It feels like Brady is trying to assemble a team of top mathematicians to crack the Riemann Hypothesis. Unfortunately, they don't seem very interested.
3 years ago, its still out there :-). Intuitivley the RH is unprovable. So much hinges on it and it works so well that there is some fundemental property of numbers that means its unprovable. If you could prove its unprovable you would be famous.
Recently I saw a TV show on PBS about the "Twin Prime Conjecture" featuring both Tom Zhang and James Maynard, two very smart guys. :) On April 2013 professor Tom (Yitang) Zhang of the University of New Hampshire has submitted a paper to the Annals of Mathematics as proof that there are an infinite pairs of prime numbers that differ by 70 million or less. This spurred a lot of activity in the field, such as the Polymath8 project, which lowered the bound to k ≤ 246 and recently to k ≤ 6 (general Elliott-Halberstam conjecture). The human mind is truly amazing being able to tackle and solve such difficult problems.
Brady, well done! This is what I like so much about Brady's videos across all his channels: Unlike many interview-based videos, Brady so often gets his subject to tell a full story arc before getting to their own contributions. It takes careful reading and prompting to make this happen, and to look so effortless and flow so naturally. Best of all is Brady's infectious excitement: He wouldn't be doing this unless he was genuinely thrilled by it. He calls himself a videographer, but I think that's his secondary talent. You can't do great edits if the material isn't there in the first place. And then there are the times when his subject goes into awesome mode, and Brady knows to just step back, be a videographer, and let things go wherever. Talk about being in the moment. Thanks!
This has been one of my favorite videos in recent times. Dr. Maynard was clear in his explanation and he looked so happy to be explaining it. That is the type of passion everyone needs to find in their life.
Such a humble and genuine young man. It is so refreshing and encouraging to see a brilliant intelectual mind with this personality . All the best wishes to Dr. James. Looking forward to see more videos of him!
of course you can't check with a computer, they're doing it the wrong way! They start at 0 and go towards infinity, but if they instead started from infinity and went backwards, by the time you reach 0 you would have checked ALL THE NUMBERS! Fields Medal please... Oh, it also works with the digits of Pi
The fact that he lives and works in Oxford, where I study, makes this video a little weird as he sounds practically identical to many of my younger lecturers and tutors. This video more than many others on numberphile felt like I was being taught a class! Very well explained
I love that Numberphile gets these top mathematicians who are still willing to start from first principles (reminding the viewer that 2 is the first prime, etc.)....
I've made a wonderful proof in which i bringed the gap down to 2 proving The Twin Primes Conjecture. However there is not enough space in the youtube comments section to write that here.
I like the way he moves when he talks, like he got a funky side 😎 but also very humble nice and smart !! thank you very much to him for his wonderful work and sharing that, so adorable human
@@j.vonhogen9650 I think you didn't get the comment above. Porriale means that James was still humble 5 years ago when he knew that he might win a fields soon.
Could Numberphile do a video about the recent Abel Prize earning work subject from Yves Meyer, the "Wavelette Theory" ? The Abel Prize is like the Nobel of maths (after Fields medal), and it was awarded less than a month ago --- it's a cool thing to make a video on, especially since there is little information about it on the Internet ! Please !
Congratulations on winning the Fields medal James. I admire these humans so much . I went to Oxford read Zoology from 1989 - 1992, I got an upper class second degree. Also got a boxing blue for boxing against Cambridge on March 12th 1992. I arrived at Oxford as a bright boy with serious memory capability and a thirst for competition in tests. I very quickly observed around me a level of student that I could only marvel at. A healthy reset of expectations. The very best of my peers were like James in the way that their ability was to them normality and like a frequency hum in their background. No frills; no interest in recognition. They were simply unbelievable in their fields.
It's really fascinating to see this kind of breakthrough when you don't understand the most fundamental principles of prime numbers other than it's only divisible with itself and 1.
Nice to see he is so humble. "Intuition" or the "inspiration", that "i am on the right track" even though you are not sure you will get to the proof, comes from the supersoul within (Sanskrit: paramātmā). We live in a virtual reality. The mind is the real cause of our suffering and happiness. Never think that "I am the doer," especially when you do not know who exactly "I am".
The set {2,3,A,B} contains all prime numbers in the natural set. Where: A={6n-1: n is natural}/{36xy-6y+6x-1: (x,y) are natural} B={6n-5: n is natural}/[{1} U { 36xy-6y-6x+1} U { 36xy+6y+6x+1}: (x,y) are natural] The formula A+2=B contains all the cases of prime sum of two primes, except 2+3=5. The set {2,3,A} contains all the Sophie Germain primes. B can't be Sophie Germain due to divisibility of 2B+1 by 3.
William White Albert Einstein published without peer review and citation in his seminal year. Do not discourage Julio. Maybe Julio has insights that give us that "quantum leap" forward. 😑😑😑😑😑😑😑😑😑
My understanding (from watching talks by Terence Tao) is that the barrier he refers to several times in this video is something called the 'parity problem'. I would love to see a video explaining what the 'parity problem' is.
1451, 1453 is a twin prime pair associated with protons and neutrons. If you divide half the difference of their masses into them you get the twin prime pair.Proton's mass = 938.272081 Mev/c2 Neutron's mass = 939.565413 Mev/c2
Cool with this channel, where you can experience stars of mathematics lecturing or telling their stories. And in this one, a Fields medal recipient, interviewed with a modest not fully erased math-blackboard and a small notice board in the background. Totally relaxed.
Watched this video just after having watched another one about the Twin Paradox. So now, I imagine two twin primes, and one of them taking a rocket, flying at the speed of light, and returning close to his twin prime. But now, their distance had become more that 2, because of Special Relativity.
The equation 6n+-1, can be used as a serial equation or a matrix. The matrix first column is odd numbers and the top row is factoials of prime numbers, 6n, 30n, 210n, 2310n and etc. The result is: 30 + 11 and 13, 30 + 29 and 31, 210 + 29 and 31, 2310 + 29 and 31 all twin primes.
