Watching a math related video strictly out of curiosity and having your general math professor Bill Dunham from 25 years ago pop up is a surprise…and finding out he’s now a well respected mathematics historian and not just some guy who endlessly suffered non-math students struggles with train problems is absolutely fantastic. Go Mules!
I would assume whatever institution his professor whom he recognized in the video taught at had a Mule as their mascot. Either that or this guy really just likes Moscow Mules, which I wouldnt blame him for.@@ArawnOfAnnwn
One big application of Mersenne primes, that came from studying perfect numbers, is a good random number generator. RNGs had been historically very bad, until the introduction of Mersenne Twister in 1997, which uses a property of Mersenne primes to prove a good randomness. The most popular version uses a Mersenne prime 2^19937 - 1 for example, hence the name MT19937. There exist much more performant RNGs than Mersenne Twister now, but Mersenne Twister is still widely used thanks to its initial impact.
Got any keywords to recommend for searching for information on these PRNGs? If there's something more performant that I can guarantee generates the same sequence regardless of platform that would give me something fun to do for a game engine I'm writing as a hobby.
As a physics undergrad. I’ve come to realize that Euler is a Titan alongside Einstein and Newton. Every single bit of modern physics has Euler to thank for providing the mathematical Tools to construct a vivid picture of the universe and its underlying principles. Absolute legend.
Penrose, Euler, and Archimedes of Syracuse try and fail to walk into a bar due to the exponential volume of proofs they collectively produce by accident on their journey from the parking lot
I took a class from Dr. Nielsen in 2009. He was a very engaging, dynamic teacher, to the point that when he wrote an answer on the board, followed by an exclamation point, someone asked, "Is that factorial or excitement?" and he responded, "EXCITEMENT!"
I do not mean, seek intend or wish to be or appear to be impertinent, but it is interesting to me that the piece contains a misuse of the word "*perfect*"(which means finished completed or accomplished). why not just call them some short(quick-to-type) word like pig, ant, or god numbers, given that perfect is taken to mean neither more nor less than any-thing-you-please? "When I use a word, Humpty Dumpty said in rather a scornful tone, ‘it means just what I choose it to mean - neither more nor less.’ ’The question is,’ said Alice, ‘whether you can make words mean so many different things.’ ’The question is,’ said Humpty Dumpty, ‘which is to be master - that’s all.” Might it be relevant that Charles Lutwidge Dodgson(aka Lewis Carroll) was also a mathematician? In what respect or particular are the "perfect numbers" spoken of in the piece finished completed or accomplished or could be *said* to be finished completed or accomplished? Various people have said that mathematics is strictly a young man's game, might that be true? Please forgive me if I am being impertinent; as there can be the arrogance of youth, so also can there be the impertinence of senescence It may be that any potential to be interested in mathematics can be snuffed out by what is called " education.
@@vhawk1951kl its a noun, no? i dont say "why is the grand canyon called the grand canyon, i dont consider it that grand". Aside from that i do think its perfect as LHS equates to RHS
@@saucenado4844 grand is an adjective meaning big or great depending on the context; you ,might say that the Rio grande is not that great, grand or big
@@saucenado4844""why is the grand canyon called the grand canyon, i don't consider it that grand", is merely you flaunting you complete innocence of any wits and learning
WOAH! Dr. Pace Nielsen was my professor for intro to proofs. I was NOT expecting him to show up in the video. He's a fantastic guy, exceptional professor, and brilliant number theorist.
He was such a great professor, and I was surprised when he explained his research. It sounded pretty useless. But at least he's now a leader in that field, haha.
I've been involved with GIMPS for about 27 years now and it's great to see us mentioned in the video. It was one of the earliest examples of using distributed computing to work on these enormous tasks, and it's been fun to learn more about the math behind it along the way and talk with all kinds of really smart people around the world in the process.
I have a micro super computer, because I both do software development, video editing and play around with AI with huge models and video games. I've just started contributing to the project; since my demands are high, I usually replace parts before it's reasonable to do so. Now I can actually put my CPU and excessive cooling to good use when I'm just watching youtube and not waiting for something to encode or data to parse. I'm already 1.2% into my first assignment.
As someone that was never good at math it blows my mind how people could and can think in ways that can actually make sense of math so abstract. And without having computers to do the crunch for them back in the days.
@Believe5inJesusChristYou may be barking up the wrong tree. This video is about people setting out to prove or disprove claims with evidence - the exact opposite of religion which asserts a claim and then uses the claim itself as evidence. "I believe that a god exists, as claimed in the Bible." "Where's your evidence?" "Look at this from the Bible..."
@@tincanblower Not only that but also "Where's your evidence?" "Look at this book written and rewritten by humans for millennia before the printing press, humans so propense to make mistakes, lie, cheat and push some ideology into the paper if that suits them" This is why the old testament God, is so different from the new testament God, they were invented and imagined by humans that add very different ideologies, about what is right and wrong.
@@Argoon1981As Sabine Hossenfelder has said, " The existence of God is not a scientific question. It can neither be proven or disproven by science. It is a philosophical question "
@@TheXuism how much do you know about Fermat? He was anything but a bragger in my Opinion. He never published any of his genious ideas, his son did it. He became one of the most famous mathematicians, but was an actually a lawyer. So mathematic was only his hobby. And you call him a bragger?
I was also thinking it's a fallacy to think because someone is working on "something that matters" that they are necessarily accomplishing anything. Given the amount of academic research fraud going on, it's hard to know whether someone got published because they found something interesting, or they are milking the system for more grant money or to get on the tenure track.
There is something so bizarre about Euclid and Euler having a collaboration. If the history of mathematics was a book of fiction, I would call this a fan service 😂
Video is well done. I'm a mathematician some of whose work has been on this topic (some of the results you put on at 23:51 are mine, and one is due to a joint paper of me with Sean Bibby and Pieter Vyncke). My apologies also for the length of this comment. I do have some quibbles about some of the history details but they are minor. (And it is possible that I'm getting some of the details wrong myself.) Descartes's construction of a spoof perfect number, shows he had a pretty good understanding of how sigma behaves. Descartes's spoof shows he had a pretty good understanding of sigma(n). Also, Descartes likely did prove that an odd perfect number must be of the form he suggested. What Euler did was a bit stronger. Euler showed that if n is an odd perfect number n= p^e m^2 where p is a prime , p does not divide m, and p and e are both 1 (mod 4). Notice that this implies Descartes's result. Regarding the Lenstra-Pomerance-Wagstaff conjecture, while it gives a specific estimate for how large the nth Mersenne prime is, there is some degree of doubt of if it is correct. We're much more confident that the conjecture is correct up to a multiplicative constant near 1. And we are much much confident that there are infinitely many Mersenne primes, even if LPW turns out to be wrong even on the order of growth of Mersenne primes. Regarding Pace's comment to high school students, I want to expand on that slightly. No one should be working on this problem with any hope of solving it any time soon. The problem is genuinely very difficult. The spoofs are in many respects a major obstruction to proving that no odd perfect numbers exist. In particular, many of the things we can prove about odd perfect numbers, also apply to spoofs. So if they were enough to prove that no odd perfect numbers existed, we would have proven that no spoofs exist, which is obvious nonsense. To use an analogy that my spouse suggested a while ago: If we are trying to convince ourselves that Bigfoot doesn't exist, but all we've done is list properties that all mammals have, we can't hope to show Bigfoot isn't real. There are few other big obstructions, one of which has a very similar flavor. But, Pace correctly notes that not that many people are working on the problem, so there may be more low hanging fruit than one would otherwise expect for aspects of the problem. For most really famous open math problems, like say the Riemann Hypothesis, or P ?= NP, lots of people have spent a lot of time thinking about aspects of it. So most mathematicians have a general attitude of not trying to bash their head against problems that a lot of other people have thought about. But in the odd perfect number situation, to some extent, the community may have overcorrected, and thus spent less time on it than they might otherwise. However, this may also be due in part to the odd perfect number problem being famous, but not by itself being very enlightening in terms of what it implies. Hundreds of papers prove theorems of the form "If the Riemann Hypothesis is true then " . And those papers are themselves very broad and varied in what follows after the then. In contrast, I'm aware of only a handful of papers with results of the form "If there are no odd perfect numbers then" and what follows after the then is always something involving divisors of a number in a somewhat straightforward fashion.
The end of your comment reminds me of my Mentor saying one time that part of him hopes someone disproves the Riemann Hypothesis just because of all the papers hes read on "if the Riemann Hypothesis is true then X" and how they'll all have to be withdrawn. He thinks its true fyi. I wouldnt call myself an odd prime "truther" but I see no reason infinitely many couldnt exist just the first one being say > 50th Fermat Number would put it out of search range for the forseeable future. Then one about every billion more digits.
wow this is crazy. prime95 is widely used for cpu benchmarks during overclocking to check temperatures and crashes. But up until today I didn't know it was calculating mersenne prime numbers. I thought it was just trying to find prime numbers for cpu stress test. great video as always
Finding primes was (and still is) its original purpose. It just so turns out that finding primes takes a lot of computation power and it is so well optimized that it can squeeze out every drop from a CPU. And if there is a fault anywhere in the CPU, it will show.
You are growing older my friend, glad to be a part of your journey along the years as you share very interesting facts with the majority of humanity, those of us that are spoofs to hidden truths in the world of science and mathematics. Thank you dearly. I will make one conjecture. If ever we find a way to time travel to the past, every single one of us following your content would love to have a day with Euler.
I knew Prime95 was to find Primes in addition to a stress test, but I had no idea of the depth of the GIMPS project. Considering the program is both so simple yet computationally intensive, to be known as one of the most intense stress tests for a computer, really speaks to the sheer computing power we have needed to go this far.
26:17 "Carl Pomerance predicts that between 10 to 2,200 and infinity, there are no more than 10 to the (power of) negative 540 perfect numbers." I'm not good at math. Can anyone tell me why that number is to the negative power instead of positive power? As far as I know, 10 ^-1 = 1/10^1 = 1/10 = 0.1 10^-2 = 1/10^2 = 1/100 = 0.01 Therefore, 10^-540 = 1/10^540) = 1/ (1 followed by 540 zeros) = 0. (539 zeros)1 10^-540 is less than 1. However, 51 perfect numbers have already been discovered, so how can the there be no more than 0. (539 zeros)1 perfect numbers in Carl Pomerance's prediction? Is there an error somewhere?
@@simon6071 10^-540 perfect numbers of the form N=pM^2 An odd perfect number must have the form N=pM^2, so there are very close to zero odd perfect numbers expected in the range 10^2200 to infinity.
Very nice video! Just a small thing, the reason why the largest known prime is almost always a Mersenne number is not because it grows so quickly (for example numbers of form 2*3^n-1 would grow quicker...), the real reason is because we have efficient test for numbers of that form so we can test them much faster (the Lucas-Lehmer primality test).
Ironically, the test is so efficient that someone skilled at arithmetic could perform it using pen and paper in some hours or days, for 15-20 digit numbers. Mersenne's "all time would not suffice" claim was likely based on trial division … the oldest and least efficient primality test. The test goes like this: Let n be an odd prime. (NOTE: a prime exponent is necessary anyway, so other than ruling out 3 = 2^2 - 1 this is w.l.o.g.) Construct a sequence S(i) with: S(1) := 4 S(k + 1) := S(k)² - 2 p := 2^n - 1 is prime if and only if S(n - 1) is divisible by p. E.g. n=3 is an odd prime, p=2^3 - 1 = 7, S(3 - 1) = S(2) = 14 = 2 * 7, therefore 7 is a Mersenne prime. Crucially, because only divisibility matters in the end, it suffices to calculate the remainders of the S(k) modulo p, which prevents the intermediate results from growing very large.
