It makes me smile thinking that, if Jared was born 300 years ago, his name would appear in textbooks and we'd probably have nothing but a single painting of him to know what he looked like. And yet here we are, watching a TH-cam video of him explaining his theorem for free.
When guys like Adam Savage talk about the magic of Numberphile, this is *exactly* the kind of video he's referring to. A young mathematician finding beauty in a famous conjecture, works in his spare time to prove it, and all throughout the video Brady is not only teasing out the points that help us laypeople understand it, but also highlighting the personality of the mathematician himself. TH-cam at its best.
@@christophersmith108 I thought of this while watching the video. I feel like he’s probably one of the only people to be able to legitimately make that claim since Erdös’s death
@@oliverwhiting7782 To get an Erdös number of 1 you need to collaborate on a paper with Erdös. As cool as it is to prove an Erdös conjecture, it is not at all the same thing. There will never be another 1.
Perfect Numberphile content. Complex but beautiful problem - simply and clearly explained. Plus an Erdős connection. More from Jared Duker Lichtman please.
What a fantastic communicator. He knew just the right tidbits to throw in to help people through his explanations. He was excited and charming. I hope to see him back here!
Absolutely. I was trying to express the same thought, but the words wouldn't come to me. I wish most professors could convey complex topics with anywhere near this clarity, studying STEM subjects would be that much easier!
btw, Jared's supervisor is (I believe) James Maynard, who has been on the channel before! As a side note, I'm super bummed that Brady came to film in my department while I was there and I didn't see him - the reason I applied to do maths at uni was the twin prime conjecture video that Brady did with James Maynard (and then I got to take his course on analytic number theory, which was super cool).
"When you discover something in math, out of humility you don't name it after yourself, you wait for your friends to do it for you, but sometimes your friends don't follow through." -- (supposedly) Richard Hamilton, who discovered Ricci flow which was the technique used to prove the Poincare conjecture
I read an article about the discovery, about him and how he's working on it since his last year of bachelors; I read his paper and now I'm watching his numberphile video interview. His explanations are so clear and precise, just like his paper! Loved this video. I had a hard time understanding Erdos sums before. Especially his proof of the constant. No idea if this is useful but how interesting! So beautiful!
I love Brady's constant need to name things after the subject he's filming. Good to see a humble young mathematician doing good work. And he's right - it's nice when there's things like this that confirm that primes are special.
Brady is such a great interviewer. He asks the questions that I dont think of, but when he does, I wonder why I didn't think to ask such an obvious question.
It makes a lot of sense to put his name on the Theorem! The Erdos-Lichtman's Primitive Set Theorem. One name for the guy that proposed and for the guy that proved it. Must have been a sensational theorem to make such a contribution to the math world.
As I understand it: Scientific etiquette is that you're not supposed to name a discovery after yourself, others have to be the first to name it after you.
Lovely clear explanation, Jared is a very nice addition to this channel. I hope he will be in more videos. Kudos to him for making the conjecture a theorem!
It makes sense that primes are the maxinal primitive set. If you were trying to generate the maxinal primitive set from scratch, what would you do? Start with 2, which rules out all multiples of 2. Pick 3, which rules out all multiples of 3. Skip 4, add 5, which rules out all multiples of 5. You're basically running the seive of Eristothenes!
I really enjoyed hearing about how this was a bit of a "candelight theorem" for Lichtman. Amazing that he took the risk and followed his true passion to prove it. Thanks for sharing and teaching us!
Probably my all time favorite Numberphile video, definitely my favorite recent video. The explanation, enthusiasm, and banter are wonderful. A modern mathematical discovery that can be simplified for the average viewer that still shares that magic that timeless proofs seem to have.
The best part of following Numberphile over the years is seeing how much math Brady has picked up. The questions he asks now are so clever and mathematical! I remember when Brady was afraid to even make conjectures!
Hard to do a video on something this hard. But I appreciate how genuinely joyful Jared is about this topic. I appreciate him being quite humble, but good to know he knows how big this work is.
What I really appericiate about Brady are the questions he asks. He is unlike any ordinary interviewer, and always asks the questions which I would be thinking of at that moment. It really requires a certain amount of skill, so I thought I'd write a comment appreciating that.
Re: the "fingerprint" number dropping as k increases until k=6 - that's reminiscent of how n-dimensional ball volumes turn out. If r=1, a 5-ball has the largest 5-dimensional measure of all the n-balls. When n=6 the n-dimensional measure tapers off and tends to 1.
The set of primes is the greedy primitive set as well. As in, if you want to build a primitive set iteratively by always picking the smallest allowed number (but not 1), then the primes is what you will end up with. Which corroborates the result from this video, that it is in some sense the primitive set with "the most small numbers".
