The Oldest Unsolved Problem in Math
ฝัง
- เผยแพร่เมื่อ 28 เม.ย. 2024
- Do odd perfect numbers exist? Head to brilliant.org/veritasium to start your free 30-day trial, and the first 200 people get 20% off an annual premium subscription.
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A massive thank you to Prof. Pace Nielsen for all his time and help with this video.
A big thank you to Dr. Asaf Karagila, Pascal Ochem, Prof. Tianxin Cai, and Prof. William Dunham for their expertise and help.
To try GIMPS out yourself: ve42.co/GIMPS
These sources were particularly helpful:
Perfect numbers via MacTutor - ve42.co/MTPerfect
Cai, T. (2022). Perfect numbers and fibonacci sequences. World Scientific. - ve42.co/Cai2022
Dunham, W. (2022). Euler: The master of us all (Vol. 22). American Mathematical Society. - ve42.co/Dunham2022
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References:
• Perfect Numbers and Me...
• Perfect Number Proof -...
Dickson, L. E. (1919). History of the Theory of Numbers.. (Vol. 1). Carnegie Institution of Washington.
Knill, O. (2007). The oldest open problem in mathematics. NEU Math Circle, December2. - ve42.co/Knill2007
Perfect number via Wikipedia - ve42.co/WikiPerfect
Introduction to Arithmetic via HalthiTrust - ve42.co/IntroArithmetic
Nicomachus of Gerasa via MacTutor - ve42.co/MTNicomachus
Sonja, B. (1988). The First Perfect Numbers and Three Types of Amicable Numbers in a Manuscript on Elementary Number Theory by Ibn Fellûs. Erdem, c. IV, 11. - ve42.co/Sonja1988
Ibn Fallus via Wikipedia - ve42.co/WikiFallus
Mersenne prime via Wikipedia - ve42.co/WikiMP
List of Known Mersenne Prime Numbers - ve42.co/ListOfMP
Marin Mersenne via MacTutor - ve42.co/MTMersenne
Leonhard Euler via Wikipedia - ve42.co/WikiEuler
Frank Nelson Cole via Wikipedia - ve42.co/WikiFNCole
GIMPS History via Mersenne.org - ve42.co/GIMPSHistory
EFF Cooperative Computing Awards via EFF - ve42.co/EFFAwards
Jonathan Pace via Primewiki - ve42.co/PWikiPace
Book with just one number sells out in Japan via BastillePost - ve42.co/PrimeBook
Predicted distribution of Mersenne primes via John D. Cook - ve42.co/JDCookMP
Euler’s Odd Perfect Numbers Theorem via Cantor's Paradise - ve42.co/EulerOPN
A Perfect (Math) Mystery via Medium - ve42.co/Machado2024
Brent, R. P., Cohen, G. L., & te Riele, H. J. (1991). Improved techniques for lower bounds for odd perfect numbers. Mathematics of Computation, 57(196), 857-868. - ve42.co/Brent1991
Ochem, P., & Rao, M. (2012). Odd perfect numbers are greater than 10¹⁵⁰⁰. Mathematics of Computation, 81(279), 1869-1877. - ve42.co/Ochem2012
Mathematicians Open a New Front on an Ancient Number Problem via Quantamagazine - ve42.co/QuantaSpoofs
Descartes number via Wikipedia - ve42.co/WikiDescartesNumber
Andersen, N., Durham, S., Griffin, M. J., Hales, J., Jenkins, P., Keck, R., ... & Wu, D. (2022). Odd, spoof perfect factorizations. Journal of Number Theory, 234, 31-47. - ve42.co/Andersen2022
Pomerance’s Heuristic that Odd Perfect Numbers are Unlikely via OddPerfect.org - ve42.co/Heuristic
Images & Video:
Clip of Piergiorgio Odifreddi - • Odifreddi da Gramellin...