17:48 Something about this quote just hit me hard, we are in the age of computers that started just a few decades ago and we often ignore how seriously revolutionary computer advancements are, something that could take years can now be done by a child with an iPad.
No doubt, this age will be remembered in history as the beginning of the computer age. It has completely transformed society in a way few technologies have before.
I just had a thought about primes. Has anyone figured 'primes' for fractions? What I mean is, instead of using whole numbers, try using a small fraction, such as 1/1298ths as your potential prime, and figure out if any two larger normal fractions multiplied together can make the smaller one. Or some other scheme using fractions to find fractional 'primes'. I'm thinking some cool new mathematical knowledge could be found, or a cool pattern.
YOUR STRATEGIES ARE SIMPLY AMAZING! I'VE TRIED SO MANY DIFFERENT APPROACHES TO BINARY OPTIONS OVER THE YEARS, BUT NOTHING HAS WORKED AS WELL AS YOUR METHODS. THANK YOU FOR SHARING YOUR KNOWLEDGE AND EXPERTISE WITH US - YOU'RE TRULY MAKING A DIFFERENCE IN THE LIVES OF SO MANY TRADERS.
One thing that is helpful about solving (or attempting to solve) such problems is that a lot of methodology is developed in the process, and methodology is always useful.
Another great thing is that it's fun to try. And that fun is a great motivation to learn the more tedious parts of mathematics. It's like when we used to say "why would I learn the multiplication tables if I have a calculator", and we had a point: what's interesting about something that's already solved? But every person I've talked about mysteries like this one are suddenly enthralled by the idea of maybe finding the answer, and that motivation to learn is priceless.
I love when people have made up their mind on something, like there is a heuristic argument for that there is no odd perfect numbers, and then faced with a reasonable counter argument, imidiately recognize that their original argument is flawed. Just listening to reason and take that logic in, it is beautiful
@@ThisHandleIsAlreadyTaken839 I love when people realize that not everyone knows how to spell or read, some didn’t go to a fancy uni, check your privilege 😠
@@hanu6158 115 have thumbsed up their message, so this is one person getting their jollies from being petty. But a spell checker is not privilege - all computers, cellphones, etc. have one.
Well, he does add that there are additional arguments that make the original heuristic argument stronger, he just doesn't specify what these arguments are (possibly implossible to explain to laymen in the space of a few minutes?)
Most sukkuna quote ever. They ask me why and if. But i do it when i like to kinda message ( admittedly finnished it few hours ago yet cant recall its quote)
When Boolean Algebra was invented in the 1840s it was purely theoretical without any possible practical use. Today it is the way the circuits in digital computers work.
Thanks I tried, but for me I need to start running through the various formula to see it working and that takes time and mental agility. I am just a bit past that right now 74 and it's nearly midnight. Take care, still very interesting.
When my professor asked us to write a program to find perfect number I was like wth is that then he gave us the formula so it was easy but never understood what it actually was until now I found only 2 6 and 28
I agree, Veritasium, Vsauce, SmarterEveryDay and Sabine Hossenfelder are prime examples of channels that make TH-cam worth using even if you wouldn't like all the ads and random stuff.
I love that you also include analog presentation methods, like the flipchard at the beginning. It not always flashy animations and short attention span bs. Your Videos are really enjoyable.
The absurdity of that 1000 page book containing that one number is that in paper form it is essentially useless, but the symbolism is so profound that people were scrambling to get a hold of a physical copy, that it sold out within days. I think this has something to do with human nature in that there is some spiritual value in having a physical copy of something, even if it is practically useless and infinitely more useful to just have a text file containing that number.
It makes a fairly decent random number generator. Flip to a page and stab your finger at a number. Just skip the first and last numbers (the first is more likely to be 1 (I think, I might be thinking of something else), and the last is odd). It's also kinda like a code pad, but less secure since there's lots of copies of it out there. To be truly secure there should only be 2 copies of a code pad. It's unbreakable though since the data is completely masked by randomness. Assuming the pad is created in a truly random manner.
In my intro to abstract math class in college, we had a final project to write a paper that had basically only two requirements: it was about an approved math-related topic and it had a proof that used concepts we were taught. I did mine on perfect numbers and Mersenne primes and gave a proof of the Euclid-Euler Theorem. It was super fun to learn and write about. It is awesome to see Veritasium cover this topic in the amazing quality he does and recognize the stuff that was talked about. I even concluded the paper like the video - it's nice to study stuff just because it's interesting, even if there's no obvious real world uses.
Terrific video. However, the part about Edouard Lucas could have been much stronger. He did not merely show M_67 was not prime, he was able to show M_127 was prime. This is the largest prime ever found without the aid of a computer. He did so using novel methods that did not rely on trial factorization, but rather exploited properties of the Fibonacci numbers. Using his methods he could test M_n for primality for all n equivalent to 3 modulo 4. These methods were further refined by D. H. Lehmer (who also should have been mentioned) so that all M_n could be tested; giving us the Lucas-Lehmer test for Mersenne primes. It is this test that makes GIMPS possible. For more informations see "Edouard Lucas and Primality Testing" by Hugh. C. Williams.
Ive never been so interested in a video that i have no idea or understanding of anything that was said, yet i want to know more even though i dont even understand the start
When he say that number theory might not have a real application in real life (but turns out can be use for encryption) I felt that, even as a guy that hate math, i realized that NO math problem is useless/don't have a real life application. And i also started to gain interest in math recently. I started to see math in this way: Solving math problems IS hard and even frustrating, but the moment you get the final result, all of that work will be worth it
My favorite bit of "useless" math at the time of its discovery are quaternions, they were discovered/invented a century before we needed it for avionics, orbital dynamics and computer graphics, yet they are integral to our civilisation now, allowing us to compute spatial rotations effortlessly. I hope this leads to a great discovery that enables even more awesome technology in the future.
Euler also worked on an interesting related problem involving "amicable numbers". Those are integers m and n where the sum of the proper divisors of m is n, and the sum of the proper divisors of n is m (so a perfect number would be where m=n). At the time, only a handful of examples were known, but Euler managed to come up with a recipe for generating many more. With one paper, the number of known pairs went from 3 to 61.
Funnily enough, though, in spite of finding some quite large amicable pairs, Euler missed the second smallest pair in existence. It was eventually found by a random nobody about a hundred years later, having been overlooked by dozens of more prolific mathematicians who had searched for amicable pairs.
Unlike the perfect numbers, there are instances of odd amicable pairs. Now, for an open question: Is there an amicable pair where one is even and the other is odd?
@@patrickmckinley8739 Interesting - I didn't know about that problem. Just to be safe, though, I'm not going to spend too much time trying to find an example that might not exist.
@@MathFromAlphaToOmega from my basic understanding, that open question would be quite analogous to the Odd Perfect Number question. Likely, the optimal known method for searching for such numbers would ALSO be running a vast network to reach insane levels of compute, for a few decades. Not likely something searchable in an individual’s free time. However, that is an assumption, unless there’s a proof that the problems have a certain equivalence. If there isn’t, then maybe there’s a different approach waiting to be found! And breakthroughs in number theory ARE things that individuals have accomplished, as illustrated by this video.
I admire this guy enough to know that when he says "WHAT BLOWS MY MIND IS" and after saying the thing he does the BOOM gesture... if I stay impassive, it means that i have missed an important chunk somewhere
I remember when I was 10, my mother and I were discussing increasing my allowance and I said "you should increase it one dollar every week". She knew it would be expensive and said "do you know how much this will cost me in the future Pedro?" and I don't know how, it just came to me on that day, yes, I just sum all the numbers, and the best way is to get the first and last one, sum, and multiply it by half the amount of weeks. I realized later this was a theorem well known. This silly (real) story is just to show how a small interaction with math can make you love it forever.
Your videos are always so crisp, clean, and educational. I absolutely love how you provide the historical progression of things without a bunch of fluff. There is no doubt you are making a positive impact in minds around the world! THANK YOU!
The real benefit of solving those kinds of problems is usually not the solved problem itself, but the insight you gained while solving it and the kinds of techniques and methods developed beeing useful in other areas where you didn't expect them to be useful. Noone knows whether the tool you invented to solve this kind of problem will suddenly crack open other problems as well in (at first glance) unrelated fields of mathmatics. Edit: Thats also the reason why proving something simply by checking all possible cases with a computer isn't very well respected by mathematicians. Sure, you may have the proof that something does/doesn't exist, but it tells you absolutly nothing about *why* it does/doesn't exist. Your understanding of the topic is still the same as befor....
Decided to give it a go using pascals triangle. I found that the row with a perfect number generates it using consecutive sum of the row number - 1. So, [(n-1)n]/2 where n is a row in pascals number. Also, I found that each row number that generates a perfect number is a multiple of the previous row numbers. And since the first row with a perfect number is row 4 (the sum being [3(4)]\2 = 6, each following row must be an even numbered row. And since it's the consecutive sum of n-1, the perfect number would also be even because the consecutive sum of an odd number is even. My only problem is proving each row with a perfect number is an even numbered row. Or at least a multiple of the previous row numbers that generate a perfect number
I love your channel so much, because the problems presented are discussed on a very nice level. Not layman's style, not lecture style, right in the middle. Awesome.
agree 100%. I tried reading about number theory when I was in college 20 years ago, before youtube, and I could only make it a couple pages into the first chapter before these textbooks seemingly go off into outer-space. Derek has done a great job of digesting and explaining. Just what I needed.
29:08 - "If you're a high schooler and you just love mathematics and you think 'I want a problem to think about', this one's a great problem to think about. And you can make progress, you can figure out new things. Yeah, don't be scared" Instructions unclear, and now I am caught in the steely grip of the Collatz Conjecture. Gee, thanks Professor Nielsen! 😂
I first learned about GIMPS in a science magazine in Bangladesh, I think in around 2012-2013. I set up GIMPS in my dad's laptop (I did not own a laptop then), and then his work computer. Finally I installed it in my laptop in 2019 when I came to the States for higher studies. Currently my dad is retired and the program only runs in my laptop. I have donated computing power to show that more than 50 numbers are not prime, still looking for one. My wife pokes fun at me when around every two to three months the LL test (or now the PRP test) on a potential number nears completion as everytime the number has turned out to be not a prime and I have been sad, and my wife finds this ritual mildly amusing. I do not even shut down my laptop. 😅 it is always on and the program is always running
I was watching this on my TV, and I had to pause so I can come to mobile to say this: I love you. There are no traditional media companies who provide anything close to the same content that you do. Thank you, and thank you, and thank you for everything that you do.
We all swim in the water of YT, and as fish say, "What is this 'water'-thing you speak of?" I watched all of Cosmos when I was a kid. Saw a few Burke's Connections in U.S.A. Just has to sink in that we are living in a golden age of science/math content. "Traditional media" don't care about math! Can't sell the soap, ha,ha!!!!
I thought it was weird for this to be uploaded at night for EST but then I remembered he just moved to Australia, so it’s still technically a normal morning upload for him
Honestly it feels weird to be awake when a big channel releases a video lmao Australia's timezone is hilariously inconvenient if you watch US or Euro stuff
I think the best case proof for this from mathematical induction and to recognize that they probably don’t exist, given the absence of them as testing grows to larger and larger numbers, taken with the unique properties that an odd perfect would have to possess.
almost cried at the end. "the only way to know for sure is to try" has always, always made so much sense to me. and i just found another one. I'm so glad to just be alive at times like these.