@@lonestarr1490 I agree. When building "optimal" sets of integers like this (depending on what restrictions you have and what metric you use to measure) going greedy is almost always a decent first attempt. It doesn't work every time, but it is usually worth trying. In this case, it did work, and I thought that was worthwhile to point out.
@@MasterHigure Worthwhile it definitely was, for without your comment I wouldn't have spotted the connection to the sieve of Eratosthenes. Erdős's conjecture feels a lot more natural to me now than it did before. So thank you ;)
Erdős, a group of math students (including myself). A blackboard. Two hours. An Erdős conjecture. His first proof of same. (Notes lost.) That man could see around mathematical corners. It was a privilege to meet him.
Indeed! Erdos was an amazing guy. He took simple concepts, saw the deeper meanings, and proposed conjectures about them. Many he proved himself, some are yet to be proven. All are interesting.
Amazing result! I’m always interested in results that suggest the primes are some kind of optimal subset of integers. Like he said, we all have this intuition that primes are special, and these results confirm that
Super cool, young mathematician and a great result as well. I was just hoping he’d elaborate a bit as to whether the known upper bound is a rational or irrational-in which case normal vs. transcendental-number. Thanks anyway 🙂🙏🏻
I’d be extremely surprised if it was rational, we have another monster-group style magic constant to wrap our heads around. To put it very unrigorously, the primes are a very fundamental set, so to have them connected to a value like 23/48 seems bizarre.
Hearing someone talk about the set of numbers with two prime factors makes me wonder if there's something clever but useless you could do with primitive sets that relates to RSA.
I did Chemistry as an undergraduate, sometimes I wish I had studied Mathematics. And then I listen to someone talking about number theory topics and I realise that maths at degree level would have been way beyond me. Fascinating, but far too demanding in rigour of abstract thought. Numberphile is a pleasant way, fifty years on from then, of musing on the beauty of mathematics. Thanks Numberphile!
Very interesting. It seems to be intuitively clear: using the primes, you get the numbers in the primitive set packed the densest. And even though, this does seem easy intuitively, the proof was pretty hard obviously.
Most of us mathematicians are extremely timid when it comes to our work and progress. We know that we're standing on the shoulders of giants. But we also know that we're helping to advance understanding and theories that, eventually, will provide somebody else an opportunity to stand on our shoulders and become the next important name in the direction we've gone. But I doubt I'll ever stand as tall as Jared. Congrats, mate!
This result sounds intuitive. If you have to replace the prime numbers with composite numbers you would have to use larger numbers. For example instead of 2 and 3 you could use 4, 6 and 9. So then when you do all this operation you would get a smaller number. 1/2log2 + 1/3log3 > 1/4log4 + 1/6log6 + 1/9log9.
Before you gave your explaination, I was thinking of something like proving that in order to have a primitive set that has different members than the primes, then the sum of that set would necessarily be smaller than the sum of the primes (notice that the whole point of the fingerprint function is to compare infinities). I hadn't thought of using probability.
Easy way to construct some non-“k-primes” primitive sets: stick arbitrary positive integer exponents on each element of the set of all primes (remove an element if you choose 0 as its exponent).
I think the only case where you end up with a maximal primative set using this idea is when you use all 1s. For 0s the element that was removed can be added back, and if you have P^x you can always add at least P^(x+1) for x > 1.
Jared said something very interesting (at 2:34 - 2:36) where he said that we can build all the unique numbers out of primes. I had never heard that before. I would have loved if Jared would have expanded on that. That would help me appreciate the set of primes more so. *Brandy,* It would be interesting if a future interview could expand on this concept, peeling back (layer by layer) how the primes are a building block of all the numbers (like the primes are some sort of foundational set of all the numbers in the universe). That would be cool to learn about. Thanks!
The basic idea is that all integers factor uniquely into their so-called "prime factors". For example, 60 = 2² × 3 × 5, and there is no other way to factor it into prime numbers.
This property is also known as the "fundamental theorem of arithmetic" - that any positive integer can be expressed as a product of primes in precisely one way (1 being represented by the special case of the empty product - not multiplying anything together).
Intuitively the set of prime numbers is the slowest growing list of numbers that form a primitive set. Each next prime is the smallest bigger number that doesn't divide or is divisible by any previous number. And the terms of the sum get smaller with bigger numbers, so you want to have as much of the small numbers in it as possible and have the smallest gap between numbers as possible.
I am impressed with your ability to see it, its is just beautiful and it continues forever and wraps on to itself in a new theroy and new sets that combines into millions of of sets. Congratulations 143.41
Something interesting about every primitive set being uniquely associated with a real number is that that means the set of primitive sets has the size of the real numbers. I'm not very knowledgeable about number theory, so I had no intuitions about how many primitive sets there are. Knowing that this is their size, it seems like it says something about them. If we imagined there some process that could produce every primitive set, then how many sets were produced would be a reflection of how the process works.