Prime numbers aside, it is refreshing to see you present your perspective on mathematical thinking. I am looking at how to inject this aspect into the high school arena since it is virtually uncatered for (at least in Australian schools). Schools are results driven and are more or less merely a set curriculum production line. My argument is that there is more - it is what YOU do and there is a way to cater for it at high school level with students of the right mindset. You have clearly expressed the "genius factor" that is lacking and not really understood in our education system. The thing is, how do we turn what you do into a television series to show how some mathematical ideas can be explored with predominately high school mathematics? Well done, so far!
i just nticed that whn they had he list going up that they missed a set of twin primes 19541 and 19543 that thy didnt highlight. i love how even in my late 20's this channel makes me fell ok to still find math interesting.
"What about if we use YOUR method" -"I think WE..." - James I enjoy he puts We instead of My or I, he shows he wants everyone to enjoy maths no matter what its called.
6ab+-a+-b. 4 equations. Any integer that can't be created this way multiplied by 6 and +-1 must be prime twin. So the problem is reduced to something like +- n such that above n+m15 in many cases. Among others n={1,2,3,5,7,10,12,17,18,23,25,30,32,33,38,40,45,47,52,58,63,70,72,77...} It is highly unlikely that at any point complexity will become so large that every number can be calculated using 4 variations of the equation yet I don't know how to prove it.
There is an Incompleteness Theorem by Kurt Godel which could apply to proving the twin prime hypothesis. Equations involving the imaginary operator might help. The equations: i^{4n + 2} = - 1 and i^{4n} = + 1 are involved. An equation: Q[N - 1] = [1/2] [Q[N] - i^{1 + Q[N]}] does not look to be provable but is axiomatic. In this equation, Q[N] is an uneven integer lying between 2^{N} and 2^{N + 1}. The integer " N " is a subscript, with N taking values: 1, 2, 3, ... N is counted down to N = 1. Inevitably, Q[N - 1] is found to be an uneven integer from this equation. With regard to twin primes, if p[k] and p[k + 1] are two primes belonging to a twin set, for example 101 and 103, then the equation Equation [1]: [p[k] - i^{1 + p[k]}] - [p[k + 1] - i^{1 + p[k + 1]}] = 0 for some twin primes and = 4 for other twin primes. Examples where this equation give the answer, zero are: 17, 19 ; 29, 31 ; 41, 43 ;101, 103 ; 149, 151 ; 197, 199 .. ad infinitum. Examples where the answer is four are: 11, 13 ; 59, 61 ; 71, 73 ; 107, 109 ... ad infinitum. The above considerations could help establish or otherwise the twin prime hypothesis. This involves bringing in the imaginary " i " into simple high school arithmetic and algebra - extra outside axiom. The equation: Q[N - 1] = [1/2] [Q[N] - i^{1 + Q[N]}] is a rearrangement of a division which is Q[N] = 2.Q[N - 1] + r[N - 1]. The term, i^{1 + Q[N]} is identifiable with the remainder, r[N - 1], which has the value minus one or plus one, only. That is r[N - 1] = - 1 or r[N - 1] = + 1.The remainder, r[N] can be chosen to be one or minus one, but has no relevance in the above context. Taking the iterations of the equation: Q[N] = 2Q[N - 1] + r[N -1], counting down, the last two equations are: Q[2] = 2Q[1] + r[N - 1] and Q[1] = 2Q[0] + r[0]. It is always found that always, Q[1] = 3, Q[0] = 1 and r[0] = 1. Algebraic elimination of Q[1], ... Q[N - 1], a very easy exercise, leaves the equation: Q[N] = q[0]2^{N} + r[0]2^{N - 1} + r[1]2^{N - 2} + ... + r[N - 1]2^{0} The q[0] and r[0] terms can be left out. Replacing the base, two by the unknown Z, gives: Equation [2]: Q[N] = Z^{N} + Z^{N - 1} + r[1]Z^{N - 2} + ... + r[N - 1]Z^{0} There are two values of r[N - 1], which are r[N - 1] + - 1 and r[N - 1] = + 1. These correspond to two values of Q[N]. In the case of two prime numbers that give the value, zero in Equation [1], above, then theses twin primes may be said to be connected, otherwise, if four results then the two primes are disconnected. The intermediate even number between any two connected integers has the value: [2 + 4n] for some integer, n. For example, [2 + 4n] =102, gives, n = 25. If Q[N] is transferred to the right hand side of Equation [2], then the equation, Equation [3]: Z^{N} + Z^{N - 1} + r[1]Z^{N - 2} + ... r[N -2].Z^{1} + r[N - 1] - Q[N] = 0 The term Z^{0} has be taken to have the value unity, one. In equation [3], taking either Q[N], corresponding to r[N - 1] = - 1, or Q[N] corresponding to r[N -1] = 1, the form of Equation [3] is the very same, and is in fact the equation representing an intermediate even number between to connected uneven integers. By inspection, Equation [3] has the factor, [Z - 2]. The correct values of r[1], ... N - 1] have to be used in Equation [2] and Equation [3], to represent the particular value of Q[N]. When Equation [3] is divided by [Z - 2] and then the value Z = 2 is substituted into the algebraic quotient, an uneven integer results. This process derived from either of two connected twin primes, say 101 or 103 can be continued indefinitely so that 2D space can be filled with even and derived uneven integers. And this is for one connected twin prime. There are many ramifications of the above.
My thoughts on the necessary approach to the solution: #1 - Keep in mind what the gap of 2 actually means. It's not just some random constant, but the smallest possible gap between primes (if you don't count 2 itself as just "prime", but it's obviously a very special and unique kind of prime, it's the starting prime). and #2 - Something very similar to the same simple approach to prove there's infinitely many primes. Just the same way you can prove there must be always be more primes, you should be able to prove there must always be more twin primes. Once you've cycle through all possible combinatorial states of all the primes up to a given point, it should be obvious not only is there another prime, but there is another prime immediately after that. The combination must be something like n1^2!*n2^2!*n3^2!... with all primes up to that point, or something, then add on 1 (or perhaps a more complexly generated constant based on the log of the number or something)
the only possible problem I can think of is whether or not you can ever truly get past the cyclical nature of the way different prime factors will begin to stack up on top of each other, sort of just out of phase of each other, and after enough numbers have filled all the phase gaps so to speak, you can expect no more twin primes.
final thought: also, it just seems like it would be EASY to prove there WEREN'T infinitely many twin primes through this line of logic, IF that happened to be the case. And so, since that doesn't seem to be the case, you could maybe somehow use that logic (combined with other logic about the primes like they behave pseudo randomly) to maybe prove there must be infinitely many
Is it possible to get a gap of 2n between consecutive primes before you get a gap of 2n-2 ? It doesn't seem so. Before we ever record a gap of 6 for instance, gaps of 4 and 2 must have occured first. Is this true? If so, say we know some prime p and we want to compute the next prime. All we need to do is record all the gaps between consecutive primes up to p. Let 2n be the largest gap recorded. The maximum next gap is 2n+2 if the answer to my initial question is yes. So there must be at least one prime between p and( p+2n+2).
