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00:00This is a video about the oldest unsolved problem in math that dates back 2000 years.
00:06Some of the brightest mathematicians of all time have tried to crack it, but all of them failed.
00:13In the year 2000, the Italian mathematician Pier Giorgio Odefredi
00:18listed it among four of the most pressing open problems at the time.
00:22Solving this problem could be as simple as finding a single number,
00:26so mathematicians have used computers and checked numbers up to 10 to the power of 2200,
00:33but so far they've come up empty-handed.
00:36Why do you think this problem has captured the imaginations of so many mathematicians?
00:41It's old, it's simple, it's beautiful, what else could you want?
00:49So the problem is this, do any odd perfect numbers exist?
00:54So what is a perfect number?
00:58Well, take the number 6 for example. You can divide it by 1, 2, 3, and 6.
01:04But let's ignore 6 because that's the number itself.
01:07And now we're left with just the proper divisors.
01:09If you add them all up, you find that they add to 6, which is the number itself.
01:15So numbers like this are called perfect.
01:17You can also try this with other numbers, like 10.
01:2110 has the proper divisors 1, 2, and 5. If you add those up, you only get 8.
01:26So 10 is not a perfect number.
01:29Now you can repeat this for all other numbers.
01:32And what you find is that most numbers either overshoot or undershoot.
01:36Between 1 and 100, only 6 and 28 are perfect numbers.
01:41Go up to 10,000 and you find the next two perfect numbers, 496 and 8,128.
01:51These were the only perfect numbers known by the ancient Greeks.
01:55And they would be the only known ones for over a thousand years.
01:59If only we could find a pattern that makes these numbers,
02:02then we could use that to predict more of them.
02:05So what do these numbers have in common?
02:07Well, one thing to notice is that each next perfect number is one digit longer than the number that came before it.
02:15Another thing they share is that the ending digit alternates between 6 and 8, which also means they are all even.
02:27But here's where things get really weird.
02:30You can write 6 as the sum of 1 plus 2 plus 3 and 28 as the sum of 1 plus 2 plus 3 plus 4 plus 5 plus 6 plus 7.
02:42And so on for the others as well.
02:46They are all just the sum of consecutive numbers.
02:50And you can think of each additional number as adding a new layer.
02:53And so these create a triangle, which is why these numbers are called triangular numbers.
02:58Also, every number except for 6 is the sum of consecutive odd cubes.
03:04So 28 is 1 cubed plus 3 cubed.
03:07496 is equal to 1 cubed plus 3 cubed plus 5 cubed plus 7 cubed.
03:13And 8,128 is equal to 1 cubed plus 3 cubed plus 5 cubed plus 7 cubed plus 9 cubed all the way up to 15 cubed.
03:22But here's the one that really blows my mind.
03:25If you write these numbers in binary, 6 becomes 1,1,0 and 28 becomes 1,1,1,0,0,0.
03:35496 becomes 1,1,1,1,0,0,0,0 and 8,128, you guessed it, it is also a string of ones followed by a series of zeros.
03:50So if you write them out, they are all just consecutive powers of 2.
04:03Now, around 300 BC, Euclid was actually thinking along similar lines when he discovered the pattern that makes these perfect numbers.
04:10Take the number 1 and double it.
04:12You get 2, now keep doubling it.
04:15You get 4, 8, 16, 32, 64 and so on.
04:20Now starting from 1, add the next number to it.
04:22So 1 plus 2 equals 3.
04:24If that adds up to a prime, then you multiply it by the last number in the sequence to get a perfect number.
04:30So 2 times 3 equals 6, the first perfect number.
04:35Now let's keep doing this.
04:36Add 1 plus 2 plus 4 and you get 7, which is again prime.
04:41So multiply it by the last number, 4, and you get 28, the next perfect number.
04:46Next, add 1 plus 2 plus 4 plus 8 equals 15.
04:51But 15 isn't prime, so we continue.
04:54Add 16 to get 31.
04:56This is prime.
04:57So you multiply it by 16 and you get 496, the third perfect number.
05:02Now you can keep doing this to find bigger and bigger perfect numbers.