Hey Veritasium, I hope you read this message. I am a high school student from Turkey, currently studying at 10th grade. I was going to talk about the equation you showed at 6:08 . I actually found that equation when I was at 8th grade, later learned from my uncle -who is a math teacher- that such equation already existed. What are your opinions about this? Thanks.
I read somewhere that Mersenne may have made a typo on the 67th number of (2^n)-1, since the 61th number of that form is prime, and 7 looks close enough to 1. All said, this can't be confirmed. Also, in that same book, I read that Cole's exact words were, "Three years of Sundays." Also, fun fact: Any Mersenne Number whose index is composite will be composite. The same cannot be said for primes, since (2^13)-1 = 8191, and (2^8191)-1 is not prime. Thus, run a test and ignore all the non-prime indexes. For those wondering, GIMPS last checked M118212673 (as of my comment), so theres a good starting point.
This is … a really interesting problem, and again we don’t know the answer, because the historical record is missing in substantial parts. First of all - lots of people made false claims about the perfect numbers: Ibn Fallûs, Cataldi, Fermat, even Euler thought at one point that 2^43-1 and 2^47-1 were primes, related to perfect numbers (they aren’t). Mersenne is really interesting because he claims perfect numbers for outrageously large exponents that he couldn’t possibly have evaluated with the resources he had to hand in 1644. While Euler in the next century is erroneously laying claim to 2^47-1, Mersenne stakes out 2^67-1, 2^127-1, and 2^257-1 as primes related to perfect numbers. It took until 1876 for Lucas to prove 2^127-1, a 39-digit number, was prime. (The other two numbers aren’t.) This was the largest prime known for about the next 75 years until the first vacuum tube computers were given the task. What, or perhaps whom, gave Mersenne the idea of claiming these colossal numbers? The following involves speculation to high powers. Mersenne’s circle of mathematicians included Pierre de Fermat, and four years earlier Fermat had written to Mersenne with a description of his method for shortcuts at finding perfect numbers: “Je trouve plusieurs abrégés pour trouver les nombres parfaits ...”. The rest of Fermat’s letter (no. XXXIX in the correspondence) is inconveniently lost - it is presumed (for example by Fletcher, for the case of whether 2^37-1 is prime) that Fermat might have shared some calculations that Mersenne could have used later. So we could speculate Fermat might have suggested 2^61-1 is prime, which Mersenne then might have misread as using the exponent 67, but unfortunately any such evidence is lost to time. However that would only raise the score to two out of five, since there were two other perfect numbers (corresponding to the primes 2^89-1, 2^107-1) in the same range missed by Mersenne. It still doesn’t provide any explanation for what method was used for making these colossal predictions.
@@Xanthe_Cat I think the method is probably "I checked a whole bunch of factors and it looks prime". That would discard most candidates and makes it quite a bit more likely to get "lucky" - which he did with 2^127 and maybe with 2^61 (if it was that) - but not with the next one
@@Cowtymsmiesznego Maybe, but we simply don’t know what method Mersenne had, and there are a number of Mersennes that are improbably difficult to factor well before getting to M257.
@@Xanthe_Cat Right - that's why he didn't factor them, just checked enough potential prime factors to take an "informed guess". Idk, it was probably more advanced than that but that's the gist of it I'd say
A couple hundred years ago, this Galois dude worked on this unsolvable geometry thing, he actually came up a solution (or whatever the appropriate expression is), and 200 years later it was found to be useful in designing cell phone antenna. Its a crazy story, and his short life should probably be made into a movie,just because its all so darn crazy
I really love the direction of this channel was heading towards, which I felt that specially videos from the last 6 months or so, it's not just sharing something amazing or interesting, but really courage who was watching to pursue something, or to realize more possibilities this world offers.
Wow, this video is absolutely fascinating and so well put together! 🎉 I really appreciate the clear explanations and the engaging visuals-such a great way to break down complex topics. It's always refreshing to come across content that makes science both informative and fun to watch! Keep up the amazing work, looking forward to seeing more! 🙌
All I can think is how mathematicians throughout history would be absolutely blown away by modern computer technology. I think they would be so proud to know that people picked up and carried their legacy and continued work on this problem. Just imagine what could have happened if Euler got his hands on Matlab or Wolfram alpha
@@skyfeelan While this is true, it's interesting to imagine what would've happened if the development of the technology could've happen within their lifespan. Impossible, of course, but it's interesting to think about.
Even if you're not a mathematician, you should give it a go if you're interested! Math problems that stump the masters get solved by a novice perspective all the time, but even if you end up retreading existing ground, you'll end up learning something cool along the way :)
Man, this video made me realise how little we think about the world. I used to think there may be a point where we learn everything from this world, but seeing this, i realise we just think very little of everything, including ourself. I want to introduce change to myself but seeing videos like this, gives me an idea of how to proceed, even though i am not mathemathician, but i hope to become so
@@mansouralshamri1387 Even though i had the desire to read more books and engage in more subjects (most of them are self taught), it will still not be enough to achieve my goal. i dreamt to become like leonardo davinci but as technology progresses, it is becoming little easier but i question that where is the world going then? To pursue things that we don't know? But it also makes them less wiser, or maybe more? Or is it the phenoemon that sapiens are unaware of? I wish that if finances were not the problem in my whole life, i can figure it by myself
@@stupiocity245 it definitely doesnt make any them less wiser, every form of new knowledge isnt bad, ever. just go ahead, experiment and find little by little how you can introduce change in yourself. as time goes by, no matter the path you went, when you look back you will realize you definitely changed
@@mansouralshamri1387 However, at some point, wisdom must kick in, to make us realize that not all of that knowledge is valuable or useful (except perhaps on trivia night). Intelligence is knowing a tomato is a fruit. Wisdom is not putting it in a fruit salad.
Do you know that a woman was given the abortion pill ,gave birth to her child and they cut the spinal cord and put the baby in the bin while he,she was alive
Once again your math videos sail way above my comprehension level, but I feel like the more exposure to these concepts you give me the closer I get to actually understanding some of it.
@@micah4628 The Sigma function , σ(n) = the sum of all the divisors of n Yes that's it! Since a perfect number is equal to the sum of all of its divisors EXCEPT for the number itself! Therefore, sigma( perfect number) = 2 * the number itself
@@1stlullaby484 thanks i sorta got that part of it, it was more the part around 14:10 i was confused with but i just rewatched it with that in mind and i got it now, thanks!
@@micah4628that's why in case of a prime number say p we get sigma(p) =1+p Because these are the only divisors of p Now you might be thinking what about -1 and -p , but in number theory people are only concerned with positive integers (this doesn't mean that other numbers don't show up, it's just a field focusing on positive ones)
@@PlayerSlotAvailablehe’s a revolutionary in math-you can look him up on your own time, but for example, he’s the one who came up with the modern notation for functions, and also came up with the most beautiful math equation (Euler’s identity).
@@PlayerSlotAvailableHe is the greatest mathematician to ever live. It’s hard to even compare him with other people in other fields. Like I can’t think of anyone having as big of an impact in their field as euler did with mathematics.
Been building computes for 20 years now and back in the day Prime95 was _the_ way to stress test your CPU. I did know it was a math test but this is the first I’ve seen it explained exactly what it was doing.
You have such a knack for presenting information that is way over my head, and I often fail to fully understand it but I'm still fascinated, and consistently placed in awe of the mystery and power of math and numbers. Thank you for your public service Destin!
i'm becoming more respectful to my teachers, when i realize i can now understand and enjoy these kind of videos.. even 15 years later after the school..
there are more perfect numbers, but they become increasingly rare and larger as you go up in value. The first four perfect numbers, as mentioned earlier, are 6, 28, 496, and 8128. The next perfect number after 8128 is 33,550,336. After that comes 8,589,869,056. Perfect numbers are connected to Mersenne primes (prime numbers of the form ). Every even perfect number can be written in the form: 2^{(p-1)} \times (2^p - 1) where is a Mersenne prime. However, it is unknown if there are any odd perfect numbers or if they exist at all. So far, only even perfect numbers have been discovered.
One thing that you overlooked with Lucas is that he helped develop an algorithm that could determine the primality of Mersenne numbers far, far faster than trial division. What is now known as the Lucas-Lehmer primality test only takes p steps (which today involves a modular square and a subtraction), where p is the exponent of the Mersenne number, which is many orders of magnitude faster than trying to divide the whole Mersenne number by every prime between 3 and its square root to see if any return a zero remainder (although the prime factors, including itself, are of the form 2kp + 1, which narrows the search a bit). Using his "Lucas sequences", he proved that 2^127 - 1 was prime, the largest prime number ever calculated by hand.
This channel is absolutely THE BEST science channel. Not only on YT but in general. I'm a primary school teacher from Poland and the amount of facts and curiosities I get from here and transfer into teaching physics, chemistry and even English is astonishing. Thank you.
I absolutely recommend you Real engineering , Mustard , Vsause , Kirzguat in nuteshell ( Idk perfect name ) , But why , SciencePhileAI , Kosmo .. there are many more who provide valuable information with the proof and good details and you can learn something new that's worth your time instead of spending time on tiktk..
Historical math videos have become my favourite type of videos on this chanel. Please continue doing them. It is not necessary to have fancy animations or graphics. Great work
The numbers 6, 28, and 496 are perfect numbers. A perfect number is a number that is equal to the sum of its proper divisors (excluding the number itself). 6: Divisors are 1, 2, 3 → 1 + 2 + 3 = 6. 28: Divisors are 1, 2, 4, 7, 14 → 1 + 2 + 4 + 7 + 14 = 28. 496: Divisors are 1, 2, 4, 8, 16, 31, 62, 124, 248 → 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496. The next perfect number after 496 is 8128. Its divisors are 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064. Adding them gives 8128. So, the next number is 8128.
I "worked" on this problem when I was a math student, but miserably failed, thanks for bringing this on youtube. Your channel is a gem man. thanks for your work. If I had to guess there is no odd perfect number but infinite even perfect numbers.
I wonder how many matching digits of Pie you could find within that book. I'd think "31415" would show up once in that string of numbers. Edit: in the 39th Mersenne prime the string of "31415" shows up 7 times, "314159" shows up twice, and "3141592" shows up once. I did use Ctrl + F to search on the website, but there is a space every 5 digits so there could be more depending on where it starts within those 5 digits and how you search for the number, but those are the ones I've found so far. Id like to search in the 50th Mersenne prime but i cannot find a website, or PDF of the book with it fully written out so it can be easily searched for.
Was doing math problems on perfect numbers, opened youtube saw the thumbnail written 6, 28, 496 recognised they're perfect numbers, couldn't stop myself from clicking on it and here I'm enjoying the video and I've to accept Derek makes videos on topic nobody could even imagine of, hats off to this guy man, incredible
Either odd perfect numbers exist, or they don’t. If they don’t, that would mean all perfect numbers are even and elegantly fit the form N = (2^p -1)*2^(p-1) with (2^p -1) prime. If they do, that means there’s some gargantuan odd perfect number somewhere out there just waiting to be discovered. And both possibilities are equally fascinating!
@@marcusscience23 There is undecidableness. When running Conway's game of life there is no algorithm that guarantees predicting it's outcome in a limited number of time. So it's kind of selecting "or" from "yes or no".