I feel like a philosophical argument that counting numbers and whole number division are a part of the real world would be fairly easy to construct. And primes are numbers for which whole number division is impossible. Integers and Real numbers however technically have little to no direct basis in reality.
All numbers are just mathematical objects that are defined as they are. You could easily say anything is part of reality just because you are thinking about it, so it is pointless to even talk about it.
+Ishaan Sabnis Statements about numbers are also quantifiable statements about physical reality. Whether you can divide a number of objects into equal piles is something that you can measure. No, numbers are not purely mental constructs, I've seen that claim before.
I think the key to this problem's solution doesn't lie in the twin primes themselves, but rather the numbers that show up between them. Between any two twin primes lies an even composite number. There is a sequence of numbers following these criteria. The sequence is 4, 6, 12, 18, 30, 42, 60, 72, 102, 108, and so on.
Such numbers are either divisible by 30 or have the rest of 12 or 18 from division by 30. If such numbers differ by 6, it indicates maximal concentration of primes - four in ten (quadruplets), starting from 11, 101, 191, 821, etc. If such numbers are not divisible by 30, it indicates possibility of twins to have cousin prime pair, twin primes belong to two triplets, The sequence of first elements is 7, 13, 37, 97, 103, 223, 307 and so on. If such numbers are divisible by 30, twin primes can make sexy primes pairs. It indicates maximal concentration of primes near the number divisible by 30. Start numbers: 23, 53, 263, 563, 593...
So I've just watched James interview on getting the Fields Medal, which make me Google what he got it for and then this was next in TH-cam from 6 years ago...
I would like to suggest, without evidence, something even more specific. That there are an infinite number of pairs of primes separated by 2, where the prime factors of the inbetween number are consecutive primes. For example, 29 and 31 and primes, and the factors of 30 are the consecutive primes 2, 3 and 5.
The next number is 2x3x5x7 = 210. 209 isn't prime, 211 is. then 210x11 = 2310. 2309 & 2311. Both of these are primes. 2nd example. Then 2310x13 = 30030. 30031 isn't prime, 30029 is Then 30030x17 = 510510. Neither 510509 and 510511 are prime Then 510510x19 = 9699690. Doesn't work. Then 9699690x23 = 223092870. Doesn't work. Then 6469693230. Doesn't work. Need a computer to take it much further. The way it looks, I don't see a third example coming, and you think there are infinite number of these? If you can prove that, then you've proven the twin prime conjecture and then some. But I think a proof would be very hard to come by, for this simple reason: eventually you're going to end up with numbers on either side that are so large, you won't be able to factor them, or even test they are prime with the computer technology we have, so you'll in effect get 'stuck'. Notice you have to start at 2, because if I took a sequence like 5x7x11 I end up with an odd number and those on either side will be even and obviously not prime.
I'm aware how rare they would likely be if I'm right, and I'm aware it would certainly be more difficult to prove than the twin prime conjecture. But if you compare it to Mersenne primes, we only know 49 of those, going up to 274,207,281 − 1 but we know those are infinite in number. And I didn't merely notice that you have to start it 2; it is by design. My original thought was on multiplying all primes up to an arbitrary value, but I realised I could phrase it more clearly by saying consecutive primes and leaving the start implicit.
If you know of a proof that there are inifinite primes of the form 2^n - 1, I'd like to see it. Again, by computation theory as we know it, you eventually end up with a number so huge you can't factor it, rendering you stuck. But if there's a proof that gets around this, excellent, I'd like to see it.
There also seem to be lots of prime quads, consisting of sets of values 30k + 11, 30k + 13, 30k + 17, and 30k + 19. The gaps are, as is to be expected, larger than the gaps between prime pairs, but the JavaScript that I am currently running to find them has reached 170 million and is still finding them.
You do not have to be a professional or even go to college to contribute to math. Study some books, papers, and videos. Find a challenging problem, you are passionate about and try to make some progress. Even as an amatuer, any progress is better than none.
All primes > 6 are Mod(N,6) = 1, or Mod(N,6)=5. (which is a brief proof). Whereas Mod-5, Mod1, are two numbers apart. So, it might be the case, as numbers get larger, that there are MORE twin primes. Just thinking out loud. As there are fewer primes as the numbers get larger.
Grzegorz Cichosz Well I know a mathemathician who worked in a engineering company. But now he is a teacher, he says he prefers being with people rather than being all day in front of a computer.
Plenty of mathematicians take time out of their academic careers to work for technology firms (or do so alongside their career if they can balance it). Some others work for government bodies on cryptography based problems
I find that my most creative time of day is the first hour of the day when I'm still feeling that the dream-making part of my brain is still partly in control.
Dekinain Janai : What constitutes "real life"? Food, reproduction and survival? Technology? More money to buy stuffs? Inner peace? Staying high on THC? Understanding the nature of the universe and reality itself?
This Conjecture is one of those examples of something you know about by feel but have trouble specifying in unambiguous terms, (that is what "Conjecture" means). But if it is re-cognised all quantization information In-form-ation is log-antilog condensation, harmonic relative-timing modulation cause-effect mechanism and the general property of ln, base e, determines states of being like Twin Primes amongst other numberness, the "Stage is Set" in e-Pi-i sync-duration resonance quantization for a Standardised Theoretical Analysis and reduction of the Measurement Problem situation to a Disproof Methodology, ie it'll take forever to solve or prove it is reasonable in Principle. But it is fun to imagine..