Brady, you've done it again! Presented a topic that is, by definition, at the very cutting edge of mathematics, in a way that a layman can follow, but not feel patronised. Well done to Jared too, for his proof, and for his clear explanations.
Strange, when I heard that the sum over primitive set is converging, I thought comparing it to sum over primes would be the way to prove it. Intuitively it makes very much sense that n-th member of any primitive set cannot be smaller than n-th prime. But I guess it's not that simple.
I think this fails when you start taking, say, numbers with exactly 2 (prime) factors. The smallest elements of this set will be much larger than the primes, but this changes relatively quickly: prod of 2 primes: 4,6,9, 10,14,15, 21,22,25, 26 compare to primes: 2,3,5, 7, 11,13, 17,19,23, 29 So what's happening? While the primes are indeed very common in small values, when taking products you have a lot more choices you can make before having to use a larger prime. Our product of 2 primes list only needed primes less than 13, while we reached 29 in the primes list.
Really cool. I loved this video. Two comments: 1) I'm not sure if he actually confirmed Brady's idea that no two primitive sets would have the same "C". 2) The graph of "C" vs K reminds me of an atomic potential function (U vs r)
5:30 This has a kind of Kolmogorov + busy beaver feeling -- It's like the reason you can't do it is because you have a membership rule which is a function that returns the members of that set (a generator, I guess), and the length of that function is finite. A "Busy Beaver" for a specific length is the universal Turing machine program of that many states which produces the largest output while still stopping eventually. These numbers get very big, very fast, but because you can copy the previous state machine and add one state that will always write a 1, move left and then run the copied part(for example), every length can get bigger, forever. The length of the output would also be an upper bound how much info you could get out for any UTM code of that length, otherwise that input would be the busy beaver for that length. If the membership rule function was implemented using a Turing machine (as a proxy for determining its Kolmogorov complexity), it's length something left a plausible distraction to pull anyone still reading this off the right track. I basically had the first 10 words before I started and if you read this far, sorry.
So, I’ve never had a problem with understanding when a conjecture is solved, but I am completely lost on what this conjecture was and what he did to solve it. May need someone not so close to the problem to explain this one better.
It makes sense that the prime set has the biggest "fingerprint" because: with the given restriction (the set being primitive), the primes are the "smallest" numbers and you pick as many as possible of them (smallest gaps in the Integers between one and the next), when you put them on the denominator for the "fingerprint" function, you get the "biggest" number possible for every step (in your infinite sum, every addend is maximized). OBVIOUSLY is very hard to prove it, props to Lichtman for his work. Just saying it makes sense to me intuitively. What i think the video got wrong is the second biggest fingerprint being the primitive set with numbers of 2 prime factors. You can very easily have a bigger fingerprint if you take the original Prime set and remove just one number (say, the millionth Prime). This is still a primitive set by definition (you have all the primes except one, nothing divides the others) and you have a fingerprint just a tiny bit lower than 1,6366... but still a lot bigger than 1,1448...
I'm no mathematician, but the way he talks about sets being finite or convergent and the convergences having a value brings to mind the Mandelbrot set...just has a hint of fractal in the air. When he talks about the "fingerprint" numbers getting larger and smaller as you go, and everything being wedged between two and then also the "k prime factor" class being special, reminds me of someone pointing out the regions of the Mandelbrot to describe what is forming those. Just a random, non-mathematical observation.
Given that pi usually shows up somewhere in these series, I tried to see if 1.6366(I've assumed that there are more digits to this number) was some factor of pi and hit upon 1 + 2/π . I love maths but I am no mathematician, so could someone check if it's that simple? Would that be significant in some way if true?
I think I have a nice intuition about it. Imagine the full set of prime numbers. If you want to add a composite number N to it, you'd have to give up all prime factors of N, so that the set remains primitive. Say N = a*b (a, b distinct prime numbers), then you can prove that 1/(N*Log(N)) < 1/(a*log(a)) + 1/(b*log(b)), that is, the series would decrease. This shows that the set of prime numbers is maximal (locally maximum, but not necessarily maximum) with respect to the series 1/(s*log s).
I was hoping the same thing.. but actually, this work with the primes and primitive sets actually shows why the Collatz Conjecture is still going to be so hard... imagine in that whole primitive set of primes there is ONE PRIME, unimaginably large, that is so far away from a power of two that the Collatz process can't get it there and it instead forms a loop ... in another words, the set of all primes is NOT a subset of the set of numbers reducible to the 4-2-1 loop. Proving that all primes yield to the Collatz conjecture process would be a step forward ... but that would be a STAGGERING undertaking...