Similar to problems like the Riemann hypothesis and 3n+1, if it's unprovable then it must be true, since it being false means there exists a counterexample. It can't be unprovable and false because there exists a defeater. If ZFC isn't strong enough to prove a result then you can keep adding axioms until it is, but it is impossible to know if any system at ZF's strength or stronger is consistent (you can prove it from stronger systems but this just pushes around the problem). Which leads right into Veritasium's video about the hole at the bottom of mathematics.
26:47 I LOVE how you were able to respond back to his argument, proves that you actually did your research and put him right back in his place that you're not just some youtuber who tells science stories and doesn't know better.
Wtf do you mean put in his place? Place of what? Being an expert in the field? Dude already admitted it's a heuristic and heuristic come with downsides. It's not a fight where people need to be put in place.
@@alex1stamford779 English isn't my first language, I meant it was when he realized he wasn't speaking to some media person who doesn't understand much
I actually love how quickly the professor realized he was having a double standard applying the heuristic and laughed about it. You only get that from arguing with smart people.
26:47 Pace Nilsen shows an incredible sign of intelligence! Not only did he immediately agree with a contradictory statement and not let his own beliefs that "Odd Perfect Numbers don't exist" overpower him, but simultaneously, he also reexamined and concluded that he had a bias. The same theory that heuristically shows Odd Perfect Numbers don't exist also shows that large, even perfect numbers don't exist. This is a true sign of intelligence, not to let your ego get in the way and search for the truth. We all can have biases, but only intelligent people will be able to look past them.
Unfortunately the scientific community fails to do this _far_ too often. Especially if that bias is either profitable or gets more funding for their projects.
@@yasyasmarangoz3577 , haha, might be. But it did feel like he believes that they don't exist, which, probabilistically, might eventually turn out to be true.
@@MrTuneslol I think this happens everywhere, but at the same time, many people in the scientific community can look past it, and that is when truly wonderful things are discovered or invented.
I've always found the subject of perfect numbers fascinating. I saw the thumbnail here, recognized what it was about, and actually dropped everything I was doing to watch. That doesn't happen often, so thank you so much for this awesome video
At 15:42, to prove that the exponent of p is of the form 4k+1, you just have to remark that the sum of the divisors of p^(4k+3) is always divisible by 4 (the powers of p modulo 4 are all 1 if p =4a+1 or alternating 1 and 3 if p=4k+3), which would make 2n divisible by 4 hence n even. The alternating 1 and 3 must be excluded because in this case the sum of the divisors of p^(4k+1) would be divisible by 4 as well. So p is congruent to 1 modulo p (Euler's proof as well).
Minor correction! Technically, 2 is a prime number so its sigma function is odd! However, in the context of this video, it's not included in the search for an odd perfect number so this doesn't need to be stated in the discussion of sigma functions ~13 minutes in, so it's only a small correction in that phrase particularly!
I am constantly amazed by your ability to explain relatively advanced mathematical concepts in such a simple way. I probably should have made this comment after your video on Goedel's incompleteness theorem, but this is a good example too. I actually enjoy your math videos more than your physics ones. Great work as always.
16:57 Idc how nerdy this makes me, but for me this feels like the mathematical version of walking away from a house while it explodes and not looking back and I love it. 😍
Yeah, while I was watching this I started thinking about all the mathematicians he mentioned as badass celebrities/superstars in some kind of drama or thriller.
I think there is some misconceptions the factors some Will be the given number Like 6 , factors of 6 are 1+2+3=6 28 factors are 1,2,4,7 and 14 . So 28= 1+2+4+7+14
There is a reason why mathematicians joke about naming theorem after the second person who discovered them because Euler discovered them first probably lol
I didn't, but you made me check what all perfect numbers' digits sum up to. And they all sum up to 1 (except the first one, being 6). That's actually something I didn't see anywhere while reading about perfect numbers.
@@1991dmj More succinctly, all perfect numbers modulo 9 equal 1, except 6. The properties of perfect numbers, the sum of its factors equal to the number itself is true independent of number base, as is my restatement of your interesting observation.
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Watching a math related video strictly out of curiosity and having your general math professor Bill Dunham from 25 years ago pop up is a surprise…and finding out he’s now a well respected mathematics historian and not just some guy who endlessly suffered non-math students struggles with train problems is absolutely fantastic. Go Mules!
Mules?
I would assume whatever institution his professor whom he recognized in the video taught at had a Mule as their mascot. Either that or this guy really just likes Moscow Mules, which I wouldnt blame him for.@@ArawnOfAnnwn
@ArawnOfAnnwn yea mules horses sheep lol....
did you have a stroke at some point, or have you always been illiterate?
(He’s a prof emeritus at Muhlenberg College…mascot is the Mule…Go Mules)
>walks up to blackboard
>multiplies 2 numbers
>walks away
>round of applause
Frank Nelson Cole was unfathomably based
Am I the only one bothered that he says AND between all the millions, billions, trillions, etc... couldn't help but mention
@@jacobe280 Yes. You are.
@@jacobe280no you’re not
Fish
@@AMPProfSquid
I love consistently understanding the first 25% of veritasium maths videos.
It was same for me, then I started studying math.
and then I went to undertand about 26%
@@slamn8917 😂.
But actually I do understand better now, almost completely. Besides the things I have no experience in.
121
One big application of Mersenne primes, that came from studying perfect numbers, is a good random number generator. RNGs had been historically very bad, until the introduction of Mersenne Twister in 1997, which uses a property of Mersenne primes to prove a good randomness. The most popular version uses a Mersenne prime 2^19937 - 1 for example, hence the name MT19937. There exist much more performant RNGs than Mersenne Twister now, but Mersenne Twister is still widely used thanks to its initial impact.
The
That actually helps a lot with understanding why RNG is multiplicative in most video games.
omg i was using that in programming, never knew why it was called MT19937 😮 my mind is blown away
@@lpc9929well said
Got any keywords to recommend for searching for information on these PRNGs? If there's something more performant that I can guarantee generates the same sequence regardless of platform that would give me something fun to do for a game engine I'm writing as a hobby.
I have a research project due tomorrow and I was really looking for something distracting.
My procrastination thanks you.
lol
Same
I’m actually early to a Veritasium video
This comment hurts
Same although it’s project about a book
As a physics undergrad. I’ve come to realize that Euler is a Titan alongside Einstein and Newton. Every single bit of modern physics has Euler to thank for providing the mathematical Tools to construct a vivid picture of the universe and its underlying principles. Absolute legend.
Penrose, Euler, and Archimedes of Syracuse try and fail to walk into a bar due to the exponential volume of proofs they collectively produce by accident on their journey from the parking lot
I will never not be disappointed that MIT's hockey team isn't the Eulers.
The Age of Unreason series clued me into how awesome Euler is (though he's a secondary character), and I've been stanning ever since.
@@Greyhawksci only like 1% of people would get it. I would bet the vast majority of people read and pronounce Euler phonetically.
There’s the old joke that so many random bits of math are named after the guy, we may as well just start calling numbers Euler letters.
So if I get a 6 on my test that means I got a perfect score.
*OMG*
Maybe?
In Switzerland, yes
Well, explain that to your parents😂😂
Tell that to germans
if it’s an AP test, then you got a 120%, which is impossible
13:25 "But Euler wasn't finished yet." I think this sentence appears in most histories of mathematical concepts.
Right? It feels like if we had found a way to keep the guy alive he would be responsible for the majority of all mathematical discoveries
Number theory concepts*
Possibly the most important mathematician in history
@@ab3040either him or Gauss
@@rogerszmodis Gauss was equal in math and science, so overall he was probably more important, but as far as just math goes I gotta give it to Euler
I took a class from Dr. Nielsen in 2009. He was a very engaging, dynamic teacher, to the point that when he wrote an answer on the board, followed by an exclamation point, someone asked, "Is that factorial or excitement?" and he responded, "EXCITEMENT!"
Sounds like the best kind of teacher.
I do not mean, seek intend or wish to be or appear to be impertinent, but it is interesting to me that the piece contains a misuse of the word "*perfect*"(which means finished completed or accomplished).
why not just call them some short(quick-to-type) word like pig, ant, or god numbers, given that perfect is taken to mean neither more nor less than any-thing-you-please?
"When I use a word, Humpty Dumpty said in rather a scornful tone, ‘it means just what I choose it to mean - neither more nor less.’
’The question is,’ said Alice, ‘whether you can make words mean so many different things.’
’The question is,’ said Humpty Dumpty, ‘which is to be master - that’s all.”
Might it be relevant that Charles Lutwidge Dodgson(aka Lewis Carroll) was also a mathematician?
In what respect or particular are the "perfect numbers" spoken of in the piece finished completed or accomplished or could be *said* to be finished completed or accomplished?
Various people have said that mathematics is strictly a young man's game, might that be true?
Please forgive me if I am being impertinent; as there can be the arrogance of youth, so also can there be the impertinence of senescence
It may be that any potential to be interested in mathematics can be snuffed out by what is called " education.
@@vhawk1951kl its a noun, no? i dont say "why is the grand canyon called the grand canyon, i dont consider it that grand". Aside from that i do think its perfect as LHS equates to RHS
@@saucenado4844 grand is an adjective meaning big or great depending on the context; you ,might say that the Rio grande is not that great, grand or big
@@saucenado4844""why is the grand canyon called the grand canyon, i don't consider it that grand", is merely you flaunting you complete innocence of any wits and learning
WOAH! Dr. Pace Nielsen was my professor for intro to proofs. I was NOT expecting him to show up in the video. He's a fantastic guy, exceptional professor, and brilliant number theorist.
He was one of my favorites!
lol that's cool
He also likes Magic the Gathering based on all the cards on his desk in the background. My kind of guy!
He seems to like playing Magic and has a Chuck Norris fact hanging on his wall, sounds like a cool guy indeed.
He was such a great professor, and I was surprised when he explained his research. It sounded pretty useless. But at least he's now a leader in that field, haha.
Thanks.
It amazes me how many things my instructors and professors failed to mention, let alone teach us. Chief
I've been involved with GIMPS for about 27 years now and it's great to see us mentioned in the video. It was one of the earliest examples of using distributed computing to work on these enormous tasks, and it's been fun to learn more about the math behind it along the way and talk with all kinds of really smart people around the world in the process.
you've been involved with gimps ? 🤨
@@Filo127you haven't watched the video?
I have a micro super computer, because I both do software development, video editing and play around with AI with huge models and video games. I've just started contributing to the project; since my demands are high, I usually replace parts before it's reasonable to do so. Now I can actually put my CPU and excessive cooling to good use when I'm just watching youtube and not waiting for something to encode or data to parse. I'm already 1.2% into my first assignment.
Do you know what a gimp suit is? If not look it up lol.@@LeVasTiaN
OG distributed computing projects were the best way to stress test overclocks back in the day. did alot of gimps, fah and seti myself.
As someone that was never good at math it blows my mind how people could and can think in ways that can actually make sense of math so abstract. And without having computers to do the crunch for them back in the days.
Crazy how humans are capable of all this, but still can't stop using plastic for everything lol. We're too intelligent for our own good xd.
@Believe5inJesusChristYou may be barking up the wrong tree.
This video is about people setting out to prove or disprove claims with evidence - the exact opposite of religion which asserts a claim and then uses the claim itself as evidence.
"I believe that a god exists, as claimed in the Bible."
"Where's your evidence?"
"Look at this from the Bible..."