What's the largest gap that we can create? There doesn't seem any reason why (at least that I can think of) gaps between primes can't be found to be arbitrarily large (since primes exist onto infinity, it would seem we could make the gaps between them as large as we want by increasing the number of primes). But if we can find a gap of any number, then how at the same time can there be infinite number of gaps of a finite size? This is truly a really difficult problem.
We can create a gap as large as we want very easily, take a number n and consider the string of numbers {n!+2,...,n!+n}, they are all composite numbers and the string is n-1 elements long, so by increasing n this gets as large as you want. You should rethink the second question, you'll see it's very clear too.
insidetrip101 Clearly there is no limit to how large the gap between primes can be. Given a positive integer n and k = n!, then k+2, k+3, k+4, etc. up to k+n are all necessarily composite, for a guaranteed gap of n-1. So we have the paradox that the number of primes is infinite, but that's not so illogical - an infinite subset can be taken from an infinite set (for example, take all the primes from the set of positive integers) and still be left with an infinite set.
The state of the art in finding big gaps formed of primes lowest as possible is this paper of Ford, Green, Konyagin, Maynard & Tao : arxiv.org/abs/1412.5029 It is an improvement over an almost 80 year-old result of Rankin. It uses almost the same method but tweaks a final argument by using (an involved version of) the improvement of Maynard concerning small gaps.
"You should rethink the second question, you'll see it's very clear too." I don't think it is. Because if we can create a gap that is arbitrarily large, then we can create a gap that goes on forever; however, that's impossible because we can also prove that there is an infinite number of primes. Thinking about infinity is never simple. For example: "So we have the paradox that the number of primes is infinite, but that's not so illogical - an infinite subset can be taken from an infinite set (for example, take all the primes from the set of positive integers) and still be left with an infinite set." You can say that, but do we actually have an understanding for what that means? I genuinely can't wrap my head around it. Yea I know we can do such a thing in number theory; but I can't get an understanding of how we can have an infinite amount of *integers* in between two primes. I hear what you guys are saying, I understand the proof that you guys are also citing, but it still doesn't sit right with me.
Nice to see how humble he is, given the massive contribution he made.
Cheers Brady! Love your work!!
John Redberg isn't he just humble bragging? ;)
No, he is just being modest
Fiqih Fandrian if he is (which I think he isn't), then he's doing an excellent job. Which itself would be something to complement him on. ;-)
LOL NOPE
It feels like Brady is trying to assemble a team of top mathematicians to crack the Riemann Hypothesis. Unfortunately, they don't seem very interested.
Your username says it all
@@aazimshahul7488 yeah disgusting
Problems like the RH are career-killers. No prizes are given for almost being right. None at all.
Or just overwhelmed.
3 years ago, its still out there :-). Intuitivley the RH is unprovable. So much hinges on it and it works so well that there is some fundemental property of numbers that means its unprovable. If you could prove its unprovable you would be famous.
James, congrats on the Fields Medal! Well deserved.
Recently I saw a TV show on PBS about the "Twin Prime Conjecture" featuring both Tom Zhang and James Maynard, two very smart guys. :)
On April 2013 professor Tom (Yitang) Zhang of the University of New Hampshire has submitted a paper to the Annals of Mathematics as proof that there are an infinite pairs of prime numbers that differ by 70 million or less. This spurred a lot of activity in the field, such as the Polymath8 project, which lowered the bound to k ≤ 246 and recently to k ≤ 6 (general Elliott-Halberstam conjecture). The human mind is truly amazing being able to tackle and solve such difficult problems.
I suspect he found it in a field.
@@apusapus71 I need to start visiting farms more often then.
Huge congratulations to James on winning the fields medal for 2022, absolutely amazing we get to watch him talk through his work
Brady, well done!
This is what I like so much about Brady's videos across all his channels: Unlike many interview-based videos, Brady so often gets his subject to tell a full story arc before getting to their own contributions. It takes careful reading and prompting to make this happen, and to look so effortless and flow so naturally.
Best of all is Brady's infectious excitement: He wouldn't be doing this unless he was genuinely thrilled by it. He calls himself a videographer, but I think that's his secondary talent. You can't do great edits if the material isn't there in the first place.
And then there are the times when his subject goes into awesome mode, and Brady knows to just step back, be a videographer, and let things go wherever. Talk about being in the moment.
Thanks!
Agreed.
Such a passionate person! Would love to see more episodes with him.
We have another one in the works.
You are one of the best channels on TH-cam! Thanks for existing!
And cute too.
Numberphile he needs a podcast episode
@@Fk67Lg get in line
This has been one of my favorite videos in recent times. Dr. Maynard was clear in his explanation and he looked so happy to be explaining it. That is the type of passion everyone needs to find in their life.
Such a humble and genuine young man. It is so refreshing and encouraging to see a brilliant intelectual mind with this personality . All the best wishes to Dr. James. Looking forward to see more videos of him!
There's not a single video on this channel that I don't like but this one I found particularly enjoyable to watch.
Gabriel's horn is amazing too!
A prime minister is a minister that is divisible by 1 and him/herself.
stylus59 but only 1 and theirself
stylus59 I heard that joke from a comedian on Conan's show.
Most prime ministers cant even
Look at Justin Trudeau.
+
Now a fields medalist!
This man has been awarded with the Fields Medal in 2022. for his contribution.
The partially erased blackboard is driving me crazy.
GildedBear , You are psychologically sick.
of course you can't check with a computer, they're doing it the wrong way! They start at 0 and go towards infinity, but if they instead started from infinity and went backwards, by the time you reach 0 you would have checked ALL THE NUMBERS! Fields Medal please...
Oh, it also works with the digits of Pi
Goryllo OK THIS is epic
Computers don’t know the definition of infinity since their memory is finite
We can start at the Parker infinity, which is not quite infinity, but kinda close
Cant start at a quantified number since it defeats the point of Infinite.
BREAKTHROUGH IN THE METHOD!
The fact that he lives and works in Oxford, where I study, makes this video a little weird as he sounds practically identical to many of my younger lecturers and tutors. This video more than many others on numberphile felt like I was being taught a class! Very well explained
I love that Numberphile gets these top mathematicians who are still willing to start from first principles (reminding the viewer that 2 is the first prime, etc.)....
I've made a wonderful proof in which i bringed the gap down to 2 proving The Twin Primes Conjecture.