Although it’s very logical and almost self-explanatory that the ‘largest’ primitive set is those of the prime numbers, it’s still impressive that he managed to prove it mathematically.
I saw this on the Quanta website. I have a math degree so this kind of thing interests me. It was a mind bogglingly simple proof. It doesn't take a math degree to understand. It may have been simple (in hindsight), but it is no less than a major achievement! Congrats to this young man for proving it, he deserves the accolades he is getting.
@@Jodabomb24 As I said, it may be simple "in hindsight", but it is a major achievement. I apologize if my wording implied anything less than an elegant and important piece of work.
@@ferretyluv To get all the "proof" reasoning yes, you do need some proof construction experience. What I wanted to convey in "non-math speak" was that his breakthrough logic is so good and understandable that that most people can say (with some explanation)... "Oh, I see what he did!" Again, I find this work amazing, beautiful, and worthy of the recognition.
@@jppagetoo Oh yeah my comment was meant to support yours. I wanted to reiterate that even when something is simple it doesn't mean it's obvious or easy to come up with.
![⭐️( ลอยกระทง 2024 ) Ver. แดนซ์ Remix BY [ ดีเจกิต รีมิกซ์ ]](http://i.ytimg.com/vi/rMWOLhgXUHc/mqdefault.jpg)
It makes me smile thinking that, if Jared was born 300 years ago, his name would appear in textbooks and we'd probably have nothing but a single painting of him to know what he looked like. And yet here we are, watching a TH-cam video of him explaining his theorem for free.
We'd probably have seen him in a nice wig though
My thoughts exactly, what a privilege
This reminds me of the fact that the only picture we have of Legendre is that one caricature.
Numberphile is also like an archive of such discoveries ( like the videos with J Maynard)
If not a wig, maybe a hastily folded dish towel {:-)
When guys like Adam Savage talk about the magic of Numberphile, this is *exactly* the kind of video he's referring to. A young mathematician finding beauty in a famous conjecture, works in his spare time to prove it, and all throughout the video Brady is not only teasing out the points that help us laypeople understand it, but also highlighting the personality of the mathematician himself. TH-cam at its best.
Unfortunately no nudity involved though
@@aceman0000099 Yes..He's gorgeous
@@philipthomey7884 haha, stop it, you two! 🤣
Adam Savage likes numberphile? He just keeps getting cooler lol
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11:15 -- I can appreciate the modesty, but "Erdös-Lichtman" is a pretty boss name for a theorem.
The Erdös-Lichtman Primitive Set Theorem. Very cool.
Whether or not this catches on (and I certainly hope that it does) does this proof mean that Jared has now, effectively, an Erdös number of 1?
I like it
@@christophersmith108 No
@@christophersmith108 I thought of this while watching the video. I feel like he’s probably one of the only people to be able to legitimately make that claim since Erdös’s death
@@oliverwhiting7782 To get an Erdös number of 1 you need to collaborate on a paper with Erdös. As cool as it is to prove an Erdös conjecture, it is not at all the same thing. There will never be another 1.
Perfect Numberphile content. Complex but beautiful problem - simply and clearly explained. Plus an Erdős connection. More from Jared Duker Lichtman please.
Make him darker and he looks like Srinivasa Ramanujan, maybe a reincarnation :-)
@@royroye1643 bit of a stretch there but he's a genius
Not simple enough for me lol
What a fantastic communicator. He knew just the right tidbits to throw in to help people through his explanations. He was excited and charming. I hope to see him back here!
Z
absolutely agree!
false.
One can easily see how well this man understands this subject by the clarity of his explanations.
Absolutely. I was trying to express the same thought, but the words wouldn't come to me. I wish most professors could convey complex topics with anywhere near this clarity, studying STEM subjects would be that much easier!
Yeah he's been working on this problem for 4 years.
btw, Jared's supervisor is (I believe) James Maynard, who has been on the channel before! As a side note, I'm super bummed that Brady came to film in my department while I was there and I didn't see him - the reason I applied to do maths at uni was the twin prime conjecture video that Brady did with James Maynard (and then I got to take his course on analytic number theory, which was super cool).
Whoa I didn't knew James Maynard was supervisor of Jared.
So are you working on the twin prime conjecture?
Numberphile Cinematic Universe is crazy connected.
I love how Brady asks smarter and smarter questions as the years go by, now being more and more knowledgeable in maths than when he started
And, don't forget, in gemstone trading.
Yeah he was asking some potent questions in this video
"When you discover something in math, out of humility you don't name it after yourself, you wait for your friends to do it for you, but sometimes your friends don't follow through."
