The Oldest Unsolved Problem in Math [31:32]

Transcript
· Intro
0:00 · This is a video about the oldest unsolved problem in math that dates back 2000 years.
0:06 · Some of the brightest mathematicians of all time have tried to crack it, but all of them failed.
0:13 · In the year 2000 the Italian mathematician, Piergiorgio Odifreddi, listed it among four of the most pressing open problems
0:21 · at the time. Solving this problem could be as simple as finding a single number.
0:26 · So mathematicians have used computers and checked numbers up to 10 to the power of 2,200,
0:33 · but so far they've come up empty handed. Why do you think this problem has captured the imaginations
0:39 · of so many mathematicians? - It's old, it's simple,
0:45 · it's beautiful. - What else could you want? So the problem is this.
0:51 · Do any odd perfect numbers exist? So what is a perfect number?
· What are perfect numbers
0:58 · Well take the number six for example. You can divide it by 1, 2, 3, and 6, but let's ignore 6
1:05 · because that's the number itself, and now we're left with just the proper divisors. If you add them all up, you find that they add to six,
1:13 · which is the number itself. So numbers like this are called perfect. You can also try this with other numbers like 10.
1:21 · 10 has the proper divisors one, two, and five. If you add those up, you only get eight.
1:26 · So 10 is not a perfect number. Now you can repeat this for all other numbers,
1:32 · and what you find is that most numbers either overshoot or undershoot between 1 and a 100,
1:38 · only 6 and 28 are perfect numbers. Go up to 10,000 and you find the next two perfect numbers 496 and 8,128.
1:51 · These were the only perfect numbers known by the ancient Greeks, and they would be the only known ones
1:57 · for over a thousand years. If only we could find a pattern that makes these numbers,
2:02 · then we could use that to predict more of them. So what do these numbers have in common?
2:08 · Well, one thing to notice is that each next perfect number is one digit longer
2:13 · than the number that came before it. Another thing they share is that the ending digit alternates
2:18 · between 6 and 8, which also means they are all even.
2:27 · But here's where things get really weird. You can write 6 as the sum of 1 plus 2 plus 3
2:35 · and 28 as the sum of one, plus 2, plus 3, plus 4 plus 5 plus 6 plus 7, and so on
2:43 · for the others as well, they're all just the sum of consecutive numbers
2:50 · and you can think of each additional number as adding a new layer. And so these create a triangle,
2:55 · which is why these numbers are called triangular numbers. Also, every number except for six is the sum
3:01 · of consecutive odd cubes. So 28 is 1 cubed plus 3 cubed.
3:07 · 496 is equal to 1 cubed plus 3 cubed plus 5 cubed plus 7 cubed.
3:13 · And 8,128 is equal to 1 cubed plus 3 cubed plus 5 cubed plus 7 cubed
3:19 · plus 9 cubed all the way up to 15 cubed. But here's the one that really blows my mind.
3:25 · If you write these numbers in binary, six becomes 110,
3:30 · and 28 becomes 11100.
3:36 · 496 becomes 111110000.
3:42 · And 8,128, you guessed it. It is also a string of ones followed by a series of zeros.
3:51 · So if you write them out, they are all just consecutive powers of two.
4:03 · What now around 300 BC Euclid was actually thinking along similar lines when he discovered the pattern
4:08 · that makes these perfect numbers. Take the number one and double it, you get two now, keep doubling it.
4:15 · You get 4, 8, 16, 32, 64, and so on. Now starting from one, add the next number to it.
4:22 · So 1 plus 2 equals 3. If that adds up to a prime, then you multiply it
4:28 · by the last number in the sequence to get a perfect number. So two times three equals six, the first perfect number.
4:35 · Now let's keep doing this. Add 1 plus 2 plus 4, and you get 7, which is again prime.
4:41 · So multiply it by the last number four, and you get 28. The next perfect number.
4:46 · Next, add 1 plus 2 plus 4 plus 8 equals 15, but 15 isn't prime, so we continue add 16 to get 31,
4:56 · this is prime. So you multiply it by 16 and you get 496. The third perfect number.