However there is not enough space in the youtube comments section to write that here.
Hahaha
Pierre de Fermat *brought
DAMN YOU FERMAT!!!
Dude, maybe you'd be happier in another line of work. Like running for office.
i dont think people realize what you just did there
How beautifully constructed and expressed his sentences are!
So happy to see James Maynard has become the fields medalist❤️
I like the way he moves when he talks, like he got a funky side 😎 but also very humble nice and smart !! thank you very much to him for his wonderful work and sharing that, so adorable human
Nice to see how humble he is, given that in 5 years he will win the Fields Medal
As a matter of fact, James just won the Fields medal a few days before you wrote that comment.
@@j.vonhogen9650 I think you didn't get the comment above.
Porriale means that James was still humble 5 years ago when he knew that he might win a fields soon.
Congratulations on the Fields Medal Professor!
I absolutely love these videos. Number theory is one of my favorite subjects.
I am very much surprised at how amazingly accurately James Maynard pronounces YiTang Zhang.
Could Numberphile do a video about the recent Abel Prize earning work subject from Yves Meyer, the "Wavelette Theory" ? The Abel Prize is like the Nobel of maths (after Fields medal), and it was awarded less than a month ago --- it's a cool thing to make a video on, especially since there is little information about it on the Internet ! Please !
Placki Plicki , No one cares for this rubbish idea. Throw it in dustbin, you fool.
Imroz, calm down douchebag.
Imroz zahan
No one cares for this rubbish comment, throw it in *the* dustbin, you fool
no idea what that is honestly, but if it was abel prize worthy there probably should be a video on it
Congratulations on winning the Fields medal James. I admire these humans so much . I went to Oxford read Zoology from 1989 - 1992, I got an upper class second degree. Also got a boxing blue for boxing against Cambridge on March 12th 1992. I arrived at Oxford as a bright boy with serious memory capability and a thirst for competition in tests. I very quickly observed around me a level of student that I could only marvel at. A healthy reset of expectations. The very best of my peers were like James in the way that their ability was to them normality and like a frequency hum in their background. No frills; no interest in recognition. They were simply unbelievable in their fields.
I hope to see the day the Riemann Hypothesis is solved...
I have found a truly marvelous solution to the Riemann Hypothesis, but my brain is too small to contain it.
+Symbiosinx what an original joke
Fresh beans
It actually is original because he said "my brain is too small to contain it" not the comment box
probably armageddon
I already solved it
Really great video... congrats to dr. Maynard and Brady, your questions always amaze me... great work as well.
I hope I'm able to prove something that is worthy of being presented on Numberphile some day.
This is one cool, down to earth mathematician. I'm a layman, but I look forward to seeing what else he can come up with.
This is great. I'll never get tired of learning about prime numbers :-)
It's really fascinating to see this kind of breakthrough when you don't understand the most fundamental principles of prime numbers other than it's only divisible with itself and 1.
It's inspiring to see a younger mathematician talk about his work!
Nice to see he is so humble. "Intuition" or the "inspiration", that "i am on the right track" even though you are not sure you will get to the proof, comes from the supersoul within (Sanskrit: paramātmā). We live in a virtual reality. The mind is the real cause of our suffering and happiness. Never think that "I am the doer," especially when you do not know who exactly "I am".
You missed a twin prime at 2:51 (19541 and 19543)
Agam Kohli They did to! I never even noticed, so thanks for pointing it out😀
19541
19543
🤩
The set {2,3,A,B} contains all prime numbers in the natural set.
Where:
A={6n-1: n is natural}/{36xy-6y+6x-1: (x,y) are natural}
B={6n-5: n is natural}/[{1} U { 36xy-6y-6x+1} U { 36xy+6y+6x+1}: (x,y) are natural]
The formula A+2=B contains all the cases of prime sum of two primes, except 2+3=5.
The set {2,3,A} contains all the Sophie Germain primes.
B can't be Sophie Germain due to divisibility of 2B+1 by 3.
Is there a website where we can find some mathematical papers? Like where do mathematicians publish their papers online?
Try Google Scholar as a starting point
arxiv.org
Ok thank you very much guys!
Julio Presidente In what language do you want the papers?
William White Albert Einstein published without peer review and citation in his seminal year. Do not discourage Julio. Maybe Julio has insights that give us that "quantum leap" forward. 😑😑😑😑😑😑😑😑😑
My understanding (from watching talks by Terence Tao) is that the barrier he refers to several times in this video is something called the 'parity problem'. I would love to see a video explaining what the 'parity problem' is.
1451, 1453 is a twin prime pair associated with protons and neutrons. If you divide half the difference of their masses into them you get the twin prime pair.Proton's mass = 938.272081 Mev/c2 Neutron's mass = 939.565413 Mev/c2
This James Maynard guy is pretty Keen
Venator Longstride an
Cool with this channel, where you can experience stars of mathematics lecturing or telling their stories. And in this one, a Fields medal recipient, interviewed with a modest not fully erased math-blackboard and a small notice board in the background. Totally relaxed.
Watched this video just after having watched another one about the Twin Paradox. So now, I imagine two twin primes, and one of them taking a rocket, flying at the speed of light, and returning close to his twin prime. But now, their distance had become more that 2, because of Special Relativity.
I would say thanks to Zhang, Maynard and Tao et al. Because I had no idea about such beautiful results involving prime numbers.
"...alot of fumbling around in the darkness before you understand how things work..."
I see
The equation 6n+-1, can be used as a serial equation or a matrix. The matrix first column is odd numbers and the top row is factoials of prime numbers, 6n, 30n, 210n, 2310n and etc. The result is: 30 + 11 and 13, 30 + 29 and 31, 210 + 29 and 31, 2310 + 29 and 31 all twin primes.
6n+-1 is not an equation.
Who's back to this after he won the fields medal?
Prime numbers aside, it is refreshing to see you present your perspective on mathematical thinking. I am looking at how to inject this aspect into the high school arena since it is virtually uncatered for (at least in Australian schools). Schools are results driven and are more or less merely a set curriculum production line. My argument is that there is more - it is what YOU do and there is a way to cater for it at high school level with students of the right mindset. You have clearly expressed the "genius factor" that is lacking and not really understood in our education system. The thing is, how do we turn what you do into a television series to show how some mathematical ideas can be explored with predominately high school mathematics? Well done, so far!