-- (supposedly) Richard Hamilton, who discovered Ricci flow which was the technique used to prove the Poincare conjecture
I read an article about the discovery, about him and how he's working on it since his last year of bachelors; I read his paper and now I'm watching his numberphile video interview. His explanations are so clear and precise, just like his paper! Loved this video. I had a hard time understanding Erdos sums before. Especially his proof of the constant. No idea if this is useful but how interesting! So beautiful!
In some sense, the interest and the beauty is the first priority in mathematics. Usefulness is not always knowable and often secondary.
Lol yeah me too I read about him on Quanta Magazine.
I look up these videos for inspiration.
I absolutely love these interviews with mathematicians talking about their work, especially the recent discoveries.
I love this. I hope Jared will become a Numberphile regular...
I love Brady's constant need to name things after the subject he's filming. Good to see a humble young mathematician doing good work. And he's right - it's nice when there's things like this that confirm that primes are special.
This guy is so humble and wholesome
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Brady is such a great interviewer. He asks the questions that I dont think of, but when he does, I wonder why I didn't think to ask such an obvious question.
Wow proving a number theory theorem in the 2020’s.. that’s quite an accomplishment. Gauss would be impressed!
I gauss he would!
It makes a lot of sense to put his name on the Theorem! The Erdos-Lichtman's Primitive Set Theorem.
One name for the guy that proposed and for the guy that proved it.
Must have been a sensational theorem to make such a contribution to the math world.
As I understand it: Scientific etiquette is that you're not supposed to name a discovery after yourself, others have to be the first to name it after you.
Lovely clear explanation, Jared is a very nice addition to this channel. I hope he will be in more videos. Kudos to him for making the conjecture a theorem!
It makes sense that primes are the maxinal primitive set. If you were trying to generate the maxinal primitive set from scratch, what would you do? Start with 2, which rules out all multiples of 2. Pick 3, which rules out all multiples of 3. Skip 4, add 5, which rules out all multiples of 5. You're basically running the seive of Eristothenes!
I really enjoyed hearing about how this was a bit of a "candelight theorem" for Lichtman. Amazing that he took the risk and followed his true passion to prove it. Thanks for sharing and teaching us!
More proof that you should follow your heart. Easier said than done though.
Probably my all time favorite Numberphile video, definitely my favorite recent video. The explanation, enthusiasm, and banter are wonderful. A modern mathematical discovery that can be simplified for the average viewer that still shares that magic that timeless proofs seem to have.
The best part of following Numberphile over the years is seeing how much math Brady has picked up. The questions he asks now are so clever and mathematical! I remember when Brady was afraid to even make conjectures!
Hard to do a video on something this hard. But I appreciate how genuinely joyful Jared is about this topic. I appreciate him being quite humble, but good to know he knows how big this work is.
What I really appericiate about Brady are the questions he asks. He is unlike any ordinary interviewer, and always asks the questions which I would be thinking of at that moment. It really requires a certain amount of skill, so I thought I'd write a comment appreciating that.
11:14 he's so humble, heartwarming to see.
Yes it should be the Erdos-Lichtman Theorem. What a beautiful idea, and another reason to love the Primes.
One who proposed the conjecture, one who proved it!
Re: the "fingerprint" number dropping as k increases until k=6 - that's reminiscent of how n-dimensional ball volumes turn out. If r=1, a 5-ball has the largest 5-dimensional measure of all the n-balls. When n=6 the n-dimensional measure tapers off and tends to 1.
Actually I think it tends to 0
I would like to suggest to name the sequence of fingerprint numbers, the Lichtman Sequence.
This guy is so down-to-earth and great at explaining such a complex problem! Very fascinating, I hope he’ll have a fantastic career!
The set of primes is the greedy primitive set as well. As in, if you want to build a primitive set iteratively by always picking the smallest allowed number (but not 1), then the primes is what you will end up with.
Which corroborates the result from this video, that it is in some sense the primitive set with "the most small numbers".
That is actually super cool
This is however very obvious and therefor less interesting dont you think? :)
@@jonasjoko294 It's nothing more than the sieve of Eratosthenes, yes. Probably what lead Erdős to his conjecture in the first place.
@@lonestarr1490 I agree. When building "optimal" sets of integers like this (depending on what restrictions you have and what metric you use to measure) going greedy is almost always a decent first attempt. It doesn't work every time, but it is usually worth trying. In this case, it did work, and I thought that was worthwhile to point out.
@@MasterHigure Worthwhile it definitely was, for without your comment I wouldn't have spotted the connection to the sieve of Eratosthenes. Erdős's conjecture feels a lot more natural to me now than it did before. So thank you ;)
"Lichtman Primitive Set Theory"... has a nice ring to it.
Erdős, a group of math students (including myself). A blackboard. Two hours. An Erdős conjecture. His first proof of same. (Notes lost.)