5:03 · Now you can keep doing this to find bigger and bigger perfect numbers, and using this we can rewrite the first three.
5:10 · So 6 equals 1 plus 2 times 2 to the power of 1 and 28 equals 1 plus 2 plus 4 times 2 squared
5:18 · and 496 equals 1 plus 2 plus 4 plus 8 plus 16 times 2 to the power of 4
5:25 · where the first term is prime. But there's a more convenient way to write this still.
5:30 · Take any sum of consecutive powers of 2. So 2 to the power of zero which is 1 plus 2 to the 1 plus 2 to the 2,
5:38 · all the way up to 2 to the n minus 1. And now because you don't know n, you don't know what
5:43 · that is equal to, but it will be equal to something. So let's call that T. Now multiply this whole equation by two.
5:50 · So 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.
5:57 · If you now subtract the first equation from the second, almost all the terms will cancel out and you're left with T equals 2 to the n minus 1.
6:06 · So you can replace this whole series with one less than the next power of 2.
6:12 · So six becomes 2 squared minus 1 times 2 to the 1. 28 becomes 2 cubed minus 1 times 2 squared,
6:19 · and 496 becomes 2 to the 5 minus 1 times 2 to the 4.
6:25 · Do you see the pattern? This number is always one more than this. So if we call this P, then Euclid formula
6:32 · that gives a perfect number is 2 to the P minus 1 times 2 to the P minus 1 whenever this is prime.
6:42 · Now, because you're multiplying it by 2 to the P minus 1, which is even, this will always give an even number.
· The history of perfect numbers
6:50 · Euclid had found a way to generate even perfect numbers, but he didn't prove that this was the only way.
6:57 · So there could be other ways to get perfect numbers, including potentially ones that are odd.
7:04 · 400 years later, the Greek philosopher nicomchaus published Introdutio Arithmetica, the standard arithmetic text
7:11 · for the next thousand years. In it, he stated five conjectures statements he believed
7:16 · to be true, but did not bother actually trying to prove. His conjectures were one,
7:23 · the nth perfect number has n digits. Two, all perfect numbers are even.
7:29 · Three, all perfect numbers end in 6 and 8 alternately. Four, Euclid algorithm produces every even perfect number.
7:37 · And five, there are infinitely many perfect numbers. For the next thousand years
7:43 · no one could prove or disprove any of these conjectures, and they were considered facts.
7:51 · But in the 13th century, Egyptian mathematician Ibn Fallus published a list with 10 perfect numbers
7:57 · and their values of P. Three of these perfect numbers turned out not to be perfect at all.
8:03 · But the remaining ones are. The fifth perfect number is eight digits long,
8:08 · which disproves Nicomachus's first conjecture. And the next thing to notice is that both the fifth
8:14 · and sixth perfect number end in a 6. So that disproves Nicomachus's third conjecture
8:19 · that all perfect numbers end in a 6 or 8 alternately. Two conjectures were proven false.
8:25 · But what about the other three? Two centuries later, the problem reached Renaissance Europe
8:32 · where they rediscovered the fifth, sixth, and seventh perfect numbers.
8:37 · So far every perfect number had Euclid's form. And the best way to find new ones was by finding the values
8:43 · of P that make 2 to the P minus 1 prime. So French polymath Marin Mersenne extensively studied
8:51 · numbers of this form. In 1644, he published his in a book
8:56 · including a list of 11 values of P for which he claimed they corresponded to primes.
9:01 · Numbers for which this is true are now called Mersenne Primes. Of his list the first seven exponents of P
9:08 · do result in primes and they correspond to the first seven perfect numbers. But for some of the larger numbers
9:14 · like 2 to the 67 minus 1, Mersenne admitted to not even checking whether they were prime.
9:20 · "To tell if a given number of 15 to 20 digits is prime or not all time would not suffice for the test."
9:30 · Mersenne discussed the problem of perfect numbers with other luminaries of the time, including Pierre de Fermat and Rene Descartes.
9:36 · In 1638, Descartes wrote to Mersenne, I think I can show that there are no even perfect numbers
9:42 · except those of Euclid. He also believed that if an odd perfect number does exist,
9:48 · it must have a special form. It must be the product of a prime and the square of a different number.