My mom is still trying to decide where to move the sofa...
I am an Identical twin, and i have been able to map out our life using simple math, you just have to know how to do it.... Its an Amazing Phenomenon
Dude, this guy won the Field's medal in 2022
i just nticed that whn they had he list going up that they missed a set of twin primes 19541 and 19543 that thy didnt highlight. i love how even in my late 20's this channel makes me fell ok to still find math interesting.
Friday nights, "phew a long week of rigorous mathematics, time to kick back, relax and think about that ole' twin prime conjecture"
James Maynard is very impressive, he has done the UK well
You missed the twin primes 19541 and 19543 at 2:51
"What about if we use YOUR method"
-"I think WE..." - James
I enjoy he puts We instead of My or I, he shows he wants everyone to enjoy maths no matter what its called.
Identical or fraternal ?
half
M.K.D. a few words.
He explains so clearly and humbly
I like this guy, more videos with him please :)
Congratulations for the Fields medal 2022 Prof Maynard!
2:52 You guys missed the pair of 19541 and 19543
6ab+-a+-b. 4 equations. Any integer that can't be created this way multiplied by 6 and +-1 must be prime twin. So the problem is reduced to something like +- n such that above n+m15 in many cases. Among others n={1,2,3,5,7,10,12,17,18,23,25,30,32,33,38,40,45,47,52,58,63,70,72,77...} It is highly unlikely that at any point complexity will become so large that every number can be calculated using 4 variations of the equation yet I don't know how to prove it.
nth
(n-1)th
(n-2)th
tamil movies
n/0th
There is an Incompleteness Theorem by Kurt Godel which could apply to proving the twin prime hypothesis. Equations involving the imaginary operator might help. The equations: i^{4n + 2} = - 1 and i^{4n} = + 1 are involved. An equation: Q[N - 1] = [1/2] [Q[N] - i^{1 + Q[N]}] does not look to be provable but is axiomatic. In this equation, Q[N] is an uneven integer lying between 2^{N} and 2^{N + 1}. The integer " N " is a subscript, with N taking values: 1, 2, 3, ...
N is counted down to N = 1. Inevitably, Q[N - 1] is found to be an uneven integer from this equation. With regard to twin primes, if p[k] and p[k + 1] are two primes belonging to a twin set, for example 101 and 103, then the equation Equation [1]: [p[k] - i^{1 + p[k]}] - [p[k + 1] - i^{1 + p[k + 1]}] = 0 for some twin primes and = 4 for other twin primes. Examples where this equation give the answer, zero are: 17, 19 ; 29, 31 ; 41, 43 ;101, 103 ; 149, 151 ; 197, 199 .. ad infinitum. Examples where the answer is four are: 11, 13 ; 59, 61 ; 71, 73 ; 107, 109 ... ad infinitum.
The above considerations could help establish or otherwise the twin prime hypothesis. This involves bringing in the imaginary " i " into simple high school arithmetic and algebra - extra outside axiom.
The equation: Q[N - 1] = [1/2] [Q[N] - i^{1 + Q[N]}] is a rearrangement of a division which is Q[N] = 2.Q[N - 1] + r[N - 1]. The term, i^{1 + Q[N]} is identifiable with the remainder, r[N - 1], which has the value minus one or plus one, only. That is r[N - 1] = - 1 or r[N - 1] = + 1.The remainder, r[N] can be chosen to be one or minus one, but has no relevance in the above context.
Taking the iterations of the equation: Q[N] = 2Q[N - 1] + r[N -1], counting down, the last two equations are: Q[2] = 2Q[1] + r[N - 1] and Q[1] = 2Q[0] + r[0].
It is always found that always, Q[1] = 3, Q[0] = 1 and r[0] = 1.
Algebraic elimination of Q[1], ... Q[N - 1], a very easy exercise, leaves the equation: Q[N] = q[0]2^{N} + r[0]2^{N - 1} + r[1]2^{N - 2} + ... + r[N - 1]2^{0} The q[0] and r[0] terms can be left out. Replacing the base, two by the unknown Z, gives: Equation [2]: Q[N] = Z^{N} + Z^{N - 1} + r[1]Z^{N - 2} + ... + r[N - 1]Z^{0}
There are two values of r[N - 1], which are r[N - 1] + - 1 and r[N - 1] = + 1. These correspond to two values of Q[N]. In the case of two prime numbers that give the value, zero in Equation [1], above, then theses twin primes may be said to be connected, otherwise, if four results then the two primes are disconnected. The intermediate even number between any two connected integers has the value: [2 + 4n] for some integer, n. For example, [2 + 4n] =102, gives, n = 25.
If Q[N] is transferred to the right hand side of Equation [2], then the equation, Equation [3]:
Z^{N} + Z^{N - 1} + r[1]Z^{N - 2} + ... r[N -2].Z^{1} + r[N - 1] - Q[N] = 0 The term Z^{0} has be taken to have the value unity, one. In equation [3], taking either Q[N], corresponding to r[N - 1] = - 1, or Q[N] corresponding to r[N -1] = 1, the form of Equation [3] is the very same, and is in fact the equation representing an intermediate even number between to connected uneven integers. By inspection, Equation [3] has the factor, [Z - 2]. The correct values of r[1], ... N - 1] have to be used in Equation [2] and Equation [3], to represent the particular value of Q[N]. When Equation [3] is divided by [Z - 2] and then the value Z = 2 is substituted into the algebraic quotient, an uneven integer results. This process derived from either of two connected twin primes, say 101 or 103 can be continued indefinitely so that 2D space can be filled with even and derived uneven integers. And this is for one connected twin prime.
There are many ramifications of the above.
Classic but still nice! Good job :-)
james recently won the Fields medal! congrats
Nice to see that Benedict Cumberbatch is interested in maths too.
Yakushii ... who?
Bendadick Cucumberpatch is a famous actor.
Tyko Brian he plays Dr.Strange
Benjamin Lehman
Yeah, Bean pick comber patch did.
Or was his name Bin trick clapper catch?