@@tincanblower Not only that but also
"Where's your evidence?"
"Look at this book written and rewritten by humans for millennia before the printing press, humans so propense to make mistakes, lie, cheat and push some ideology into the paper if that suits them"
This is why the old testament God, is so different from the new testament God, they were invented and imagined by humans that add very different ideologies, about what is right and wrong.
@@tincanblower It's a bot. There's a lot of them on TH-cam that exist just to quote verses.
@@Argoon1981As Sabine Hossenfelder has said, " The existence of God is not a scientific question. It can neither be proven or disproven by science. It is a philosophical question "
The answers is C
???
@@zhaoyuanlow8154 it’s a joke
Haven’t picked C in a while - best logic
I thought it was Q
No it’s A
When Euler says "it's most difficult", it's gotta be impossible.
"I have discovered a truly marvelous proof of this, which this margin is too narrow to contain."
this guy is the biggest bragger in human history.@@BixbyConsequence
No it’s a joke reference to fermats last theorem lol
@@BixbyConsequenceThat was Fermat
@@TheXuism how much do you know about Fermat?
He was anything but a bragger in my Opinion.
He never published any of his genious ideas, his son did it. He became one of the most famous mathematicians, but was an actually a lawyer. So mathematic was only his hobby.
And you call him a bragger?
That point at the end, about the value in doing math, felt like the thesis statement every veritasium math problem video. Hats off.
your feelings are irrational
I was also thinking it's a fallacy to think because someone is working on "something that matters" that they are necessarily accomplishing anything. Given the amount of academic research fraud going on, it's hard to know whether someone got published because they found something interesting, or they are milking the system for more grant money or to get on the tenure track.
@@Fire_Axus your comment is perfectly odd
Where’s the proof
There is something so bizarre about Euclid and Euler having a collaboration.
If the history of mathematics was a book of fiction, I would call this a fan service 😂
Eu(clid x ler)
Imagine the noises the readers would make if Gauss joined in!
@@Xezlec Math : No Way Home
Oiclid and Yooler
Maybe, "I reincarnated into math genius, Euler, and continue my own legacy. Yes, I was Euclid."
21:15 As of Oct 2024, largest known prime is now 2^136,279,841 - 1
Video is well done. I'm a mathematician some of whose work has been on this topic (some of the results you put on at 23:51 are mine, and one is due to a joint paper of me with Sean Bibby and Pieter Vyncke). My apologies also for the length of this comment.
I do have some quibbles about some of the history details but they are minor. (And it is possible that I'm getting some of the details wrong myself.) Descartes's construction of a spoof perfect number, shows he had a pretty good understanding of how sigma behaves. Descartes's spoof shows he had a pretty good understanding of sigma(n).
Also, Descartes likely did prove that an odd perfect number must be of the form he suggested. What Euler did was a bit stronger. Euler showed that if n is an odd perfect number n= p^e m^2 where p is a prime , p does not divide m, and p and e are both 1 (mod 4). Notice that this implies Descartes's result.
Regarding the Lenstra-Pomerance-Wagstaff conjecture, while it gives a specific estimate for how large the nth Mersenne prime is, there is some degree of doubt of if it is correct. We're much more confident that the conjecture is correct up to a multiplicative constant near 1. And we are much much confident that there are infinitely many Mersenne primes, even if LPW turns out to be wrong even on the order of growth of Mersenne primes.
Regarding Pace's comment to high school students, I want to expand on that slightly. No one should be working on this problem with any hope of solving it any time soon. The problem is genuinely very difficult. The spoofs are in many respects a major obstruction to proving that no odd perfect numbers exist. In particular, many of the things we can prove about odd perfect numbers, also apply to spoofs. So if they were enough to prove that no odd perfect numbers existed, we would have proven that no spoofs exist, which is obvious nonsense. To use an analogy that my spouse suggested a while ago: If we are trying to convince ourselves that Bigfoot doesn't exist, but all we've done is list properties that all mammals have, we can't hope to show Bigfoot isn't real. There are few other big obstructions, one of which has a very similar flavor.
But, Pace correctly notes that not that many people are working on the problem, so there may be more low hanging fruit than one would otherwise expect for aspects of the problem. For most really famous open math problems, like say the Riemann Hypothesis, or P ?= NP, lots of people have spent a lot of time thinking about aspects of it. So most mathematicians have a general attitude of not trying to bash their head against problems that a lot of other people have thought about. But in the odd perfect number situation, to some extent, the community may have overcorrected, and thus spent less time on it than they might otherwise.
However, this may also be due in part to the odd perfect number problem being famous, but not by itself being very enlightening in terms of what it implies. Hundreds of papers prove theorems of the form "If the Riemann Hypothesis is true then " . And those papers are themselves very broad and varied in what follows after the then. In contrast, I'm aware of only a handful of papers with results of the form "If there are no odd perfect numbers then" and what follows after the then is always something involving divisors of a number in a somewhat straightforward fashion.
The end of your comment reminds me of my Mentor saying one time that part of him hopes someone disproves the Riemann Hypothesis just because of all the papers hes read on "if the Riemann Hypothesis is true then X" and how they'll all have to be withdrawn.
He thinks its true fyi.
I wouldnt call myself an odd prime "truther" but I see no reason infinitely many couldnt exist just the first one being say > 50th Fermat Number would put it out of search range for the forseeable future. Then one about every billion more digits.
1×1=2
Do you know any papers that rely on the existence of odd perfect numbers?
@@Featherless1keep going...
2x2=4=2+2
4:03 "Euclid was actually thinking along similar lines"
Euclid: calculates perfect numbers with actual lines
Euclid god of math
I disagree
Foreal?
Beautiful pun
@@supremelordoftheuniverse5449why?
wow this is crazy. prime95 is widely used for cpu benchmarks during overclocking to check temperatures and crashes. But up until today I didn't know it was calculating mersenne prime numbers. I thought it was just trying to find prime numbers for cpu stress test. great video as always
It is used for stress testing overclocks because it is sensitive to mistakes in the calculation caused by overclocking too much.
Damn thats interesting
It says this during the test.
Finding primes was (and still is) its original purpose. It just so turns out that finding primes takes a lot of computation power and it is so well optimized that it can squeeze out every drop from a CPU. And if there is a fault anywhere in the CPU, it will show.
@@fulgerion you probably also read EULA’s 💀
You are growing older my friend, glad to be a part of your journey along the years as you share very interesting facts with the majority of humanity, those of us that are spoofs to hidden truths in the world of science and mathematics. Thank you dearly.
I will make one conjecture. If ever we find a way to time travel to the past, every single one of us following your content would love to have a day with Euler.
As a computer and math enthusiast I'm so disappointed I didn't know what Prime 95 was for, other than a OC stress test tool.
I knew Prime95 was to find Primes in addition to a stress test, but I had no idea of the depth of the GIMPS project. Considering the program is both so simple yet computationally intensive, to be known as one of the most intense stress tests for a computer, really speaks to the sheer computing power we have needed to go this far.
Read this as “as a computer who is also a math enthusiast” at first and had to think for a second lmao
26:17 "Carl Pomerance predicts that between 10 to 2,200 and infinity, there are no more than 10 to the (power of) negative 540 perfect numbers."
I'm not good at math. Can anyone tell me why that number is to the negative power instead of positive power?
As far as I know,
10 ^-1 = 1/10^1 = 1/10 = 0.1
10^-2 = 1/10^2 = 1/100 = 0.01
Therefore, 10^-540 = 1/10^540) = 1/ (1 followed by 540 zeros) = 0. (539 zeros)1
10^-540 is less than 1. However, 51 perfect numbers have already been discovered, so how can the there be no more than 0. (539 zeros)1 perfect numbers in Carl Pomerance's prediction? Is there an error somewhere?
@@simon6071 10^-540 perfect numbers of the form N=pM^2
An odd perfect number must have the form N=pM^2, so there are very close to zero odd perfect numbers expected in the range 10^2200 to infinity.
Very nice video! Just a small thing, the reason why the largest known prime is almost always a Mersenne number is not because it grows so quickly (for example numbers of form 2*3^n-1 would grow quicker...), the real reason is because we have efficient test for numbers of that form so we can test them much faster (the Lucas-Lehmer primality test).
I must mention that 3^n -1 is always even so none of those are prime.
But about the test I think you are right.
@@mehrabnikoofaraz233Thanks for correction, I've changed it to different example to avoid confusion.
Ironically, the test is so efficient that someone skilled at arithmetic could perform it using pen and paper in some hours or days, for 15-20 digit numbers. Mersenne's "all time would not suffice" claim was likely based on trial division … the oldest and least efficient primality test.
The test goes like this:
Let n be an odd prime. (NOTE: a prime exponent is necessary anyway, so other than ruling out 3 = 2^2 - 1 this is w.l.o.g.)
Construct a sequence S(i) with:
S(1) := 4
S(k + 1) := S(k)² - 2
p := 2^n - 1 is prime if and only if S(n - 1) is divisible by p.
E.g. n=3 is an odd prime, p=2^3 - 1 = 7, S(3 - 1) = S(2) = 14 = 2 * 7, therefore 7 is a Mersenne prime.
Crucially, because only divisibility matters in the end, it suffices to calculate the remainders of the S(k) modulo p, which prevents the intermediate results from growing very large.
@@TruthNerdsClear and informative. Thank you.
It’s because it’s both: it’s fast-growing but _also_ easy relatively to check.
17:48 Something about this quote just hit me hard, we are in the age of computers that started just a few decades ago and we often ignore how seriously revolutionary computer advancements are, something that could take years can now be done by a child with an iPad.
No doubt, this age will be remembered in history as the beginning of the computer age. It has completely transformed society in a way few technologies have before.
Same, I literally shed a tear.
I remember when a computer beating a human at chess was newsworthy.
Now realize that LLMs dont even come close to representing that increase in the efficiency of labour....
I just had a thought about primes. Has anyone figured 'primes' for fractions? What I mean is, instead of using whole numbers, try using a small fraction, such as 1/1298ths as your potential prime, and figure out if any two larger normal fractions multiplied together can make the smaller one. Or some other scheme using fractions to find fractional 'primes'. I'm thinking some cool new mathematical knowledge could be found, or a cool pattern.
YOUR STRATEGIES ARE SIMPLY AMAZING! I'VE TRIED SO MANY DIFFERENT APPROACHES TO BINARY OPTIONS OVER THE YEARS, BUT NOTHING HAS WORKED AS WELL AS YOUR METHODS. THANK YOU FOR SHARING YOUR KNOWLEDGE AND EXPERTISE WITH US - YOU'RE TRULY MAKING A DIFFERENCE IN THE LIVES OF SO MANY TRADERS.
One thing that is helpful about solving (or attempting to solve) such problems is that a lot of methodology is developed in the process, and methodology is always useful.
Another great thing is that it's fun to try. And that fun is a great motivation to learn the more tedious parts of mathematics. It's like when we used to say "why would I learn the multiplication tables if I have a calculator", and we had a point: what's interesting about something that's already solved?
But every person I've talked about mysteries like this one are suddenly enthralled by the idea of maybe finding the answer, and that motivation to learn is priceless.
I sometimes wonder what else could be invented or discovered if the productivity is redirected to some other endeavours.
Exactly, this whole quest spawned Prime95, which has helped me overclock PCs for years now.
The methodology is a crucial component in math, sometimes even more than the answer itself.