Euclid’s Elements 1 via Claymath - ve42.co/CM1
Euclid’s Elements 2 via Claymath - ve42.co/CM2
Euclid’s Elements 3 via Claymath - ve42.co/CM3
Diophanti - ve42.co/Diophanti
Gauss book - ve42.co/GaussDis
Euler’s Archive 1 - ve42.co/Euler1
Euler’s Archive 2 - ve42.co/Euler2
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Directed by Casper Mebius
Written by Casper Mebius and Derek Muller
Edited by Peter Nelson
Illustrated by Jakub Misiek
Animated by Fabio Albertelli, Ivy Tello, David Szakaly, Alondra Vitae, Alex Drakoulis, and Leigh Williamson
Filmed by Derek Muller, Raquel Nuno, and Peter Nelson
Additional research by Aaron Santos, Camilla Machado, and Gregor Čavlović
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Music from Epidemic Sound
>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
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
@@rogerszmodis6913 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
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 saw this exact comment at least 24 hours ago, does that mean i time traveled?? Or did you delete your prev post and reposted
I have an important question
Somebody said that The reason Gödel was able to show that math is incomplete [ that is there are true statements which can never be proven] is because he assumed that math is consistent (Meaning he assumed it's free of contradictions,
So what the hell is happening!!??
If this other guy is right, then Gödel's proof of incompleteness seems completely flawed
@BarryBarrington-zc6lz
As someone's who's 21... sounds _surreal!_ I even feel like congratulating you, lol. 🫱🏻🫲🏾
You forgot to end your parentheses. 😉@@1stlullaby484
@@1stlullaby484it's a form of mathematical proof known as proof by contradiction. Gödel showed that if you assume math is consistent and all true statements can be proven, obviously false statements (contradictions) arise.
A simple example is a proof for the non-existence of a largest integer. We assume two things:
1. You can increment any integer to create a larger integer.
2. There exists a largest integer.
If you apply assumption 1 to assumption 2, you end up with an integer that is larger than the supposed "largest integer". Therefore, one of the assumptions is false.
Math is a hell of a drug
It will mad u
68th like
@@WaddieJoe you mean (n-1)?
@@oliverthomas3134you sound like your on drugs
@@RAGHAV4882a-s-s
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?
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
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.
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.
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?
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."
This is a superb analysis. It got complicated as we progressed but I was amazed en route at all the ways of writing perfect numbers, and the history of the area.
Your videos are always so crisp, clean, and educational
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
What's also really cool is that if you divide the perfect number (at least the first four) by the last number in the line of numbers that make it then divide the perfect number by it, the result keeps doubling. To explain: 6 is 1+2+3, 6/3 is 2 or 2^1. 28 is 1+2+3+4+5+6+7, 28/7 is 4 or 2^2. 496 is 1+2++3...30+31, 496/31 is 16 or 2^4 or 4^2. 8128 is 1+2+3+...127+127, 8128/127 is 64 or 2^6 or 8^2. I don't know if the other perfect numbers fit that, but the first four do and I think that's funky
Bro its literally told in the video... Altho slightly differently, but its there...
Cuz 1+...+127 is 127*182/2 and that the euclids representation too
Let's say P is a perfect number.
Any series 1+2+3+4+... +n is n terms long and on average (n + 1)/2, so the sum is n * (n + 1) / 2.
So P = n * (n + 1) / 2.
Another thing we notice, is that all the series are (3, 7, 31, 127) in length. Those are powers of 2, minus 1, so let's say n = 2^m - 1.
Now, you say you divide the perfect number by the last number in the series, that would be n.
So, divide P by n simply gives:
P / n = n * (n + 1) / 2 / n = (n + 1) / 2 = (2^m - 1 + 1) / 2 = 2^m / 2 = 2^(m-1).
In other words, the power of 2 you end up with after dividing by that last number, is m - 1.
Let's look at 6 again. It it the sum of 3 numbers, n = 3, m = 2 (2^2 - 1 = 3).
So P / n is 2^(m - 1) = 2^(2 - 1) = 2^1 = 2.
For 28 n is 7, m is 3, so P / n = 2^2 = 4.
For 496 n is 31, m is 5, so P / n = 2^4 = 16.
For 828 n is 127, m is 7, so P / n = 2^6 = 64.
There is not even really a pattern there. And it doesn't work anymore for the next one: P = 33550336.
Congrats on making such a topic so enjoyable and interesting throughout the whole video. Wow!
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.
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 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..
This channel is one of the most unfettered, beautifully conceived, brilliantly executed channels on this platform.
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.
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.
I had a fun watch, definitely amazing to think about! I've been fascinated with numbers and problems since grade school and has been thinking about problems with patterns like this ever since. Not that I am any good at it nor am I sure when trying to come up with formulas based on these patterns. And sometimes, I tend to simplify these kind of problems based on what they look at. With that, I also think there is no odd perfect number for the fact that these perfect numbers we currently have all have the factor "2" which obviously makes it divisible by 2.