9:55 · If he was right, these would easily have been the biggest breakthroughs on the problem since Euclid 2000 years earlier.
10:01 · But Descartes couldn't prove either of those statements. Instead, he wrote "As for me, I judge that one can find
10:07 · real odd perfect numbers. But whatever method you use, it takes a long time
10:13 · to look for these." Around a hundred years later at the St. petersburg Academy,
10:18 · the Prussian mathematician Christian Goldbach met a 20-year-old math prodigy.
10:23 · The two stayed in touch corresponding by mail, and in 1729, Goldbach introduced this young man
10:29 · to the work of Fermat. At first, he seemed indifferent, but after a little more prodding by Goldbach
10:35 · he became passionate about number theory and he spent the next 40 years working on different problems in the field
10:41 · among them was the problem of perfect numbers. This Prodigy's name was Leonhard Euler.
10:48 · Euler picked up where Descartes had left off, but with more success. In doing so, he made three breakthroughs on this problem.
10:56 · First in 1732, he discovered the eighth perfect number, which he had done
11:01 · by verifying that 2 to the 31 minus 1 is prime. Just as Mersenne had predicted.
· The sigma function
11:09 · For his other two breakthroughs, he invented a new weapon, the sigma function. All this function does is it takes all the divisors
11:16 · of a number, including the number itself and adds them up. So take any number, say six, sum up all its divisors
11:24 · and you get 12, which is twice the number we started with. And this will be true for all perfect numbers.
11:31 · The Sigma function of a perfect number will always give twice the number itself because the sigma function includes the number
11:37 · as one of its divisors. Now this may seem like a small change, but it ends up being extremely powerful.
11:44 · So let's look at a few examples. Take a prime number like seven.
11:49 · Now, because it's prime, you can't rearrange it into a rectangle, therefore the only divisors are one and the prime itself.
11:56 · So Sigma seven is 1 plus 7, which is equal to 8. Now, to keep things easier to follow,
12:02 · we'll just stick to the numbers. But what if instead of seven, you had seven cubed?
12:07 · Well, again, the sum of the divisors is really simple. It's just 1 plus 7 plus 7 squared plus 7 cubed.
12:14 · Now let's use it on a different number, say 20. The sum of its divisors is 1 plus 2 plus 4 plus 5
12:21 · plus 10 plus 20, which equals 42. But you can also write this as 1 plus 2 plus 4 times 1 plus 5.
12:30 · And this is what really makes the sigma function so powerful. If you have a number that is made up of other numbers
12:36 · that don't share factors with each other, then you can split up the sigma function into the sigma functions of the prime powers
12:41 · that make it up. So sigma of 2 squared times sigma 5 is equal to sigma 20.
12:48 · And since any number can be written as the product of prime powers, you can split up the sigma function
12:53 · of any composite number into the sigma functions of its prime powers.
12:59 · With his new function in hand, Euler achieved his second breakthrough and did what Descartes couldn't.
13:04 · He proved that every even perfect number has Euclid's form. This Euclid-Euler theorem solved a 1600-year-old problem
13:13 · and proved Nicomachus's fourth conjecture. Math historian William Dunham called it
13:19 · the greatest mathematical collaboration in history. But Euler wasn't finished yet.
13:25 · He also wanted to solve the problem of odd perfect numbers. So for his third breakthrough, he set out
13:30 · to prove Descartes other statement that every odd perfect number must have a specific form.
13:37 · Because if an odd perfect number does exist, you know two things first n is odd.
13:43 · And second sigma of n equals 2n. Now any number n, you can write as a product
13:49 · of different prime numbers and each prime can be to some power. So let's take that and put it into Euler sigma function.
13:57 · So you get sigma of n equals sigma of all of those primes to their powers, which equals 2n.
14:05 · But since all of these factors are primes, you can actually split up the sigma function into the sigmas of the individual prime powers.