I love when people have made up their mind on something, like there is a heuristic argument for that there is no odd perfect numbers, and then faced with a reasonable counter argument, imidiately recognize that their original argument is flawed. Just listening to reason and take that logic in, it is beautiful
I love when people spell immediately correctly
Absolutely😊
@@ThisHandleIsAlreadyTaken839 I love when people realize that not everyone knows how to spell or read, some didn’t go to a fancy uni, check your privilege 😠
@@hanu6158 115 have thumbsed up their message, so this is one person getting their jollies from being petty. But a spell checker is not privilege - all computers, cellphones, etc. have one.
Well, he does add that there are additional arguments that make the original heuristic argument stronger, he just doesn't specify what these arguments are (possibly implossible to explain to laymen in the space of a few minutes?)
I loved the last note here. So many people get bogged down with the “why”. Sometimes “I want to” is enough of a reason.
Why is the only irrelevant question in math.
Sisyphus
Most sukkuna quote ever.
They ask me why and if. But i do it when i like to kinda message ( admittedly finnished it few hours ago yet cant recall its quote)
@@ItsJustKaya Why are you writing like this?
When Boolean Algebra was invented in the 1840s it was purely theoretical without any possible practical use.
Today it is the way the circuits in digital computers work.
Thanks I tried, but for me I need to start running through the various formula to see it working and that takes time and mental agility. I am just a bit past that right now 74 and it's nearly midnight. Take care, still very interesting.
Finding perfect numbers is one of the first algorithm assignments you get in a computer Science degree. I never knew it was such an old idea.
Clearly you didn't watch the video, it's an even idea.
@@Dranzer_Panzerthat’s a prime quality comment
When my professor asked us to write a program to find perfect number I was like wth is that then he gave us the formula so it was easy but never understood what it actually was until now I found only 2 6 and 28
@lucashershberger623 wonder away.
@lucashershberger623 Circumstantial evidence, maybe
This channel is one of the greatest argument in favour of TH-cam as a wonderful medium of learning.
channels like these are why I love TH-cam in general
I agree, Veritasium, Vsauce, SmarterEveryDay and Sabine Hossenfelder are prime examples of channels that make TH-cam worth using even if you wouldn't like all the ads and random stuff.
@@MikkoRantalainen "prime" examples
Asianometry
You didn’t really learn anything
You just watched a video for entertainment and will forget everything the moment you click on a different video
I love that you also include analog presentation methods, like the flipchard at the beginning. It not always flashy animations and short attention span bs. Your Videos are really enjoyable.
The absurdity of that 1000 page book containing that one number is that in paper form it is essentially useless, but the symbolism is so profound that people were scrambling to get a hold of a physical copy, that it sold out within days. I think this has something to do with human nature in that there is some spiritual value in having a physical copy of something, even if it is practically useless and infinitely more useful to just have a text file containing that number.
A book containing the largest known prime and a text file containing the largest known prime are actually equally useless.
It makes a fairly decent random number generator. Flip to a page and stab your finger at a number. Just skip the first and last numbers (the first is more likely to be 1 (I think, I might be thinking of something else), and the last is odd).
It's also kinda like a code pad, but less secure since there's lots of copies of it out there. To be truly secure there should only be 2 copies of a code pad. It's unbreakable though since the data is completely masked by randomness. Assuming the pad is created in a truly random manner.
@@falconerd343Benford's Law. One Time Pad.
Imagine how much energy and computation went into making that book.
actually there were just not many copies actually printed. he completely made up the part about it being a top seller on amazon.
In my intro to abstract math class in college, we had a final project to write a paper that had basically only two requirements: it was about an approved math-related topic and it had a proof that used concepts we were taught. I did mine on perfect numbers and Mersenne primes and gave a proof of the Euclid-Euler Theorem. It was super fun to learn and write about. It is awesome to see Veritasium cover this topic in the amazing quality he does and recognize the stuff that was talked about. I even concluded the paper like the video - it's nice to study stuff just because it's interesting, even if there's no obvious real world uses.
Terrific video. However, the part about Edouard Lucas could have been much stronger. He did not merely show M_67 was not prime, he was able to show M_127 was prime. This is the largest prime ever found without the aid of a computer. He did so using novel methods that did not rely on trial factorization, but rather exploited properties of the Fibonacci numbers. Using his methods he could test M_n for primality for all n equivalent to 3 modulo 4. These methods were further refined by D. H. Lehmer (who also should have been mentioned) so that all M_n could be tested; giving us the Lucas-Lehmer test for Mersenne primes. It is this test that makes GIMPS possible. For more informations see "Edouard Lucas and Primality Testing" by Hugh. C. Williams.
a very important observation - good
love me some gimps
I was half expecting the end of this to be one of those "For more information, Google 'Two Girls One Cup'." Sort of jokes.
Ooo ah....your so smart.but are you wise?
😂@@warrior4christ777
Ive never been so interested in a video that i have no idea or understanding of anything that was said, yet i want to know more even though i dont even understand the start
When he say that number theory might not have a real application in real life (but turns out can be use for encryption) I felt that, even as a guy that hate math, i realized that NO math problem is useless/don't have a real life application. And i also started to gain interest in math recently. I started to see math in this way:
Solving math problems IS hard and even frustrating, but the moment you get the final result, all of that work will be worth it
Does (e^i(pi)) + 1 = 0 really have a real life application? I didn’t think so.
@@Tryh4rd3rryes, computer graphics using the polar plane, and complex numbers being solutions to other equations with application
@Tryh4rd3rr bruh this is one of the most useful ones
@@sirgryphon7212 fr
My favorite bit of "useless" math at the time of its discovery are quaternions, they were discovered/invented a century before we needed it for avionics, orbital dynamics and computer graphics, yet they are integral to our civilisation now, allowing us to compute spatial rotations effortlessly.
I hope this leads to a great discovery that enables even more awesome technology in the future.
Thanks for sharing this 😊
Toilet flow direction is important.
You sound really smart. Sincerely.
@@Whiterioot Thanks, I try my best.
@@Soken50 congratulations on trying your best to sound really smart, which is what you just agreed with @Whiterioot about. 👍
They lowkey tricked me with the outro at 16:25 I was so disappointed for a second 😂
I was so relieved it was finnally over. BUT IT WASNT
What da faq you doing here ?
Fr Minecraft TH-camr on math 😮
@@ruskcoderso what?
Everyone enjoys Veritasium whether they like maths or not
I was looking for this comment..
Thanks to your indicator settings, everything became much clearer
Euler also worked on an interesting related problem involving "amicable numbers". Those are integers m and n where the sum of the proper divisors of m is n, and the sum of the proper divisors of n is m (so a perfect number would be where m=n). At the time, only a handful of examples were known, but Euler managed to come up with a recipe for generating many more. With one paper, the number of known pairs went from 3 to 61.
That's like really cool, especially considering that these are also pretty big, like the numbers in the 61st pair are well over 2.5 million!
Funnily enough, though, in spite of finding some quite large amicable pairs, Euler missed the second smallest pair in existence. It was eventually found by a random nobody about a hundred years later, having been overlooked by dozens of more prolific mathematicians who had searched for amicable pairs.
Unlike the perfect numbers, there are instances of odd amicable pairs. Now, for an open question: Is there an amicable pair where one is even and the other is odd?
@@patrickmckinley8739 Interesting - I didn't know about that problem. Just to be safe, though, I'm not going to spend too much time trying to find an example that might not exist.
@@MathFromAlphaToOmega from my basic understanding, that open question would be quite analogous to the Odd Perfect Number question.
Likely, the optimal known method for searching for such numbers would ALSO be running a vast network to reach insane levels of compute, for a few decades. Not likely something searchable in an individual’s free time.
However, that is an assumption, unless there’s a proof that the problems have a certain equivalence. If there isn’t, then maybe there’s a different approach waiting to be found! And breakthroughs in number theory ARE things that individuals have accomplished, as illustrated by this video.
I admire this guy enough to know that when he says "WHAT BLOWS MY MIND IS" and after saying the thing he does the BOOM gesture... if I stay impassive, it means that i have missed an important chunk somewhere
When Derek's mind is blown, everybody's mind is blown!
I remember when I was 10, my mother and I were discussing increasing my allowance and I said "you should increase it one dollar every week". She knew it would be expensive and said "do you know how much this will cost me in the future Pedro?" and I don't know how, it just came to me on that day, yes, I just sum all the numbers, and the best way is to get the first and last one, sum, and multiply it by half the amount of weeks. I realized later this was a theorem well known. This silly (real) story is just to show how a small interaction with math can make you love it forever.
When I was a child, I would add numbers from 1 to 10
Then 1 to 20 and similarly till 100
And I used to do the same method that you mentioned
The ending I imaged went, along the lines of, 'how a small interaction with Mum can make you grateful for the sacrifices we caused her to endure.'
You two have mathematical minds.
Your videos are always so crisp, clean, and educational. I absolutely love how you provide the historical progression of things without a bunch of fluff. There is no doubt you are making a positive impact in minds around the world! THANK YOU!
37
@@satriorukito 42. At least, that's what Douglas Adams tells us.
When even Euler goes "this is a most difficult problem" I think everyone else can basically just pack it in and not even bother trying
No! That's the most golden flag possible for an interesting problem.
*went sunbathing*
Yes ! Math is so beautiful@@reapicus557
Yeah, when the going gets tough, the tough GIVE UP!
that's proof enough for me tbh
The real benefit of solving those kinds of problems is usually not the solved problem itself, but the insight you gained while solving it and the kinds of techniques and methods developed beeing useful in other areas where you didn't expect them to be useful. Noone knows whether the tool you invented to solve this kind of problem will suddenly crack open other problems as well in (at first glance) unrelated fields of mathmatics.
Edit: Thats also the reason why proving something simply by checking all possible cases with a computer isn't very well respected by mathematicians. Sure, you may have the proof that something does/doesn't exist, but it tells you absolutly nothing about *why* it does/doesn't exist. Your understanding of the topic is still the same as befor....
its the journey as they say
'The real treasure is the friends you made along the way'
Well, i don't think knowing if there is an odd perfect number would help anywhere
@@rishikeshwaghyes, especially the friends from 2000 years ago who wrote about perfect numbers
mathematicians should be banned from using computers
Decided to give it a go using pascals triangle. I found that the row with a perfect number generates it using consecutive sum of the row number - 1. So, [(n-1)n]/2 where n is a row in pascals number. Also, I found that each row number that generates a perfect number is a multiple of the previous row numbers. And since the first row with a perfect number is row 4 (the sum being [3(4)]\2 = 6, each following row must be an even numbered row. And since it's the consecutive sum of n-1, the perfect number would also be even because the consecutive sum of an odd number is even. My only problem is proving each row with a perfect number is an even numbered row. Or at least a multiple of the previous row numbers that generate a perfect number
I love your channel so much, because the problems presented are discussed on a very nice level. Not layman's style, not lecture style, right in the middle. Awesome.
your feelings are irrational
Your "right in the middle" maybe. For an amoeba like me, he lost me after like 3 mins 🤣🤣
I'll just be over here licking the window 😂
Math was my best subject in school, I made an A in calculus. But it's hard for me to follow sometimes
agree 100%. I tried reading about number theory when I was in college 20 years ago, before youtube, and I could only make it a couple pages into the first chapter before these textbooks seemingly go off into outer-space. Derek has done a great job of digesting and explaining. Just what I needed.
29:08 - "If you're a high schooler and you just love mathematics and you think 'I want a problem to think about', this one's a great problem to think about. And you can make progress, you can figure out new things. Yeah, don't be scared"
Instructions unclear, and now I am caught in the steely grip of the Collatz Conjecture.