That man could see around mathematical corners. It was a privilege to meet him.
Indeed! Erdos was an amazing guy. He took simple concepts, saw the deeper meanings, and proposed conjectures about them. Many he proved himself, some are yet to be proven. All are interesting.
primo classic numberphile content. reminds me of old interviews with James Maynard before he went on to the big time leagues.
Amazing result! I’m always interested in results that suggest the primes are some kind of optimal subset of integers. Like he said, we all have this intuition that primes are special, and these results confirm that
I like how embarrassed he seemed to be when Brady pushed him, inadvertently, into a position of implicitly comparing himself to Erdos.
This guy is great. I hope he can come back and explain more math for us.
Super cool, young mathematician and a great result as well. I was just hoping he’d elaborate a bit as to whether the known upper bound is a rational or irrational-in which case normal vs. transcendental-number. Thanks anyway 🙂🙏🏻
Glupost
I’m interested to know that as well but it’s likely like many cool constants that we don’t know
I’d be extremely surprised if it was rational, we have another monster-group style magic constant to wrap our heads around. To put it very unrigorously, the primes are a very fundamental set, so to have them connected to a value like 23/48 seems bizarre.
Hearing someone talk about the set of numbers with two prime factors makes me wonder if there's something clever but useless you could do with primitive sets that relates to RSA.
I did Chemistry as an undergraduate, sometimes I wish I had studied Mathematics. And then I listen to someone talking about number theory topics and I realise that maths at degree level would have been way beyond me. Fascinating, but far too demanding in rigour of abstract thought. Numberphile is a pleasant way, fifty years on from then, of musing on the beauty of mathematics. Thanks Numberphile!
Very interesting. It seems to be intuitively clear: using the primes, you get the numbers in the primitive set packed the densest.
And even though, this does seem easy intuitively, the proof was pretty hard obviously.
Great idea that you bring the solver of conjecture
Most of us mathematicians are extremely timid when it comes to our work and progress. We know that we're standing on the shoulders of giants. But we also know that we're helping to advance understanding and theories that, eventually, will provide somebody else an opportunity to stand on our shoulders and become the next important name in the direction we've gone.
But I doubt I'll ever stand as tall as Jared. Congrats, mate!
for sure colonel dookie
It's almost romantic how Jared discusses this, beautiful mathematics that I do not understand in the slightest. Lovely and wholesome video :)
What a lovely mathematician, such a great energy and enthusiasm. And as always, Brady's questions are so clever and interesting.
Exactly the kind of content I love from this channel. Thank you!
What a wonderful clear and precise definition and speaker - Numberphille we want more from this expert!!
This result sounds intuitive. If you have to replace the prime numbers with composite numbers you would have to use larger numbers. For example instead of 2 and 3 you could use 4, 6 and 9. So then when you do all this operation you would get a smaller number. 1/2log2 + 1/3log3 > 1/4log4 + 1/6log6 + 1/9log9.
Before you gave your explaination, I was thinking of something like proving that in order to have a primitive set that has different members than the primes, then the sum of that set would necessarily be smaller than the sum of the primes (notice that the whole point of the fingerprint function is to compare infinities). I hadn't thought of using probability.
So glad that conjectures like these can be found proof for! Congratulations :D
This guy is amazing. It's so obvious that his mind is full of genius.
Hey Brady, I like how you are getting better and better all the time in the mathematical way of thinking. It shows in the questions you ask :)
6:30 and I was looking for that comment
Easy way to construct some non-“k-primes” primitive sets: stick arbitrary positive integer exponents on each element of the set of all primes (remove an element if you choose 0 as its exponent).
I think the only case where you end up with a maximal primative set using this idea is when you use all 1s. For 0s the element that was removed can be added back, and if you have P^x you can always add at least P^(x+1) for x > 1.
@@JayTheYggdrasil p^x would divide p^(x+1), though
Erdös-Lichtman Theorum, Erdös-Lichtman Constant, and Erdös-Lichtman Fingerprint Numbers.
Jared said something very interesting (at 2:34 - 2:36) where he said that we can build all the unique numbers out of primes. I had never heard that before. I would have loved if Jared would have expanded on that. That would help me appreciate the set of primes more so.
*Brandy,* It would be interesting if a future interview could expand on this concept, peeling back (layer by layer) how the primes are a building block of all the numbers (like the primes are some sort of foundational set of all the numbers in the universe). That would be cool to learn about. Thanks!
The basic idea is that all integers factor uniquely into their so-called "prime factors". For example, 60 = 2² × 3 × 5, and there is no other way to factor it into prime numbers.
This property is also known as the "fundamental theorem of arithmetic" - that any positive integer can be expressed as a product of primes in precisely one way (1 being represented by the special case of the empty product - not multiplying anything together).