14:12 · Now one thing to notice is that if you have a prime number raised to an odd power, for example seven to the power of 1,
14:18 · then the sigma function will be even because 1 plus 7 equals 8, you'll always get an even number
14:25 · because odd plus odd is even if the prime number is instead raised to an even power like seven squared,
14:33 · then the sigma function returns an odd number. Sigma of 7 squared equals 1 plus 7 plus 7 squared,
14:40 · which equals 57. Because odd plus odd plus odd equals odd.
14:46 · So if you have the sigma function of an odd prime raised to an odd power, it will give an even number.
14:52 · If instead it's raised to an even power, you get an odd number. And this is where Euler's genius insight comes in
15:00 · because here on the right side you've got 2 times n where n is an odd perfect number, and 2 is even.
15:08 · Well, what that means is that on the left side there must only be one even number because if there were two even numbers,
15:14 · you could factor out four. But that means you should also be able to factor out four on the right side, which you can't
15:21 · because n is odd and there's only a single 2 here. So only one of these sigmas here can give an even number,
15:29 · which means that there is exactly one prime that is to an odd power and all the others must be to an even power
15:36 · just as Descartes had predicted. Now, Euler refined the form a bit more
15:42 · and showed that an odd perfect number must satisfy this condition, but even Euler couldn't prove
15:49 · whether they existed or not. He wrote "Whether there are any odd perfect numbers is a most difficult question."
15:56 · For the next 150 years very little progress was made and no new perfect numbers were discovered.
16:04 · English mathematician Peter Barlow wrote that Euler eighth perfect number "Is the greatest that ever will be discovered
16:10 · for as they are merely curious without being useful, it is not likely that any person will ever attempt
16:16 · to find one beyond it." But Barlow was wrong.
16:23 · Mathematicians kept pursuing these elusive perfect numbers and most started with Mersenne's list of proposed primes.
16:31 · The next on his list was 2 to the 67 minus 1. So far, Mersenne had done an excellent job.
16:37 · He had included Euler's eighth perfect number while avoiding others like 29 that turned out not to lead to a perfect number,
16:45 · but 230 years after Mersenne published his list, Edouard Lucas proved that 2 to the 67 minus 1 was not prime,
16:53 · although he was unable to find its factors. 27 years later, Frank Nelson Cole gave a talk
17:00 · to the American mathematical society without saying a word, he walked to one side of the blackboard
17:06 · and wrote down 2 to the 67 minus 1 equals 147,573,952,589,676,412,927.
17:22 · He then walked to the other side of the blackboard and multiplied 193,707,721 times
17:30 · 761,838,257,287
17:36 · giving the same answer. He sat down without saying a word and the audience erupted in applause.
17:44 · He later admitted it took him three years working on Sundays to solve this. A modern computer could solve this in less than a second.
17:53 · From 500 BC until 1952 people had discovered just 12 Mersenne primes
17:58 · and therefore only 12 perfect numbers. The main difficulty was checking whether large Mersenne numbers were actually prime.
18:06 · But in 1952, American mathematician Raphael Robinson wrote a computer program to perform this task
18:12 · and he ran it on the fastest computer at the time, the SWAC.
18:18 · Within 10 months, he found the next five Mersenne primes and so corresponding perfect numbers.
18:24 · And over the next 50 years, new Mersenne primes were discovered in rapid succession, all using computers.
18:31 · The largest Mersenne prime at the end of 1952 was 2 to the power of 2,281 minus 1,
18:38 · which is 687 digits long. By the end of 1994, the largest Mersenne prime was 2 to the power of 859,433 minus 1,
18:48 · which is 258,716 digits long.
· The Great Internet
18:53 · Since these numbers were getting so astronomically large, the task of finding numerous end primes became
18:58 · more and more difficult even for supercomputers. So in 1996,
19:04 · computer scientist George Woltman launched the Great Internet Mersenne Prime Search or GIMPS.
19:09 · GIMPS distributes the work over many computers allowing anyone to volunteer their computer power
19:15 · to help search for Mersenne primes. The project has been highly successful so far,
19:20 · having discovered 17 new Mersenne primes, 15 of which were the largest known primes at that time.
19:26 · And the best part, if your computer discovers a new Mersenne prime, you'll be listed as its discoverer,
19:33 · adding yourself to a list that includes some of the best mathematicians of all time. There's even a $250,000 prize
19:40 · for the first billion-digit prime.