Gee, thanks Professor Nielsen! 😂
Hey after 8128 is the next perfect number 41,328?
@@harshrajveermaran5792 No. The next perfect number is with p = 13, so 2¹²(2¹³ - 1) = 33550336
@@harshrajveermaran5792no it's 33,550,336.
Veritasium already did a video on Collatz 🫡
What if there is only one odd perfect number, and it's the only number at which Collatz Conjecture fails? 😳
I first learned about GIMPS in a science magazine in Bangladesh, I think in around 2012-2013. I set up GIMPS in my dad's laptop (I did not own a laptop then), and then his work computer. Finally I installed it in my laptop in 2019 when I came to the States for higher studies. Currently my dad is retired and the program only runs in my laptop. I have donated computing power to show that more than 50 numbers are not prime, still looking for one. My wife pokes fun at me when around every two to three months the LL test (or now the PRP test) on a potential number nears completion as everytime the number has turned out to be not a prime and I have been sad, and my wife finds this ritual mildly amusing. I do not even shut down my laptop. 😅 it is always on and the program is always running
I think I earned about them from watching Pulp Fiction...
Nice 🤜🤛
Awesome
Thank you for your service
This sounds like crypto mining lol
20:18 The book’s editor deserves a raise for proofreading and making sure all the numbers are correct!
17:37 ish
"he gave a talk" "without saying a word" thats a new level of genius
Based genius
Based AF braa
Actions speak louder than words
Nelson Cole is the main Character!
*Drops chalk and walks off stage
I was watching this on my TV, and I had to pause so I can come to mobile to say this: I love you. There are no traditional media companies who provide anything close to the same content that you do. Thank you, and thank you, and thank you for everything that you do.
💯 agree
We all swim in the water of YT, and as fish say, "What is this 'water'-thing you speak of?"
I watched all of Cosmos when I was a kid. Saw a few Burke's Connections in U.S.A. Just has to sink in that we are living in a golden age of science/math content. "Traditional media" don't care about math! Can't sell the soap, ha,ha!!!!
I thought it was weird for this to be uploaded at night for EST but then I remembered he just moved to Australia, so it’s still technically a normal morning upload for him
When did he move from LA?
Can confirm. It's midday here in 🌏
Fr I’m about to sleep soon
It's evening for me
Honestly it feels weird to be awake when a big channel releases a video lmao
Australia's timezone is hilariously inconvenient if you watch US or Euro stuff
I think the best case proof for this from mathematical induction and to recognize that they probably don’t exist, given the absence of them as testing grows to larger and larger numbers, taken with the unique properties that an odd perfect would have to possess.
16:17 Peter Barlow's statement awakened the mathematician in me until this transition
17:41
I choose to believe he dropped the chalk like it was a mic and just walked out, dapping up a few mathematicians on the way.
Imagine he just wrote some random ass numbers and it didn't even multiply to the original
😅u
almost cried at the end. "the only way to know for sure is to try" has always, always made so much sense to me. and i just found another one. I'm so glad to just be alive at times like these.
bro, that's literally part of the foundation of all of science and mathematics.
@@annoy4nce648 Damn the takeaway from this video though - now I have a burning desire to actually go try something that might be a dud XP
🫂 we brothers should make our own country
These comments are extremely weird.
@@DasAntiNaziBroetchenyou aint lie my boy 😂😂😂
Hey Veritasium, I hope you read this message. I am a high school student from Turkey, currently studying at 10th grade. I was going to talk about the equation you showed at 6:08 . I actually found that equation when I was at 8th grade, later learned from my uncle -who is a math teacher- that such equation already existed. What are your opinions about this? Thanks.
I read somewhere that Mersenne may have made a typo on the 67th number of (2^n)-1, since the 61th number of that form is prime, and 7 looks close enough to 1.
All said, this can't be confirmed.
Also, in that same book, I read that Cole's exact words were, "Three years of Sundays."
Also, fun fact: Any Mersenne Number whose index is composite will be composite. The same cannot be said for primes, since (2^13)-1 = 8191, and (2^8191)-1 is not prime.
Thus, run a test and ignore all the non-prime indexes.
For those wondering, GIMPS last checked M118212673 (as of my comment), so theres a good starting point.
This is … a really interesting problem, and again we don’t know the answer, because the historical record is missing in substantial parts.
First of all - lots of people made false claims about the perfect numbers: Ibn Fallûs, Cataldi, Fermat, even Euler thought at one point that 2^43-1 and 2^47-1 were primes, related to perfect numbers (they aren’t).
Mersenne is really interesting because he claims perfect numbers for outrageously large exponents that he couldn’t possibly have evaluated with the resources he had to hand in 1644. While Euler in the next century is erroneously laying claim to 2^47-1, Mersenne stakes out 2^67-1, 2^127-1, and 2^257-1 as primes related to perfect numbers. It took until 1876 for Lucas to prove 2^127-1, a 39-digit number, was prime. (The other two numbers aren’t.) This was the largest prime known for about the next 75 years until the first vacuum tube computers were given the task.
What, or perhaps whom, gave Mersenne the idea of claiming these colossal numbers? The following involves speculation to high powers.
Mersenne’s circle of mathematicians included Pierre de Fermat, and four years earlier Fermat had written to Mersenne with a description of his method for shortcuts at finding perfect numbers: “Je trouve plusieurs abrégés pour trouver les nombres parfaits ...”. The rest of Fermat’s letter (no. XXXIX in the correspondence) is inconveniently lost - it is presumed (for example by Fletcher, for the case of whether 2^37-1 is prime) that Fermat might have shared some calculations that Mersenne could have used later. So we could speculate Fermat might have suggested 2^61-1 is prime, which Mersenne then might have misread as using the exponent 67, but unfortunately any such evidence is lost to time. However that would only raise the score to two out of five, since there were two other perfect numbers (corresponding to the primes 2^89-1, 2^107-1) in the same range missed by Mersenne. It still doesn’t provide any explanation for what method was used for making these colossal predictions.
@@Xanthe_Cat I think the method is probably "I checked a whole bunch of factors and it looks prime". That would discard most candidates and makes it quite a bit more likely to get "lucky" - which he did with 2^127 and maybe with 2^61 (if it was that) - but not with the next one
@@Cowtymsmiesznego Maybe, but we simply don’t know what method Mersenne had, and there are a number of Mersennes that are improbably difficult to factor well before getting to M257.
@@Xanthe_Cat Right - that's why he didn't factor them, just checked enough potential prime factors to take an "informed guess". Idk, it was probably more advanced than that but that's the gist of it I'd say
A couple hundred years ago, this Galois dude worked on this unsolvable geometry thing, he actually came up a solution (or whatever the appropriate expression is), and 200 years later it was found to be useful in designing cell phone antenna. Its a crazy story, and his short life should probably be made into a movie,just because its all so darn crazy
This Galois dude 😅
I really love the direction of this channel was heading towards, which I felt that specially videos from the last 6 months or so, it's not just sharing something amazing or interesting, but really courage who was watching to pursue something, or to realize more possibilities this world offers.
Wow, this video is absolutely fascinating and so well put together! 🎉 I really appreciate the clear explanations and the engaging visuals-such a great way to break down complex topics. It's always refreshing to come across content that makes science both informative and fun to watch! Keep up the amazing work, looking forward to seeing more! 🙌
All I can think is how mathematicians throughout history would be absolutely blown away by modern computer technology. I think they would be so proud to know that people picked up and carried their legacy and continued work on this problem. Just imagine what could have happened if Euler got his hands on Matlab or Wolfram alpha
on the contrary, matlab or wolfram alpha might not exist without Euler discoveries
@@skyfeelan very true, it's because of these number theory why supercomputer turned out to be super... math is the foundation of everything 🎉
US would be bombimg mars by now.
@@skyfeelan While this is true, it's interesting to imagine what would've happened if the development of the technology could've happen within their lifespan. Impossible, of course, but it's interesting to think about.
I wonder if they would be even more shocked at how much we still can't solve...
As the co-discoverer of the first GIMPS prime (the 35th), I wasn't even aware of this unsolved problem...!
-Joel Armengaud
whgats a GIMPS prime
What a waste of time. Look…
There isn’t an odd one.
This is now officially solved.
@@PaulDeanBumgarner Is the joke that you pretend to be a boomer? Cuz "Bumgarner" surely can't be a real name.
Bro is real
@@DasAntiNaziBroetchenI've seen both Bumgardner and Baumgartner, I'm sure Bumgarner exists somewhere
Not me watching thinking I’m gonna try to solve this while eating hot cheetos
Ghost pepper, Cheeteeeeeeeaeeeaeaeaeaeaeaeaeaeaeaeæéêēêåeeeaeaeaeaeaea
this comment just blew my mind🤯 doing this exact thing while high
Nah it’s alright. Better an attempt at solving it, than not trying at all ❤
Even if you're not a mathematician, you should give it a go if you're interested!
Math problems that stump the masters get solved by a novice perspective all the time, but even if you end up retreading existing ground, you'll end up learning something cool along the way :)
That's so inspiring haha thanks@CananaMan
There is a little * at M49 to M51. Not all candidates >M48 and
Man, this video made me realise how little we think about the world. I used to think there may be a point where we learn everything from this world, but seeing this, i realise we just think very little of everything, including ourself. I want to introduce change to myself but seeing videos like this, gives me an idea of how to proceed, even though i am not mathemathician, but i hope to become so
The more we learn, the more we realise how little we know
@@mansouralshamri1387 Even though i had the desire to read more books and engage in more subjects (most of them are self taught), it will still not be enough to achieve my goal. i dreamt to become like leonardo davinci but as technology progresses, it is becoming little easier but i question that where is the world going then? To pursue things that we don't know? But it also makes them less wiser, or maybe more? Or is it the phenoemon that sapiens are unaware of? I wish that if finances were not the problem in my whole life, i can figure it by myself
@@stupiocity245 it definitely doesnt make any them less wiser, every form of new knowledge isnt bad, ever. just go ahead, experiment and find little by little how you can introduce change in yourself. as time goes by, no matter the path you went, when you look back you will realize you definitely changed
@@mansouralshamri1387 However, at some point, wisdom must kick in, to make us realize that not all of that knowledge is valuable or useful (except perhaps on trivia night).
Intelligence is knowing a tomato is a fruit.
Wisdom is not putting it in a fruit salad.
Do you know that a woman was given the abortion pill ,gave birth to her child and they cut the spinal cord and put the baby in the bin while he,she was alive
Once again your math videos sail way above my comprehension level, but I feel like the more exposure to these concepts you give me the closer I get to actually understanding some of it.
For real, same here.
same cause before these typa videos id be lost a minute or two in, this one i was perfectly following up until the sigma function stuff
@@micah4628 The Sigma function ,
σ(n) = the sum of all the divisors of n
Yes that's it!
Since a perfect number is equal to the sum of all of its divisors EXCEPT for the number itself! Therefore, sigma( perfect number) = 2 * the number itself
@@1stlullaby484 thanks i sorta got that part of it, it was more the part around 14:10 i was confused with but i just rewatched it with that in mind and i got it now, thanks!
@@micah4628that's why in case of a prime number say p we get
sigma(p) =1+p
Because these are the only divisors of p
Now you might be thinking what about -1 and -p , but in number theory people are only concerned with positive integers (this doesn't mean that other numbers don't show up, it's just a field focusing on positive ones)
10:45 I feel that calling Euler a "prodigy" is a bit of an understatement.