My thoughts on the necessary approach to the solution: #1 - Keep in mind what the gap of 2 actually means. It's not just some random constant, but the smallest possible gap between primes (if you don't count 2 itself as just "prime", but it's obviously a very special and unique kind of prime, it's the starting prime). and #2 - Something very similar to the same simple approach to prove there's infinitely many primes. Just the same way you can prove there must be always be more primes, you should be able to prove there must always be more twin primes.
Once you've cycle through all possible combinatorial states of all the primes up to a given point, it should be obvious not only is there another prime, but there is another prime immediately after that. The combination must be something like n1^2!*n2^2!*n3^2!... with all primes up to that point, or something, then add on 1 (or perhaps a more complexly generated constant based on the log of the number or something)
the only possible problem I can think of is whether or not you can ever truly get past the cyclical nature of the way different prime factors will begin to stack up on top of each other, sort of just out of phase of each other, and after enough numbers have filled all the phase gaps so to speak, you can expect no more twin primes.
final thought: also, it just seems like it would be EASY to prove there WEREN'T infinitely many twin primes through this line of logic, IF that happened to be the case. And so, since that doesn't seem to be the case, you could maybe somehow use that logic (combined with other logic about the primes like they behave pseudo randomly) to maybe prove there must be infinitely many
I didn't know that the guy from Tool was so gifted at mathematics
Is it possible to get a gap of 2n between consecutive primes before you get a gap of 2n-2 ? It doesn't seem so. Before we ever record a gap of 6 for instance, gaps of 4 and 2 must have occured first. Is this true? If so, say we know some prime p and we want to compute the next prime. All we need to do is record all the gaps between consecutive primes up to p. Let 2n be the largest gap recorded. The maximum next gap is 2n+2 if the answer to my initial question is yes. So there must be at least one prime between p and( p+2n+2).
Wow, this was a great video :D
Thanks
Glad to see him get awarded with the Fields medal this year!
Prime numbers make me uncomfortable
Prime numbers make me wet, and that makes me uncomfortable
I feel like a philosophical argument that counting numbers and whole number division are a part of the real world would be fairly easy to construct. And primes are numbers for which whole number division is impossible. Integers and Real numbers however technically have little to no direct basis in reality.
All numbers are just mathematical objects that are defined as they are. You could easily say anything is part of reality just because you are thinking about it, so it is pointless to even talk about it.
you must be fun at parties.
+Ishaan Sabnis
Statements about numbers are also quantifiable statements about physical reality. Whether you can divide a number of objects into equal piles is something that you can measure. No, numbers are not purely mental constructs, I've seen that claim before.
I think the key to this problem's solution doesn't lie in the twin primes themselves, but rather the numbers that show up between them. Between any two twin primes lies an even composite number. There is a sequence of numbers following these criteria. The sequence is 4, 6, 12, 18, 30, 42, 60, 72, 102, 108, and so on.
Such numbers are either divisible by 30 or have the rest of 12 or 18 from division by 30.
If such numbers differ by 6, it indicates maximal concentration of primes - four in ten (quadruplets), starting from 11, 101, 191, 821, etc.
If such numbers are not divisible by 30, it indicates possibility of twins to have cousin prime pair, twin primes belong to two triplets, The sequence of first elements is 7, 13, 37, 97, 103, 223, 307 and so on.
If such numbers are divisible by 30, twin primes can make sexy primes pairs. It indicates maximal concentration of primes near the number divisible by 30. Start numbers: 23, 53, 263, 563, 593...
I love Numberphile :)
So I've just watched James interview on getting the Fields Medal, which make me Google what he got it for and then this was next in TH-cam from 6 years ago...
Congratulations to this man for proving the Duffin - Schaeffer conjecture (do you guys see the news?)
He annouced a proof with another mathematician, it hasn't been reviewed yet
@@cptn_n3m012 oh, I see.... Thanks for pointing that out
Hmm, he looks pretty smart, maybe he would win a fields medal for his contributions to number theory
is there any upper bound on gaps between two primes?
No. They can be as far apart as one demands.
Take k=N! and numbers k+2, k+2,... k+N. They are all composite.
Now pick any large N as you wish.
Yes that's what the video is all about
246
This man proves that if you really really want to believe the Conjecture has not been solved already...then you can! I offer my congratulations.
We can thank gödel who proved that we can t prove some things.
How do you know about that? Lol
Who else watching again after it was announced that James Maynard will receive Fields Medal this year (2022)?
I would like to suggest, without evidence, something even more specific. That there are an infinite number of pairs of primes separated by 2, where the prime factors of the inbetween number are consecutive primes. For example, 29 and 31 and primes, and the factors of 30 are the consecutive primes 2, 3 and 5.
The next number is 2x3x5x7 = 210. 209 isn't prime, 211 is.
then 210x11 = 2310. 2309 & 2311. Both of these are primes. 2nd example.
Then 2310x13 = 30030. 30031 isn't prime, 30029 is
Then 30030x17 = 510510. Neither 510509 and 510511 are prime
Then 510510x19 = 9699690. Doesn't work.
Then 9699690x23 = 223092870. Doesn't work.
Then 6469693230. Doesn't work.
Need a computer to take it much further.
The way it looks, I don't see a third example coming, and you think there are infinite number of these? If you can prove that, then you've proven the twin prime conjecture and then some.
But I think a proof would be very hard to come by, for this simple reason: eventually you're going to end up with numbers on either side that are so large, you won't be able to factor them, or even test they are prime with the computer technology we have, so you'll in effect get 'stuck'.
Notice you have to start at 2, because if I took a sequence like 5x7x11 I end up with an odd number and those on either side will be even and obviously not prime.
I'm aware how rare they would likely be if I'm right, and I'm aware it would certainly be more difficult to prove than the twin prime conjecture. But if you compare it to Mersenne primes, we only know 49 of those, going up to 274,207,281 − 1 but we know those are infinite in number.
And I didn't merely notice that you have to start it 2; it is by design. My original thought was on multiplying all primes up to an arbitrary value, but I realised I could phrase it more clearly by saying consecutive primes and leaving the start implicit.
If you know of a proof that there are inifinite primes of the form 2^n - 1, I'd like to see it.
Again, by computation theory as we know it, you eventually end up with a number so huge you can't factor it, rendering you stuck. But if there's a proof that gets around this, excellent, I'd like to see it.