There's something heartwarming about seeing the quote at 9:26 and knowing how far we've come since then. I feel like past nerds would be so happy for and jealous of us for the technology we have to use for our own pursuits of knowledge haha
Edit: Not just that quote but this WHOLE VIDEO goes to show how computers were an absolute game changer. And a game changer built on the accomplishments of every genius before them 😭
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.
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.
Veritasium is an unbelievable treasure to humanity, thank you for your curiosity, your humility, and your obvious love and passion for crafting such incredibly high quality videos, they have enriched my life, and countless others around the world.
Incredible and fascinating! Even though I am fond of maths, I never heard of this problem... Besides, happily surprised to see how many times this video was viewed... :-)
Excellent job that did this TH-camr! Congrats!!
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
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
Such a great video!
I love your animations, its so easy to follow.
HELP! I need some conversions. I need all of the following each into Exatons and Kilotons!!
30 Megatons
3 Gigatons
22 Gigatons
48 Gigatons
15 Teratons
4 Petatons
8 Petatons
60 Exatons
400 Exatons
This show how dumb i am
omg exactly 37 likes !1!1
@GhostieTheML what it mean sir
@@GhostieTheML37…
@@GhostieTheMLwell it’s at 69 now
I am not smart but I still ended up watching the entire video
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?)
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.
A brilliant number theorist, sure, but would you say he's a perfect number theorist?
@@ES-54321 good one
@@ES-54321 even then.. would he be considered a brilliant perfect number theorist or even a perfect perfect number theorist or maybe a perfect even perfect number theorist?..
@@ES-54321da dum dun tssss
@@ES-54321 😂😂
The calculation itself is the application. In IT we use prime95 to stress test a machine, for example for overclocking or checking if the hardware is faulty.
28 years later you single handedly taught me how formulas are made!!!!
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 "
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..
To add to your question on the use for discovering these numbers. They will be eventually be used to quantify the compression and decompression mechanics of energy and matter in this Universe
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 💀
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
It's good to know that there are more and more vloger balancing the traditional media thanks.
I loved the end message.
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.
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.
This is the reason why i love this channel ❤
this video makes me want to solve some paradoxes I was knot going to tangle with
...that just came out of know where
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.
I assumed they were all just scrambling to buy gag gifts for their mathematician loved ones
Imagine how much energy and computation went into making that book.
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..
I'd like to thank you for making me aware of GIMPS. I'm donating some of my cpu power overnight now.
Love how this guy at the end has a cabinet full of commander decks on top.
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? 😳
One thing I'm curious about is how Descartes was able to guess the correct form for odd perfect numbers, when he of course didn't have any examples to work with, and he didn't have the language of the sigma function and its multiplicative property.
Very likely Descartes was aware that sigma was multiplicative. His spoof example doesn't make a lot of sense without it. One could try to do something similar just by thinking about pretending what is prime and counting things, as well as thinking about what is happening in terms of parity without that, but that would be tough.
Beautiful video, thank you. Celebrating the curiosity of humankind.
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
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
Its crazy how deep and complex maths is.
I thought this was going to be about the Goldbach Conjecture. But great video as usual!
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.
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. 👍
been using prime 95 for years for cpu stress testing and tuning, had no idea it was for this.
i like how after computers were made and they got to a good enough point to do fast calculations we stopped trying to imporve or find a different formula now we just rely on doing the calculations instead of trying to make the calculation process quicker or easier
For the even perfect numbers, there isn’t a different formula - that is exactly what Euler proved in the 18th century.
In terms of testing, you might have thought the only thing different nowadays is the computation power available, however this is untrue, since this video doesn’t mention any of the techniques used for searching for these perfect numbers (in terms of half an hour this would have added another few minutes of run-time).
This combines the ancient method of factoring (trial division by sieved primes), pre-computer methods (Lucas’ primality test, strengthened in the 1930s by D. H. Lehmer), and modern factoring methods (such as Pollard’s p-1 method or Lenstra’s elliptical curve method of factorisation). The GIMPS project combines all of these (as well as Fermat PRobable Prime testing, owing to greater reliability over the Lucas-Lehmer test); it’s not merely brute force computations.
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
Even though I'm pretty sure there's no better single word that could be applied, I agree.
@@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).