Numbers, theorems, conjectures all clearly being felt as almost a physical thing. Absolutely wonderful.
This was an excellent and very entertaining video. Congratulations on this great result!
Erdős-Lichtman Theorum, sounds about right 🙂
Intuitively the set of prime numbers is the slowest growing list of numbers that form a primitive set. Each next prime is the smallest bigger number that doesn't divide or is divisible by any previous number. And the terms of the sum get smaller with bigger numbers, so you want to have as much of the small numbers in it as possible and have the smallest gap between numbers as possible.
I am impressed with your ability to see it, its is just beautiful and it continues forever and wraps on to itself in a new theroy and new sets that combines into millions of of sets. Congratulations 143.41
This is great.
Also, loving to hear more of Erdős, not much people know of him inspite him being great scientist and a great man.
what about primitive sets with complex numbers?
Something interesting about every primitive set being uniquely associated with a real number is that that means the set of primitive sets has the size of the real numbers. I'm not very knowledgeable about number theory, so I had no intuitions about how many primitive sets there are. Knowing that this is their size, it seems like it says something about them. If we imagined there some process that could produce every primitive set, then how many sets were produced would be a reflection of how the process works.
Brady, you've done it again! Presented a topic that is, by definition, at the very cutting edge of mathematics, in a way that a layman can follow, but not feel patronised. Well done to Jared too, for his proof, and for his clear explanations.
Strange, when I heard that the sum over primitive set is converging, I thought comparing it to sum over primes would be the way to prove it. Intuitively it makes very much sense that n-th member of any primitive set cannot be smaller than n-th prime. But I guess it's not that simple.
I suppose that wouldn't work because primitive sets aren't necessarily composed only if primes.
I think this fails when you start taking, say, numbers with exactly 2 (prime) factors. The smallest elements of this set will be much larger than the primes, but this changes relatively quickly:
prod of 2 primes: 4,6,9, 10,14,15, 21,22,25, 26
compare to primes: 2,3,5, 7, 11,13, 17,19,23, 29
So what's happening? While the primes are indeed very common in small values, when taking products you have a lot more choices you can make before having to use a larger prime. Our product of 2 primes list only needed primes less than 13, while we reached 29 in the primes list.
Erdös was also a very skilled chess player. I’m also a chess fanatic, hence why I know this.
Nice result! I didn't know about this Erdös conjecture. Fascinaring! Since that Paul Erdös was the most prolificus contemporany mathematician.
What a fascinating video to watch! I enjoyed every bit of it! Thank you! ♥️
Dude brady's underapreciated, he really asks some good questions throughout the video
One of the best presenters on the channel. Would be great if he became a regular.
Really cool. I loved this video. Two comments: 1) I'm not sure if he actually confirmed Brady's idea that no two primitive sets would have the same "C".
2) The graph of "C" vs K reminds me of an atomic potential function (U vs r)
5:30 This has a kind of Kolmogorov + busy beaver feeling -- It's like the reason you can't do it is because you have a membership rule which is a function that returns the members of that set (a generator, I guess), and the length of that function is finite.
A "Busy Beaver" for a specific length is the universal Turing machine program of that many states which produces the largest output while still stopping eventually. These numbers get very big, very fast, but because you can copy the previous state machine and add one state that will always write a 1, move left and then run the copied part(for example), every length can get bigger, forever. The length of the output would also be an upper bound how much info you could get out for any UTM code of that length, otherwise that input would be the busy beaver for that length.
If the membership rule function was implemented using a Turing machine (as a proxy for determining its Kolmogorov complexity), it's length something left a plausible distraction to pull anyone still reading this off the right track.
I basically had the first 10 words before I started and if you read this far, sorry.
Brady deserves the privilege of naming mathematical objects because his work has contributed to the world's understanding of what is BEAUTIFUL!!
He has an eerie resemblance to Ramanujan
I love this guy! So intelligent and well articulated. We demand more!
So, I’ve never had a problem with understanding when a conjecture is solved, but I am completely lost on what this conjecture was and what he did to solve it. May need someone not so close to the problem to explain this one better.
It makes sense that the prime set has the biggest "fingerprint" because:
with the given restriction (the set being primitive),
the primes are the "smallest" numbers
and you pick as many as possible of them (smallest gaps in the Integers between one and the next),
when you put them on the denominator for the "fingerprint" function, you get the "biggest" number possible for every step (in your infinite sum, every addend is maximized).
OBVIOUSLY is very hard to prove it, props to Lichtman for his work. Just saying it makes sense to me intuitively.
What i think the video got wrong is the second biggest fingerprint being the primitive set with numbers of 2 prime factors.