19:45 · In 2017 Church Deacon John Pace discovered the 50th Mersenne Prime by using GIMPS.
19:52 · The number 2 to the 77,232,917 minus 1
19:57 · is more than 23 million digits long, and it was also the largest known prime at the time.
20:04 · To celebrate this achievement the Japanese publishing house, Nanairosha published this book,
20:10 · "The Largest Prime number of 2017." And all it is is that number spread over 719 glorious pages.
20:20 · It's wild. The size of this font is so tiny. The book quickly rose to the number one spot on Amazon
20:27 · and sold out in four days. A year later, the 51st Mersenne Prime was discovered.
20:34 · It's 2 to the 82,589,933 minus 1,
20:42 · and this number has 24,000,860 2048 digits.
20:49 · But there's something I enjoy about the absurdity, like there is knowledge in here,
20:56 · but it's not the kind of knowledge that anyone's ever gonna read out of a book. But in some way it's nice that there's this physical artifact
21:02 · that like has the number, if ever we lost all the prime numbers. You know, someone could find this book
21:08 · be like, here's the big one. As of today, this is still the largest known prime.
21:15 · And since numbers of this form grow so rapidly, the largest Mersenne Prime is almost always the largest known prime.
21:24 · Computers have been incredibly successful at finding new Mersenne primes and their corresponding perfect numbers,
21:31 · but we've still only found 51 so far. So you might suspect that there are
21:36 · only a finite number of them, which would mean that Nicomachus's fifth conjecture would be false,
21:42 · that there aren't infinitely many perfect numbers, but that might not be the case.
21:48 · The Lenstra and Pomerance Wagstaff conjecture predicts how many Mersenne primes should appear
21:53 · based on how large P is. Now this is the actual data the conjecture performs remarkably well.
22:01 · But more importantly, it predicts that there are infinitely many Mersenne primes and so infinitely many even perfect numbers.
22:08 · The Mersenne primes are just so large and rare that they take a lot of time and computer resources to find.
· Odd Perfect Numbers
22:17 · But a conjecture is not a proof. And up until this day, this problem shares the title of oldest unsolved problem in math
22:24 · with the other open problem. Do any odd perfect numbers exist?
22:30 · The easiest way to solve this problem is by finding an example. So maybe we could just check different odd numbers
22:37 · and see if one of them is perfect. That's exactly what researchers tried in 1991.
22:43 · By using a smart algorithm called a factor chain, they were able to show that if an odd perfect number does exist, it must be larger than 10
22:51 · to the power of 300. 21 years later, Pascal Ochem and Michael Rao
22:56 · raised that lower bound to 10 to the 1,500 with recent progress pushing that number up to 10
23:02 · to the 2,200. With numbers that large, it's unlikely that a computer will find one anytime soon.
23:10 · So we'll need to get smart. What would a proof look like? Like how could we actually prove this?
23:15 · - I think the main idea that people have been trying to approach this problem with is coming up with
23:21 · more and more conditions odd perfect numbers have to satisfy, it's called this web
23:26 · of conditions where it has to have 10 prime factors now that we know and maybe thousands
23:33 · of non distinct prime factors and has to be bigger than 10 to the 3000. And it has to do all these different things
23:40 · and we hope that eventually there's just so many conditions that can strain the numbers so much that they can't exist.
23:47 · - Since Euler, mathematicians have kept adding new conditions to this web. - But so far it hasn't worked.
23:55 · - But there might be another path. When Descartes was looking for odd perfect numbers, he came across 198,585,576,189,
24:06 · which you can factor as 3 squared times 7 squared times 11 squared times 13 squared times 22021.
24:15 · Put this into Euler sigma function and you find it is equal to two times the original number.
24:20 · In other words, it is perfect. That is if 22021 were prime, but it's not
24:28 · because it is equal to 19 squared times 61. And filling that in shows that it is not perfect.
24:34 · Numbers like this that are very close to being odd perfect numbers are called spoofs.
24:40 · Spoofs are a larger group of numbers. So odd perfect numbers share all properties of spoofs
24:47 · and then a few extra ones. And the goal is to find properties of spoofs that ultimately prevent them from being odd perfect numbers.