05:08And using this, we can rewrite the first 3.
05:11So 6 equals 1 plus 2 times 2 to the power of 1.
05:14And 28 equals 1 plus 2 plus 4 times 2 squared.
05:18And 496 equals 1 plus 2 plus 4 plus 8 plus 16 times 2 to the power of 4.
05:25Where the first term is prime.
05:27But there's a more convenient way to write this still.
05:30Take any sum of consecutive powers of 2.
05:33So 2 to the power of 0, which is 1, plus 2 to the 1, plus 2 to the 2,
05:38all the way up to 2 to the n minus 1.
05:40And now because you don't know n, you don't know what that is equal to.
05:44But it will be equal to something, so let's call that t.
05:47Now multiply this whole equation by 2.
05:50So you get 2 to the 1 plus 2 to the 2, all the way up to 2 to the n, and this is equal to 2t.
05:57If you now subtract the first equation from the second, almost all the terms will cancel out.
06:02And you're left with t equals 2 to the n minus 1.
06:06So you can replace this whole series with 1 less than the next power of 2.
06:12So 6 becomes 2 squared minus 1 times 2 to the 1.
06:1628 becomes 2 cubed minus 1 times 2 squared.
06:19And 496 becomes 2 to the 5 minus 1 times 2 to the 4.
06:25Do you see the pattern?
06:27This number is always one more than this.
06:29So if we call this p, then Euclid's formula that gives a perfect number is
06:342 to the p minus 1 times 2 to the p minus 1, whenever this is prime.
06:40Now because you're multiplying it by 2 to the p minus 1, which is even, this will always give an even number.
06:50Euclid had found a way to generate even perfect numbers.
06:54But he didn't prove that this was the only way.
06:57So there could be other ways to get perfect numbers, including potentially ones that are odd.
07:04400 years later, the Greek philosopher Nicomachus published Introductio Arithmetica,
07:09the standard arithmetic text for the next thousand years.
07:13In it, he stated five conjectures, statements he believed to be true,
07:17but did not bother actually trying to prove.
07:20His conjectures were,
07:221. The nth perfect number has n digits.
07:262. All perfect numbers are even.
07:293. All perfect numbers end in 6 and 8, alternately.
07:334. Euclid's algorithm produces every even perfect number.
07:375. There are infinitely many perfect numbers.
07:42For the next thousand years, no one could prove or disprove any of these conjectures,
07:47and they were considered facts.
07:51But in the 13th century, Egyptian mathematician Ibn Fallis published a list with 10 perfect numbers
07:57and their values of p.
07:593 of these perfect numbers turned out not to be perfect at all, but the remaining ones are.
08:05The 5th perfect number is 8 digits long, which disproves Nicomachus' first conjecture.
08:12And the next thing to notice is that both the 5th and 6th perfect number end in a 6,
08:17so that disproves Nicomachus' third conjecture, that all perfect numbers end in a 6 or 8 alternately.
08:23Two conjectures were proven false, but what about the other three?
08:29Two centuries later, the problem reached Renaissance Europe,
08:32where they rediscovered the 5th, 6th, and 7th perfect numbers.
08:37So far, every perfect number had Euclid's form,
08:40and the best way to find new ones was by finding the values of p that make 2 to the p minus 1 prime.
08:47So, French polymath Marin Mersenne extensively studied numbers of this form.
08:53In 1644, he published his results in a book,
08:56including a list of 11 values of p for which he claimed they corresponded to primes.
09:01Numbers for which this is true are now called Mersenne primes.
09:06Of his list, the first seven exponents of p do result in primes,
09:10and they correspond to the first seven perfect numbers.
09:12But for some of the larger numbers, like 2 to the 67 minus 1,
09:16Mersenne admitted to not even checking whether they were prime.
09:20To tell if a given number of 15 to 20 digits is prime or not,
09:24all time would not suffice for the test.
09:30Mersenne discussed the problem of perfect numbers with other luminaries of the time,
09:33including Pierre de Fermat and René Descartes.
09:36In 1638, Descartes wrote to Mersenne,
09:39I think I can show that there are no even perfect numbers except those of Euclid.