You can very easily have a bigger fingerprint if you take the original Prime set and remove just one number (say, the millionth Prime). This is still a primitive set by definition (you have all the primes except one, nothing divides the others) and you have a fingerprint just a tiny bit lower than 1,6366... but still a lot bigger than 1,1448...
Brady “Eh… would it be that divided by 2?”
Lichtman *encouraging smile*
The constant is similar to the golden ratio, that is beautiful 🙂
I'm no mathematician, but the way he talks about sets being finite or convergent and the convergences having a value brings to mind the Mandelbrot set...just has a hint of fractal in the air. When he talks about the "fingerprint" numbers getting larger and smaller as you go, and everything being wedged between two and then also the "k prime factor" class being special, reminds me of someone pointing out the regions of the Mandelbrot to describe what is forming those. Just a random, non-mathematical observation.
More from Jared please.
Given that pi usually shows up somewhere in these series, I tried to see if 1.6366(I've assumed that there are more digits to this number) was some factor of pi and hit upon 1 + 2/π . I love maths but I am no mathematician, so could someone check if it's that simple? Would that be significant in some way if true?
1.6366 is suspiciously close to 1+2/pi 🤔
And...
why did u know that?
@@aayushadlakha7425 he has wolfram alpha implanted in his brain
I think I have a nice intuition about it. Imagine the full set of prime numbers. If you want to add a composite number N to it, you'd have to give up all prime factors of N, so that the set remains primitive. Say N = a*b (a, b distinct prime numbers), then you can prove that 1/(N*Log(N)) < 1/(a*log(a)) + 1/(b*log(b)), that is, the series would decrease. This shows that the set of prime numbers is maximal (locally maximum, but not necessarily maximum) with respect to the series 1/(s*log s).
Hey nice. I read about this guy on Quanta Magazine. Cool to see a Numberphile video on him.
One of the most interresting video from numberfile !
"Lichtman-Theorem" it is!
BTW, Lichtman could literally translated to Lightman: He shed light on the conjecture 😊
Fantastic episode. A topic way above my level of expertise but somehow, I got the gist. Thank-you.
I loved the explanation of the problem, but hoped for more detail on how the proof was ultimately constructed :)
Amazing. I don't know why but I have the feeling Jared could solve the collatz conjecture
I was hoping the same thing.. but actually, this work with the primes and primitive sets actually shows why the Collatz Conjecture is still going to be so hard... imagine in that whole primitive set of primes there is ONE PRIME, unimaginably large, that is so far away from a power of two that the Collatz process can't get it there and it instead forms a loop ... in another words, the set of all primes is NOT a subset of the set of numbers reducible to the 4-2-1 loop. Proving that all primes yield to the Collatz conjecture process would be a step forward ... but that would be a STAGGERING undertaking...
I suggest this guy be assigned Erdos number 1. Any person who proves Erdos conjecture deserves it for sure.
We should also compel him to propose a new conjecture (well, he does in the preprint iirc). So we keep the chain going.
0 !
or -1/12 🤣
@@harriehausenman8623 only when he proves rieman's conjecture
I like the idea of honorary Erdos numbers.
Hope to see jared again on the channel! Great vid
2:50 Very nice question here, Brady!
Brady you are the best interviewer the world has ever seen
You can tell his mind is working so fast, I would have to slow the video down to absorb it all
Please never stop uploading videos
As a mathematician I know is quite an outrageous thing to say that your field is better than the others
It intuitively makes sense that the set of numbers with no non-trivial divisors would have the highest density and therefore the highest sum
Brady has a knack for naming things.
Although it’s very logical and almost self-explanatory that the ‘largest’ primitive set is those of the prime numbers, it’s still impressive that he managed to prove it mathematically.
I saw this on the Quanta website. I have a math degree so this kind of thing interests me. It was a mind bogglingly simple proof. It doesn't take a math degree to understand. It may have been simple (in hindsight), but it is no less than a major achievement! Congrats to this young man for proving it, he deserves the accolades he is getting.
The proof may be simple, but if the proof had been simple *to find*, then presumably it wouldn't have taken so many decades!
I read the arxiv link and yes, you do need a math degree to understand it. It’s less dense than most math proofs, but no less complex.
@@Jodabomb24 As I said, it may be simple "in hindsight", but it is a major achievement. I apologize if my wording implied anything less than an elegant and important piece of work.
@@ferretyluv To get all the "proof" reasoning yes, you do need some proof construction experience. What I wanted to convey in "non-math speak" was that his breakthrough logic is so good and understandable that that most people can say (with some explanation)... "Oh, I see what he did!" Again, I find this work amazing, beautiful, and worthy of the recognition.
@@jppagetoo Oh yeah my comment was meant to support yours. I wanted to reiterate that even when something is simple it doesn't mean it's obvious or easy to come up with.