数学中最古老的未解之谜：奇完全数之问

This is a video about the oldest unsolved problem in math that dates back 2000 years.

Some of the brightest mathematicians of all time have tried to crack it, but all of them failed.

In the year 2000 the Italian mathematician, Piergiorgio Odifreddi, listed it among four of the most pressing open problems at the time.

Solving this problem could be as simple as finding a single number.

So mathematicians have used computers and checked numbers up to 10 to the power of 2,200, but so far they've come up empty handed.

Why do you think this problem has captured the imaginations of so many mathematicians?

It's old, it's simple, it's beautiful.

What else could you want?

So the problem is this. Do any odd perfect numbers exist?

So what is a perfect number?

Well take the number six for example.

You can divide it by 1, 2, 3, and 6, but let's ignore 6 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, which is the number itself.

So numbers like this are called perfect.

You can also try this with other numbers like 10.

10 has the proper divisors one, two, and five.

If you add those up, you only get eight.

So 10 is not a perfect number.

Now you can repeat this for all other numbers, and what you find is that most numbers either overshoot or undershoot between 1 and a 100, only 6 and 28 are perfect numbers.

Go up to 10,000 and you find the next two perfect numbers 496 and 8,128.

These were the only perfect numbers known by the ancient Greeks, and they would be the only known ones for over a thousand years.

If only we could find a pattern that makes these numbers, then we could use that to predict more of them.

So what do these numbers have in common?

Well, one thing to notice is that each next perfect number is one digit longer than the number that came before it.

Another thing they share is that the ending digit alternates between 6 and 8, which also means they are all even.

But here's where things get really weird.

You can write 6 as the sum of 1 plus 2 plus 3 and 28 as the sum of one, plus 2, plus 3, plus 4 plus 5 plus 6 plus 7, and so on for the others as well, they're all just the sum of consecutive numbers

and you can think of each additional number as adding a new layer.

And so these create a triangle, which is why these numbers are called triangular numbers.

Also, every number except for six is the sum of consecutive odd cubes.

So 28 is 1 cubed plus 3 cubed.

496 is equal to 1 cubed plus 3 cubed plus 5 cubed plus 7 cubed.

And 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.

But here's the one that really blows my mind.

If you write these numbers in binary, six becomes 110,

and 28 becomes 11100.

496 becomes 111110000.

And 8,128, you guessed it.

It is also a string of ones followed by a series of zeros.

So if you write them out, they are all just consecutive powers of two.

What now around 300 BC Euclid was actually thinking along similar lines when he discovered the pattern that makes these perfect numbers.

Take the number one and double it, you get two now, keep doubling it.

You get 4, 8, 16, 32, 64, and so on.

Now starting from one, add the next number to it.

So 1 plus 2 equals 3.

If that adds up to a prime, then you multiply it by the last number in the sequence to get a perfect number.

So two times three equals six, the first perfect number.

Now let's keep doing this.

Add 1 plus 2 plus 4, and you get 7, which is again prime.

So multiply it by the last number four, and you get 28.

The next perfect number.

Next, add 1 plus 2 plus 4 plus 8 equals 15, but 15 isn't prime, so we continue add 16 to get 31,

this is prime.

So you multiply it by 16 and you get 496.

The third perfect number.

Now you can keep doing this to find bigger and bigger perfect numbers, and using this we can rewrite the first three.

So 6 equals 1 plus 2 times 2 to the power of 1

and 28 equals 1 plus 2 plus 4 times 2 squared

and 496 equals 1 plus 2 plus 4 plus 8 plus 16 times 2 to the power of 4 where the first term is prime.

But there's a more convenient way to write this still.

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, all the way up to 2 to the n minus 1.

And now because you don't know n, you don't know what that is equal to, but it will be equal to something.

So let's call that T.

Now multiply this whole equation by two.

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.

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.

So you can replace this whole series with one less than the next power of 2.

So six becomes 2 squared minus 1 times 2 to the 1.

28 becomes 2 cubed minus 1 times 2 squared,

and 496 becomes 2 to the 5 minus 1 times 2 to the 4.

Do you see the pattern?

This number is always one more than this.

So if we call this P, then Euclid formula that gives a perfect number is 2 to the P minus 1 times 2 to the P minus 1 whenever this is prime.

Now, because you're multiplying it by 2 to the P minus 1, which is even, this will always give an even number.

Euclid had found a way to generate even perfect numbers, but he didn't prove that this was the only way.

So there could be other ways to get perfect numbers, including potentially ones that are odd.