09:44He also believed that if an odd perfect number does exist, it must have a special form.
09:50It must be the product of a prime and the square of a different number.
09:54If he was right, these would easily have been the biggest breakthroughs on the problem
09:58since Euclid 2,000 years earlier. But Descartes couldn't prove either of those statements.
10:04Instead, he wrote,
10:05As for me, I judge that one can find real, odd, perfect numbers.
10:09But whatever method you use, it takes a long time to look for these.
10:15Around 100 years later, at the St. Petersburg Academy, the Prussian mathematician Christian Goldbach
10:20met a 20-year-old math prodigy. The two stayed in touch, corresponding by mail.
10:26And in 1729, Goldbach introduced this young man to the work of Fermat.
10:31At first, he seemed indifferent, but after a little more prodding by Goldbach,
10:35he became passionate about number theory, and he spent the next 40 years working on
10:39different problems in the field. Among them was the problem of perfect numbers.
10:45This prodigy's name was Leonard Euler.
10:48Euler picked up where Descartes had left off, but with more success.
10:52In doing so, he made three breakthroughs on this problem.
10:56First, in 1732, he discovered the 8th perfect number, which he had done by verifying that
11:022 to the 31 minus 1 is prime, just as Mersenne had predicted.
11:09For his other two breakthroughs, he invented a new weapon, the sigma function.
11:14All this function does is it takes all the divisors of a number, including the number itself,
11:18and adds them up. So take any number, say 6, sum up all its divisors, and you get 12, which is twice
11:26the number we started with. And this will be true for all perfect numbers. The sigma function of a
11:32perfect number will always give twice the number itself, because the sigma function includes the
11:37number as one of its divisors. Now this may seem like a small change, but it ends up being extremely
11:43powerful. So let's look at a few examples.
11:47Take a prime number, like 7. Now because it's prime, you can't rearrange it into a rectangle.
11:52Therefore, the only divisors are 1 and the prime itself. So sigma 7 is 1 plus 7, which is equal to 8.
12:00Now to keep things easier to follow, we'll just stick to the numbers.
12:04But what if instead of 7, you had 7 cubed? Well again, the sum of the divisors is really simple.
12:10It's just 1 plus 7 plus 7 squared plus 7 cubed. Now let's use it on a different number, say 20.
12:17The sum of its divisors is 1 plus 2 plus 4 plus 5 plus 10 plus 20, which equals 42.
12:24But you can also write this as 1 plus 2 plus 4 times 1 plus 5. And this is what really makes the
12:31sigma function so powerful. If you have a number that is made up of other numbers that don't share
12:36factors with each other, then you can split up the sigma function into the sigma functions of the
12:41prime powers that make it up. So sigma 2 squared times sigma 5 is equal to sigma 20. And since any
12:49number can be written as the product of prime powers, you can split up the sigma function of any
12:54composite number into the sigma functions of its prime powers.
12:59With his new function in hand, Euler achieved his second breakthrough and did what Descartes
13:04couldn't. He proved that every even perfect number has Euclid's form. This Euclid-Euler theorem
13:11solved a 1600-year-old problem and proved Nicomachus' fourth conjecture. Math historian William Dunham called it
13:19the greatest mathematical collaboration in history. But Euler wasn't finished yet. He also wanted to
13:26solve the problem of odd perfect numbers. So for his third breakthrough, he set out to prove Descartes'
13:32other statement that every odd perfect number must have a specific form. Because if an odd perfect number
13:39does exist, you know two things. First, n is odd. And second, sigma of n equals 2n. Now, any number n you
13:48can write as a product of different prime numbers. And each prime can be to some power. So let's take
13:55that and put it into Euler's sigma function. So you get sigma of n equals sigma of all of those primes to
14:02their powers, which equals 2n. But since all of these factors are primes, you can actually split
14:08up the sigma function into the sigmas of the individual prime powers. Now, one thing to notice
14:13is that if you have a prime number raised to an odd power, for example, 7 to the power of 1,
14:18then the sigma function will be even, because 1 plus 7 equals 8. You'll always get an even number
14:25because odd plus odd is even. If the prime number is instead raised to an even power, like 7 squared,
14:33then the sigma function returns an odd number. Sigma of 7 squared equals 1 plus 7 plus 7 squared,
14:40which equals 57. Because odd plus odd plus odd equals odd. So if you have the sigma function of an odd
14:48prime raised to an odd power, it will give an even number. If instead it's raised to an even power,
14:54you get an odd number. And this is where Euler's genius insight comes in. Because here on the right
15:01side, you've got 2 times n, where n is an odd perfect number, and 2 is even. Well, what that means
15:09is that on the left side, there must only be one even number. Because if there were two even numbers,
15:14you could factor out 4. But that means you should also be able to factor out 4 on the right side,
15:20which you can't because n is odd, and there's only a single 2 here. So only one of these sigmas here
15:28can give an even number, which means that there is exactly one prime that is to an odd power,
15:33and all the others must be to an even power, just as Descartes had predicted.