Sorry, my mistake. I thought that was proven, but it is merely conjectured. I must have misremembered the content of an earlier Numberphile video.
There also seem to be lots of prime quads, consisting of sets of values 30k + 11, 30k + 13, 30k + 17, and 30k + 19. The gaps are, as is to be expected, larger than the gaps between prime pairs, but the JavaScript that I am currently running to find them has reached 170 million and is still finding them.
Proof by javascript. I like it.
I wish I was a mathematician sometimes.
Just take a mathcourse or go study by yourself
You do not have to be a professional or even go to college to contribute to math. Study some books, papers, and videos. Find a challenging problem, you are passionate about and try to make some progress. Even as an amatuer, any progress is better than none.
very good video, thank you Brady and the mathamtitions for your work and generosity
I proved that no two prime numbers differ by 7.
I'm a famous mathematician now :)
Heliocentric post your proof
email me so that I can also be famous.. thanks in advance.
Heliocentric 2017 2027 differ by 10 both are primes
-5 is a prime? do we count negative integers also? just asking cuz I really don't know>
Heliocentric oh oops i misunderstood
All primes > 6 are Mod(N,6) = 1, or Mod(N,6)=5. (which is a brief proof). Whereas Mod-5, Mod1, are two numbers apart. So, it might be the case, as numbers get larger, that there are MORE twin primes. Just thinking out loud. As there are fewer primes as the numbers get larger.
Do professional mathematicians have any other job aside from teaching at university?
Grzegorz Cichosz They can work in engineering companies
Oh really? I thought that some engineers work their not mathematicians
Grzegorz Cichosz Well I know a mathemathician who worked in a engineering company. But now he is a teacher, he says he prefers being with people rather than being all day in front of a computer.
Plenty of mathematicians take time out of their academic careers to work for technology firms (or do so alongside their career if they can balance it). Some others work for government bodies on cryptography based problems
Grzegorz Cichosz I don't know if you are aware, but there is a lot of math involved in engineering
Also pretty much every field of hard science
And now he’s got a Fields medal for his work :)
Just how many channels does Brady have for crying out loud?!
Docobonbon The one how many do you know?
1.Numberphile
2.Computerphile
3.Sixty Symbols
4.Periodic Table of Videos
5.Objectivity
There are probably more, but those are the ones I know about.
James Maynard just won the field medal this year.
Primes-Probably Really Ingenious Maybe Easy Solution
Please no one like this comment. It's not funny or creative, just scroll past. Thanks
Damnit
YOU CAN'T TELL ME WHAT TO DO
Ewe can't tell me what to do. True.
I dunno, it's funnier than the "I have a proof of the twin prime conjecture, but it's too big to put here" which has 200 likes.
I find that my most creative time of day is the first hour of the day when I'm still feeling that the dream-making part of my brain is still partly in control.
This makes me ask the very pressing question: What the hell am I doing with my life??
Dekinain Janai : What constitutes "real life"? Food, reproduction and survival? Technology? More money to buy stuffs? Inner peace? Staying high on THC? Understanding the nature of the universe and reality itself?
calm down bro. drink milk. you'll be fine.
@@subh1 just THC
This Conjecture is one of those examples of something you know about by feel but have trouble specifying in unambiguous terms, (that is what "Conjecture" means).
But if it is re-cognised all quantization information In-form-ation is log-antilog condensation, harmonic relative-timing modulation cause-effect mechanism and the general property of ln, base e, determines states of being like Twin Primes amongst other numberness, the "Stage is Set" in e-Pi-i sync-duration resonance quantization for a Standardised Theoretical Analysis and reduction of the Measurement Problem situation to a Disproof Methodology, ie it'll take forever to solve or prove it is reasonable in Principle. But it is fun to imagine..
They missed a twin in the list: 19541, 19543
Duane Schuh , well done old man. :-)
Congratulations to Dr. Maynard on the Field's medal!
What's the largest gap that we can create?
There doesn't seem any reason why (at least that I can think of) gaps between primes can't be found to be arbitrarily large (since primes exist onto infinity, it would seem we could make the gaps between them as large as we want by increasing the number of primes). But if we can find a gap of any number, then how at the same time can there be infinite number of gaps of a finite size?
This is truly a really difficult problem.
We can create a gap as large as we want very easily, take a number n and consider the string of numbers {n!+2,...,n!+n}, they are all composite numbers and the string is n-1 elements long, so by increasing n this gets as large as you want.
You should rethink the second question, you'll see it's very clear too.
insidetrip101 Clearly there is no limit to how large the gap between primes can be. Given a positive integer n and k = n!, then k+2, k+3, k+4, etc. up to k+n are all necessarily composite, for a guaranteed gap of n-1. So we have the paradox that the number of primes is infinite, but that's not so illogical - an infinite subset can be taken from an infinite set (for example, take all the primes from the set of positive integers) and still be left with an infinite set.
The state of the art in finding big gaps formed of primes lowest as possible is this paper of Ford, Green, Konyagin, Maynard & Tao : arxiv.org/abs/1412.5029
It is an improvement over an almost 80 year-old result of Rankin. It uses almost the same method but tweaks a final argument by using (an involved version of) the improvement of Maynard concerning small gaps.
"You should rethink the second question, you'll see it's very clear too."
I don't think it is. Because if we can create a gap that is arbitrarily large, then we can create a gap that goes on forever; however, that's impossible because we can also prove that there is an infinite number of primes.
Thinking about infinity is never simple. For example:
"So we have the paradox that the number of primes is infinite, but that's not so illogical - an infinite subset can be taken from an infinite set (for example, take all the primes from the set of positive integers)
and still be left with an infinite set."
You can say that, but do we actually have an understanding for what that means? I genuinely can't wrap my head around it. Yea I know we can do such a thing in number theory; but I can't get an understanding of how we can have an infinite amount of *integers* in between two primes.
I hear what you guys are saying, I understand the proof that you guys are also citing, but it still doesn't sit right with me.
Arbitrarily large != infinite
Thank you Dr. Maynard.
Keep this comment with prime number of likes !
Heliocentric 17
Heliocentric 19!
23!
I was going to like but it's 31 so I'm very sorry
29 ^^/