24:55 · For example, one condition of odd perfect numbers is that they can't be divided by 105.
25:01 · So if you find that spoofs must be divisible by 105, then this would prove that odd perfect numbers can't exist.
25:08 · In 2022 Pace Nielsen and a team at BYU found 21 spoof numbers including Descartes number,
25:15 · and while they discovered some new properties of spoofs, they didn't find any that rule out odd, perfect numbers.
25:23 · So how large would an odd perfect number have to be? - They don't exist.
25:29 · - You don't think odd perfect numbers exist? - No, they don't exist.
25:36 · I wish they did. That'd be really cool if if there was just this one gigantic odd, perfect number out in the universe.
25:42 · They don't exist. No. - How are you convinced that they don't exist?
25:47 · - There is something called a heuristic argument where it's not a proof.
25:53 · So if we had a proof, we'd be done. It's just an argument from, okay, we think primes occur this often of this type.
26:02 · And you put that those pieces of information together and you think, okay, on average
26:07 · how many numbers should be perfect. - This argument, which was made by Carl Pomerance predicts
26:12 · that between 10 to the 2,200 and infinity, there are no more than 10 to the negative 540 perfect numbers
26:20 · of the form N equals pm squared - With odd perfect numbers
26:25 · the heuristic says we shouldn't expect any. We've searched high enough now that we think
26:33 · we have enough evidence they shouldn't exist anymore. - My understanding is this heuristic argument.
26:40 · It also predicts that there are no large perfect numbers even or odd.
26:46 · So... That's true. So there's a downside.
26:53 · Yeah, there's a downside because it says there shouldn't be large, even perfect numbers and we actually expect there
26:59 · to be infinitely many. And so, okay, so... why do I believe the heuristic in this case
27:05 · and not this case? You're right. Am I being hypocritical about that? There are other aspects you can add on
27:12 · to the heuristic and make it stronger. Let me put it that way. But you're right, it's not a proof. - For now this is still the oldest unsolved problem in math.
27:21 · Euler was right when he said whether there are any odd perfect numbers is a most difficult question.
27:29 · So are there any applications of this problem? - I can say no.
27:37 · - Now, many people may think that if there are no applications to the real world, then there's no point studying it.
27:44 · Why should anyone care about some old unsolved problem? But I think that's the wrong approach.
27:50 · For more than 2000 years, number theory had no real world applications. It was just mathematicians following their curiosity
27:57 · and solving problems they found interesting proving one result after another and building a foundation of useless mathematics.
28:04 · But then in the 20th century, we realized that we could take this foundation and base our cryptography on it.
28:11 · This is what protects everything from text messages to government secrets. - Whenever you have a group
28:17 · of people put their minds towards a problem, something good's gonna come out of it.
28:22 · If it's only, if it's only at the beginning, this doesn't work. Okay, well, as Edison said, I learned 999 ways
28:31 · of not making a light bulb. Eventually I got a good way to do it. It's the same with math.
28:36 · You have a problem and you throw your mind at it and others do too. And you come up with new ideas
28:41 · and eventually something good comes from that process. - Einstein's general relativity was built
28:47 · on non-Euclidean geometries, geometries that were developed as intellectual curiosities without foresight
28:53 · of how they would one day change the way we understand the universe. How many people do you think are working on the problem
29:00 · of perfect numbers right now? - I'd guess around 10 people currently
29:06 · have papers in the area, 10 to 15. If you're a high schooler and you just love mathematics
29:12 · 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.
29:19 · Yeah, don't be scared. Hundreds of people have thought about this problem for thousands of years.
29:25 · What can I do? You can do something. - Why should you do math if you don't know
29:31 · that it will lead anywhere? Well, because doing the math is the only way to know for sure.
29:37 · You can't tell in advance what the outcome will be. Like this problem might turn out to be a dud.
29:43 · We might solve it and it might not mean anything to anyone, or it could turn out to be remarkably helpful.
29:50 · [ad text redacted]
31:22 · ...So I wanna thank Brilliant for sponsoring this video and I want to thank you for watching.

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