15:40Now Euler refined the form a bit more and showed that an odd perfect number must satisfy this condition.
15:46But even Euler couldn't prove whether they existed or not. He wrote,
15:51whether there are any odd perfect numbers is a most difficult question.
15:56For the next 150 years, very little progress was made, and no new perfect numbers were discovered.
16:04English mathematician Peter Barlow wrote that Euler's eighth perfect number is the greatest
16:08that ever will be discovered. For as they are merely curious without being useful,
16:13it is not likely that any person will ever attempt to find one beyond it.
16:20But Barlow was wrong. Mathematicians kept pursuing these elusive perfect numbers.
16:27And most started with Mersenne's list of proposed primes. The next on his list was 2 to the 67 minus 1.
16:35So far, Mersenne had done an excellent job. He had included Euler's eighth perfect number,
16:40while avoiding others, like 29, that turned out not to lead to a perfect number.
16:45But 230 years after Mersenne published his list,
16:48Eduard Luca proved that 2 to the 67 minus 1 was not prime, although he was unable to find its factors.
16:5727 years later, Frank Nelson Cole gave a talk to the American Mathematical Society.
17:02Without saying a word, he walked to one side of the blackboard and wrote down,
17:072 to the 67 minus 1 equals 147,573,952,589,676,412,927.
17:21He then walked to the other side of the blackboard and multiplied 193,707,721 times 761,838,257,287,
17:36giving the same answer. He sat down without saying a word, and the audience erupted in applause.
17:42He later admitted it took him three years working on Sundays to solve this.
17:48A modern computer could solve this in less than a second.
17:53From 500 BC until 1952, people had discovered just 12 Mersenne primes, and therefore only 12 perfect
18:00numbers. The main difficulty was checking whether large Mersenne numbers were actually prime.
18:06But in 1952, American mathematician Raphael Robinson wrote a computer program to perform this task,
18:12and he ran it on the fastest computer at the time, the SWAC. Within 10 months, he found the next 5
18:20Mersenne primes, and so corresponding perfect numbers. And over the next 50 years, new Mersenne primes were
18:26discovered in rapid succession, all using computers. The largest Mersenne prime at the end of 1952 was
18:342 to the power of 2281 minus 1, which is 687 digits long. By the end of 1994, the largest Mersenne prime
18:42was 2 to the power of 859,433 minus 1, which is 258,716 digits long. Since these numbers were getting so
18:54astronomically large, the task of finding new Mersenne primes became more and more difficult, even for
19:00supercomputers.
19:02So in 1996, computer scientist George Woltman launched the Great Internet Mersenne prime search, or GIMPS.
19:10GIMPS distributes the work over many computers, allowing anyone to volunteer their computer power to help
19:15search for Mersenne primes. The project has been highly successful so far, having discovered 17 new Mersenne primes,
19:2315 of which were the largest known primes at that time. And the best part? If your computer discovers
19:29a new Mersenne prime, you'll be listed as its discoverer, adding yourself to a list that includes
19:34some of the best mathematicians of all time. There's even a $250,000 prize for the first billion-digit prime.