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
17:28 How you read the numbers is an art lol
If I view this search for the odd perfect number in a transcendental way, it seems oddly aesthetic and meaningful to me, but I can't exactly say why.
Is it that some of the greatest minds in history play the ball to each other over the course of centuries?
Is it because we are really protruding deeper into the mysteries of the universe, getting closer to it's very fabric?
Is it because we can?
Is it because people try to solve the problem despite the fact that it is seemingly impossible?
It might not have a real life application (which is also highly uncertain, there might be useful new techniques discovered in the process) but it certainly makes as much sense as meditating, dancing or creating art.
It's a question so difficult to answer that the attempts to do so have shown people the limits of their technology, which humans have nevertheless persisted in trying to answer for almost the entirety of recorded history: despite there being no obvious use for the answer to this question. So yes, I think it's fair to say it is aesthetic. The fact that we do this, says more about us humans than it does about numbers or the universe. For all we know, the concept of a perfect number has no meaning in nature at all.
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
26:37 : Fantastic how you "caught" his argument from flying! 😂
It seems likely to be that the heuristic is actually JUST for odd perfect #'s, and the mathematician was briefly confused/incorrect.
26:51
@@FeeblePenguin Not quite. Veratasium is correct here. The basic form of the heuristic does imply there are only finitely many even perfect numbers. There are some variants that partially avoid this but only partially. One way of thinking about it is that the power of 2 themselves are the culprit and allow a pattern to occur that would otherwise be extremely unlikely. But they allow things to line up just right to avoid the heuristic's probabilistic estimates.
Although it would raise the question if infinity exists in the first place.@@joshuazelinsky5213
37!
20:30 - Derek miming reading the book was hilarious
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 love the bit at 21:02 that says "If we ever lost all the prime numbers, someone could find this book, and be like, here's a big one."
I just think it's hilarious to imagine some archaeologist coming across a book and going, "Is this just a bunch of numbers? no, wait. IT'S THE ONE WE'VE BEEN SEARCHING FOR!"
After all this years, I have all of them.
Interesting how even though this is way beyond me I still find it enjoyable to watch
Please make a video on the new study going on about the age of the universe!
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
11:10 Euler named the function after himself
The Euler Totient Function...😮
look up how many things are named after Euler
Sigma Eular 😅
Well if you discover new function i think you earned the right to named it after yourself
he meant sigma guys, chill :)
I'm sure there's one lurking out there silently chuckling to itself; then again- I think we'll make contact with an alien species before we find it.
I am mathematician, and I learn this from you ! Thank you !
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
20:59 Imagine having a time machine and just randomly handing this book to some mathematician in the old days lmao
Almost EVERY Veritasium video has me for the first half....then totally loses me!
5:04
I dont know if anyone noticed but 6 and 28 are the first 3 digits of tau, or pi*2, meaning that not only 6 and 28 are true numbers, but also digits of pi*2
Does it hold true? If so, it could be an indication that there is an infinitely large amount of perfect numbers 🤔 I’m on my way to go look up tau and our known list of perfect numbers…
Edited: it doesn’t hold true, but it’s still neat!
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
@@grissee 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.
@@grissee 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...
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 😂😂😂
Veritasium, i dont remember what it was but something in the algorithms or something, makes me feel like this problem is somehow connected to 3x+1, maby we(or i) can rewatch the video and see what it was that caught my attention, it might of been a video i watched on the #37, although i can't remember maby we(or i) can check it out. (love the videos btw, keep up the hard work)
what software did you use to make this video?, visual effects and animation are so smooth, especially the infographics and tables part
16:17 Peter Barlow's statement awakened the mathematician in me until this transition
16:15 Damn Derek you've never tripped me up so hard in the middle of a video before
Love these math videos!
This video feels like a cup of hot chocolate in Christmas Eve, I enjoyed it very much. Thank you!
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.
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 😅
16:20
You got me there😂😂
I got lost around euler's 3rd breakthrough but still kept watching 😂
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
Euler surprises me every single time. He has been an absolute genius.
You can subtract consecutive square numbers by adding their square roots
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!
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.
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.
Prine decomposition proces a =p 2:35 wherre p^n -p where 2^3 -2 =6 , lolv😅 p= 2 p=3 p=5
I clicked on the video and immediately solved it. But my solution is too long to fit into this comment.
r/iamverysmart
Thanks, Fermat!
yeah, sure.
Same
ㅋㅋㅋㅋ