Yeah Magnus Carlson was just good at Chess at 20 pales to the understatement that 20 year old Euler was just a prodigy
@cf-yg4bd I was about to throw one back at you then realized I legitimately can’t think of one either. Well said.
What is special about them? It is my first time seeing their name.
@@PlayerSlotAvailablehe’s a revolutionary in math-you can look him up on your own time, but for example, he’s the one who came up with the modern notation for functions, and also came up with the most beautiful math equation (Euler’s identity).
@@PlayerSlotAvailableHe is the greatest mathematician to ever live. It’s hard to even compare him with other people in other fields. Like I can’t think of anyone having as big of an impact in their field as euler did with mathematics.
Finally a better TITLE!!
Changed from "Do odd perfect numbers exist"
To "The Oldest Unsolved Problem in Math"
Perfect!❤
I use prime95 a lot for stability tests and DID NOT know the history behind prime95. I felt chills when it was shown. Thanks!
Is it a good stress test?
@@96thelycan Yeah it's one of the best
@@96thelycan Yes. So is linpak. But prime95 is actually contributing to some collective goal.
19:10
Been building computes for 20 years now and back in the day Prime95 was _the_ way to stress test your CPU. I did know it was a math test but this is the first I’ve seen it explained exactly what it was doing.
You have such a knack for presenting information that is way over my head, and I often fail to fully understand it but I'm still fascinated, and consistently placed in awe of the mystery and power of math and numbers. Thank you for your public service Destin!
Chemistry TH-cam
i'm becoming more respectful to my teachers, when i realize i can now understand and enjoy these kind of videos.. even 15 years later after the school..
there are more perfect numbers, but they become increasingly rare and larger as you go up in value. The first four perfect numbers, as mentioned earlier, are 6, 28, 496, and 8128.
The next perfect number after 8128 is 33,550,336.
After that comes 8,589,869,056.
Perfect numbers are connected to Mersenne primes (prime numbers of the form ). Every even perfect number can be written in the form:
2^{(p-1)} \times (2^p - 1)
where is a Mersenne prime. However, it is unknown if there are any odd perfect numbers or if they exist at all. So far, only even perfect numbers have been discovered.
One thing that you overlooked with Lucas is that he helped develop an algorithm that could determine the primality of Mersenne numbers far, far faster than trial division. What is now known as the Lucas-Lehmer primality test only takes p steps (which today involves a modular square and a subtraction), where p is the exponent of the Mersenne number, which is many orders of magnitude faster than trying to divide the whole Mersenne number by every prime between 3 and its square root to see if any return a zero remainder (although the prime factors, including itself, are of the form 2kp + 1, which narrows the search a bit).
Using his "Lucas sequences", he proved that 2^127 - 1 was prime, the largest prime number ever calculated by hand.
This channel is absolutely THE BEST science channel. Not only on YT but in general. I'm a primary school teacher from Poland and the amount of facts and curiosities I get from here and transfer into teaching physics, chemistry and even English is astonishing. Thank you.
I absolutely recommend you Real engineering , Mustard , Vsause , Kirzguat in nuteshell ( Idk perfect name ) , But why , SciencePhileAI , Kosmo ..
there are many more who provide valuable information with the proof and good details and you can learn something new that's worth your time instead of spending time on tiktk..
Shout out to P(r)oland my favorite country
Numberphile is a similar channel, but you probably know that 😊
Historical math videos have become my favourite type of videos on this chanel. Please continue doing them. It is not necessary to have fancy animations or graphics. Great work
The numbers 6, 28, and 496 are perfect numbers. A perfect number is a number that is equal to the sum of its proper divisors (excluding the number itself).
6: Divisors are 1, 2, 3 → 1 + 2 + 3 = 6.
28: Divisors are 1, 2, 4, 7, 14 → 1 + 2 + 4 + 7 + 14 = 28.
496: Divisors are 1, 2, 4, 8, 16, 31, 62, 124, 248 → 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496.
The next perfect number after 496 is 8128.
Its divisors are 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064. Adding them gives 8128.
So, the next number is 8128.
I "worked" on this problem when I was a math student, but miserably failed, thanks for bringing this on youtube. Your channel is a gem man. thanks for your work. If I had to guess there is no odd perfect number but infinite even perfect numbers.
I wonder how many matching digits of Pie you could find within that book. I'd think "31415" would show up once in that string of numbers.
Edit: in the 39th Mersenne prime the string of "31415" shows up 7 times, "314159" shows up twice, and "3141592" shows up once. I did use Ctrl + F to search on the website, but there is a space every 5 digits so there could be more depending on where it starts within those 5 digits and how you search for the number, but those are the ones I've found so far. Id like to search in the 50th Mersenne prime but i cannot find a website, or PDF of the book with it fully written out so it can be easily searched for.
Was doing math problems on perfect numbers, opened youtube saw the thumbnail written 6, 28, 496 recognised they're perfect numbers, couldn't stop myself from clicking on it and here I'm enjoying the video and I've to accept Derek makes videos on topic nobody could even imagine of, hats off to this guy man, incredible
LLM. The AI knows when your vector is projected on it's own vector.
Personally i clicked because of the funny white pyramid
the principle is so simple, but the way to get the answer is very hard! That is very interesting.
Either odd perfect numbers exist, or they don’t. If they don’t, that would mean all perfect numbers are even and elegantly fit the form N = (2^p -1)*2^(p-1) with (2^p -1) prime. If they do, that means there’s some gargantuan odd perfect number somewhere out there just waiting to be discovered. And both possibilities are equally fascinating!
Third possibility: it's indeterminate
@@RichardHennigan Indeterminate how? There either is or isn't.
@@marcusscience23it could be one of those that can never be proved or disproved
the incompleteness trap card
@@marcusscience23 There is undecidableness. When running Conway's game of life there is no algorithm that guarantees predicting it's outcome in a limited number of time. So it's kind of selecting "or" from "yes or no".
Similar to problems like the Riemann hypothesis and 3n+1, if it's unprovable then it must be true, since it being false means there exists a counterexample. It can't be unprovable and false because there exists a defeater. If ZFC isn't strong enough to prove a result then you can keep adding axioms until it is, but it is impossible to know if any system at ZF's strength or stronger is consistent (you can prove it from stronger systems but this just pushes around the problem). Which leads right into Veritasium's video about the hole at the bottom of mathematics.
26:47 I LOVE how you were able to respond back to his argument, proves that you actually did your research and put him right back in his place that you're not just some youtuber who tells science stories and doesn't know better.
That was kind of awesome
Wtf do you mean put in his place? Place of what? Being an expert in the field?
Dude already admitted it's a heuristic and heuristic come with downsides. It's not a fight where people need to be put in place.
@@alex1stamford779 English isn't my first language, I meant it was when he realized he wasn't speaking to some media person who doesn't understand much
He's a professor I think. Not just some random youtuber. 😅
I actually love how quickly the professor realized he was having a double standard applying the heuristic and laughed about it. You only get that from arguing with smart people.
26:47 Pace Nilsen shows an incredible sign of intelligence! Not only did he immediately agree with a contradictory statement and not let his own beliefs that "Odd Perfect Numbers don't exist" overpower him, but simultaneously, he also reexamined and concluded that he had a bias.
The same theory that heuristically shows Odd Perfect Numbers don't exist also shows that large, even perfect numbers don't exist.
This is a true sign of intelligence, not to let your ego get in the way and search for the truth. We all can have biases, but only intelligent people will be able to look past them.
Exactly
I thought he was joking with that assumption anyway.
Unfortunately the scientific community fails to do this _far_ too often. Especially if that bias is either profitable or gets more funding for their projects.
@@yasyasmarangoz3577 , haha, might be. But it did feel like he believes that they don't exist, which, probabilistically, might eventually turn out to be true.
@@MrTuneslol I think this happens everywhere, but at the same time, many people in the scientific community can look past it, and that is when truly wonderful things are discovered or invented.
I've always found the subject of perfect numbers fascinating. I saw the thumbnail here, recognized what it was about, and actually dropped everything I was doing to watch. That doesn't happen often, so thank you so much for this awesome video
At 15:42, to prove that the exponent of p is of the form 4k+1, you just have to remark that the sum of the divisors of p^(4k+3) is always divisible by 4 (the powers of p modulo 4 are all 1 if p =4a+1 or alternating 1 and 3 if p=4k+3), which would make 2n divisible by 4 hence n even. The alternating 1 and 3 must be excluded because in this case the sum of the divisors of p^(4k+1) would be divisible by 4 as well. So p is congruent to 1 modulo p (Euler's proof as well).
dude, i dont know what're you talking about but i agree.
NERD
Minor correction! Technically, 2 is a prime number so its sigma function is odd! However, in the context of this video, it's not included in the search for an odd perfect number so this doesn't need to be stated in the discussion of sigma functions ~13 minutes in, so it's only a small correction in that phrase particularly!
I am constantly amazed by your ability to explain relatively advanced mathematical concepts in such a simple way.
I probably should have made this comment after your video on Goedel's incompleteness theorem, but this is a good example too.
I actually enjoy your math videos more than your physics ones. Great work as always.
No one asked
@@StoneTheCr0w and?
I am helpless at math, but always find these complex maths fascinating and just wonderful. Amazing what some minds can do!
Same, I watch these hoping something will drop and I will get it. So far, nothing!
Same Bro!
16:57 Idc how nerdy this makes me, but for me this feels like the mathematical version of walking away from a house while it explodes and not looking back and I love it. 😍
Yeah, while I was watching this I started thinking about all the mathematicians he mentioned as badass celebrities/superstars in some kind of drama or thriller.
The story is likely romanticised.
wrg, some tech, math etc s k , write that s k, doesn tmatter, no nerx etc nmw
heh, nerd
@zenmkultra are you... are you new here? This is the Veritasium youtube channel
I think there is some misconceptions the factors some Will be the given number
Like 6 , factors of 6 are 1+2+3=6
28 factors are 1,2,4,7 and 14 .
So 28= 1+2+4+7+14
Euler seems to have his hand in everything. What a remarkable man.
10:49 I always love his cheeky expression in the portrait.
There is a reason why mathematicians joke about naming theorem after the second person who discovered them because Euler discovered them first probably lol
When he said there was a prodigy who gave contribution in this perfect number after fermat i whispered, "Is it Euler?"
And yes it was. Obviously.
I love the sigma function half a minute later
@@canyoupoopI did the same thing! I thought “there’s no way it’s Euler… Nope, it was Euler”
Euler and Gauss those two show up everywhere
6:38 - yup, remember that Numberphile video where Matt Parker tried to differentiate between 2^(n-1) and (2^n)-1 :)
Parker primes?
Your videos are always so crisp, clean, and educational
This girl knows how to break down complex topics! Love it!
Did anyone else notice that in the number at 24:10 198,585,576,189, every set of three digits adds to 18?
I know that at least one person did.
@@jimdecamp7204 same
Finally a person that knows this fact!
I didn't, but you made me check what all perfect numbers' digits sum up to. And they all sum up to 1 (except the first one, being 6). That's actually something I didn't see anywhere while reading about perfect numbers.
@@1991dmj More succinctly, all perfect numbers modulo 9 equal 1, except 6. The properties of perfect numbers, the sum of its factors equal to the number itself is true independent of number base, as is my restatement of your interesting observation.
Another maths banger from this channel! I love science but I am a Mathematics and Stats major - please keep this content going!