19:42In 2017, church deacon John Pace discovered the 50th Mersenne prime by using GIMPS. The number,
19:522 to the 77,232,917 minus 1, is more than 23 million digits long, and it was also the largest known prime
20:03at the time.
20:04To celebrate this achievement, the Japanese publishing house Nanai Rosha published this book,
20:10the largest prime number of 2017. And all it is, is that number spread over 719 glorious pages.
20:20It's wild. The size of this font is so tiny. The book quickly rose to the number one spot on Amazon,
20:27and sold out in four days. A year later, the 51st Mersenne prime was discovered.
20:34It's 2 to the 82,589,933 minus 1. And this number has 24,862,048 digits.
20:49But there's something I enjoy about the absurdity. Like, there is knowledge in here,
20:56but it's not the kind of knowledge that anyone's ever going to read out of a book.
20:59But in some way, it's nice that there's this physical artifact that like has the number.
21:04If ever we lost all the prime numbers, you know, someone could find this book and be like,
21:09here's a big one.
21:11As of today, this is still the largest known prime. And since numbers of this form grow so rapidly,
21:18the largest Mersenne prime is almost always the largest known prime.
21:24Computers have been incredibly successful at finding new Mersenne primes and their corresponding
21:30perfect numbers. But we've still only found 51 so far. So you might suspect that there are only a
21:37finite number of them, which would mean that Nicomachus's fifth conjecture would be false,
21:42that there aren't infinitely many perfect numbers. But that might not be the case.
21:48The Leinstra-Pomerantz-Wagstaff conjecture predicts how many Mersenne primes should appear
21:53based on how large P is. Now, this is the actual data. The conjecture performs remarkably well.
22:01But more importantly, it predicts that there are infinitely many Mersenne primes,
22:05and so infinitely many even perfect numbers. The Mersenne primes are just so large and rare
22:11that they take a lot of time and computer resources to find.
22:16But a conjecture is not a proof. And up until this day, this problem shares the title of oldest
22:22unsolved problem in math with the other open problem. Do any odd perfect numbers exist?
22:28The easiest way to solve this problem is by finding an example. So maybe we could just
22:35check different odd numbers and see if one of them is perfect. That's exactly what researchers
22:42tried in 1991. By using a smart algorithm called a factor chain, they were able to show that if an
22:47odd perfect number does exist, it must be larger than 10 to the power of 300.
22:5321 years later, Pascal Oakham and Michael Rau raised that lower bound to 10 to the 1500,
23:00with recent progress pushing that number up to 10 to the 2200.
23:05With numbers that large, it's unlikely that a computer will find one anytime soon.
23:10So we'll need to get smart.
23:12What would a proof look like? Like how could we actually prove this?
23:16I think the main idea that people have been trying to approach this problem with is coming up with more
23:21and more conditions odd perfect numbers have to satisfy. It's called this web of conditions where
23:27it has to have 10 prime factors now that we know and maybe thousands of non-distinct prime factors and
23:35has to be bigger than 10 to the 3000 and it has to do all these different things. And we hope that
23:41eventually there's just so many conditions they can strain the numbers so much that they can't exist.
23:47Since Euler, mathematicians have kept adding new conditions to this web.
23:52But so far it hasn't worked.
23:54But there might be another path.
23:57When Descartes was looking for odd perfect numbers, he came across 198,585,576,189,
24:06which you can factor as 3 squared times 7 squared times 11 squared times 13 squared times 22,021.
24:15Put this into Euler's sigma function and you find it is equal to 2 times the original number.
24:20In other words, it is perfect.
24:23That is, if 22,021 were prime, but it's not because it is equal to 19 squared times 61.
24:31And filling that in shows that it is not perfect.
24:34Numbers like this that are very close to being odd perfect numbers are called spoofs.
24:39Spoofs are a larger group of numbers.
24:43So odd perfect numbers share all properties of spoofs and then a few extra ones.
24:48And the goal is to find properties of spoofs that ultimately prevent them from being odd perfect numbers.
24:55For example, one condition of odd perfect numbers is that they can't be divided by 105.
