, they forgot to mention this in the video.

Interesting that you even go there, I give you thumbs up for that. I used to think that considering other bases is something special - it is not, and it is due to the limitations of our brains. other bases are only useful for small numbers - as we are talking about infinity, it is pretty much useless. And "even" and "odd" are defined as divisible by two or not, so it is free of bases.

Twin prime triad conjecture? If nobody has made that conjecture yet, I would like to claim it.

Props to you, that was kind of hilarious.

Furthermore, I believe just yesterday they confirmed "reducing the bound to under 5 million," as I just read on wikipedia. That's fairly substantial!

I think it goes like this: AVERAGE gap between two primes (P and P+1) = ln(P) ln(P)=70,000,000 => P= e^(70,000,000) So, for primes larger than P=e^(70,000,000) there should be plenty pairs with larger gaps . In other words your first choice :)