25:00So if you find that spoofs must be divisible by 105, then this would prove that odd perfect numbers can't exist.
25:08In 2022, Pace Nielsen and a team at BYU found 21 spoof numbers, including Descartes number.
25:15And while they discovered some new properties of spoofs, they didn't find any that rule out odd perfect numbers.
25:22So how large would an odd perfect number have to be?
25:27They don't exist.
25:29You don't think odd perfect numbers exist?
25:31No, they don't exist.
25:36I wish they did. That'd be really cool.
25:38If there was just this one gigantic odd perfect number out in the universe, they don't exist. No.
25:44How are you convinced that they don't exist?
25:46There is there's something called a heuristic argument where it's not a proof.
25:53So if we had a proof, we'd be done.
25:55It's just an argument from, okay, we think primes occur this often of this type.
26:01And you put that those pieces of information together and you think, okay, on average,
26:07how many numbers should be perfect?
26:09This argument, which was made by Karl Pomerantz, predicts that between 10 to the 2200 and infinity,
26:15there are no more than 10 to the negative 540 perfect numbers of the form n equals pm squared.
26:24With odd perfect numbers, the heuristic says we shouldn't expect any.
26:29We've searched high enough now that we think we have enough evidence they shouldn't exist anymore.
26:36My understanding is this heuristic argument, it also predicts that there are no large perfect numbers, even or odd.
26:48That's true.
26:51So there's a downside.
26:52Yeah, there's a downside because it says there shouldn't be large, even perfect numbers,
26:58and we actually expect there to be infinitely many.
27:01And so, okay, so why do I believe the heuristic in this case and not this case?
27:06You're right.
27:07Am I being hypocritical about that?
27:10There are other aspects you can add on to the heuristic and make it stronger, let me put it that way.
27:15But you're right, it's not a proof.
27:16For now, this is still the oldest unsolved problem in math.
27:21Euler was right when he said whether there are any odd perfect numbers is a most difficult question.
27:29So are there any applications of this problem?
27:32I can say no.
27:37Now, many people may think that if there are no applications to the real world,
27:41then there's no point studying it.
27:44Why should anyone care about some old unsolved problem?
27:48But I think that's the wrong approach.
27:50For more than 2000 years, number theory had no real world applications.
27:54It was just mathematicians following their curiosity and solving problems they found interesting,
27:59proving one result after another and building a foundation of useless mathematics.
28:04But then in the 20th century, we realized that we could take this foundation and base our cryptography on it.
28:11This is what protects everything from text messages to government secrets.
28:16Whenever you have a group of people put their minds towards a problem,
28:20something good is going to come out of it.
28:22If it's only at the beginning, this doesn't work.
28:26Okay, well, as Edison said, I learned 999 ways of not making a light bulb.
28:32Eventually, I got a good way to do it.
28:35It's the same with math.
28:37You have a problem, and you throw your mind at it, and others do too,
28:40and you come up with new ideas, and eventually something good comes from that process.
28:44Einstein's general relativity was built on non-Euclidean geometries.
28:49Geometries that were developed as intellectual curiosities,
28:52without foresight of how they would one day change the way we understand the universe.
28:56How many people do you think are working on the problem of perfect numbers right now?
29:01I'd guess around 10 people currently have papers in the area, 10 to 15.
29:08If you're a high schooler, and you just love mathematics, and you think,
29:12I want a problem to think about, this one's a great problem to think about,
29:16and you can make progress, you can figure out new things.
29:19Yeah, don't be scared.
29:21Hundreds of people have thought about this problem for thousands of years.
29:25What can I do?
29:27You can do something.
29:28Why should you do math if you don't know that it will lead anywhere?
29:33Well, because doing the math is the only way to know for sure.
29:37You can't tell in advance what the outcome will be.
29:41Like, this problem might turn out to be a dud.
29:43We might solve it, and it might not mean anything to anyone.
29:47Or it could turn out to be remarkably helpful.
29:50The only way to know for sure is to try.
29:59In today's world, it often feels like you've got to choose between following your curiosity
30:03and building real skills you can apply.
30:05But the truth is, it's essential to do both.
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