400 years later, the Greek philosopher nicomchaus published Introdutio Arithmetica, the standard arithmetic text for the next thousand years.

In it, he stated five conjectures statements he believed to be true, but did not bother actually trying to prove.

Two, all perfect numbers are even.

Three, all perfect numbers end in 6 and 8 alternately.

Four, Euclid algorithm produces every even perfect number.

And five, there are infinitely many perfect numbers.

For the next thousand years no one could prove or disprove any of these conjectures, and they were considered facts.

But in the 13th century, Egyptian mathematician Ibn Fallus published a list with 10 perfect numbers and their values of P.

Three of these perfect numbers turned out not to be perfect at all.

But the remaining ones are.

The fifth perfect number is eight digits long, which disproves Nicomachus's first conjecture.

And the next thing to notice is that both the fifth and sixth perfect number end in a 6.

So that disproves Nicomachus's third conjecture that all perfect numbers end in a 6 or 8 alternately.

Two conjectures were proven false.

But what about the other three?

Two centuries later, the problem reached Renaissance Europe where they rediscovered the fifth, sixth, and seventh perfect numbers.

So far every perfect number had Euclid's form.

And the best way to find new ones was by finding the values of P that make 2 to the P minus 1 prime.

So French polymath Marin Mersenne extensively studied numbers of this form.

In 1644, he published his in a book including a list of 11 values of P for which he claimed they corresponded to primes.

Numbers for which this is true are now called Mersenne Primes.

Of his list the first seven exponents of P do result in primes and they correspond to the first seven perfect numbers.

But for some of the larger numbers like 2 to the 67 minus 1, Mersenne admitted to not even checking whether they were prime.

"To tell if a given number of 15 to 20 digits is prime or not all time would not suffice for the test."

Mersenne discussed the problem of perfect numbers with other luminaries of the time, including Pierre de Fermat and Rene Descartes.

In 1638, Descartes wrote to Mersenne, I think I can show that there are no even perfect numbers except those of Euclid.

He also believed that if an odd perfect number does exist, it must have a special form.

It must be the product of a prime and the square of a different number.

If he was right, these would easily have been the biggest breakthroughs on the problem since Euclid 2000 years earlier.

But Descartes couldn't prove either of those statements.

Instead, he wrote "As for me, I judge that one can find real odd perfect numbers. But whatever method you use, it takes a long time to look for these."

Around a hundred years later at the St. petersburg Academy, the Prussian mathematician Christian Goldbach met a 20-year-old math prodigy.

The two stayed in touch corresponding by mail, and in 1729, Goldbach introduced this young man to the work of Fermat.

At first, he seemed indifferent, but after a little more prodding by Goldbach he became passionate about number theory

and he spent the next 40 years working on different problems in the field among them was the problem of perfect numbers.

This Prodigy's name was Leonhard Euler.

Euler picked up where Descartes had left off, but with more success.

In doing so, he made three breakthroughs on this problem.

First in 1732, he discovered the eighth perfect number, which he had done by verifying that 2 to the 31 minus 1 is prime.

Just as Mersenne had predicted.

For his other two breakthroughs, he invented a new weapon, the sigma function.

All this function does is it takes all the divisors of a number, including the number itself and adds them up.

So take any number, say six, sum up all its divisors and you get 12, which is twice the number we started with.

And this will be true for all perfect numbers.

The Sigma function of a perfect number will always give twice the number itself because the sigma function includes the number as one of its divisors.

Now this may seem like a small change, but it ends up being extremely powerful.

So let's look at a few examples.

Take a prime number like seven.

Now, because it's prime, you can't rearrange it into a rectangle, therefore the only divisors are one and the prime itself.

So Sigma seven is 1 plus 7, which is equal to 8.

Now, to keep things easier to follow, we'll just stick to the numbers.

But what if instead of seven, you had seven cubed?

Well, again, the sum of the divisors is really simple.

It's just 1 plus 7 plus 7 squared plus 7 cubed.

Now let's use it on a different number, say 20.

The sum of its divisors is 1 plus 2 plus 4 plus 5 plus 10 plus 20, which equals 42.

But you can also write this as 1 plus 2 plus 4 times 1 plus 5.

And this is what really makes the sigma function so powerful.

If you have a number that is made up of other numbers that don't share factors with each other, then you can split up the sigma function into the sigma functions of the prime powers that make it up.

So sigma of 2 squared times sigma 5 is equal to sigma 20.

And since any number can be written as the product of prime powers, you can split up the sigma function of any composite number into the sigma functions of its prime powers.

With his new function in hand, Euler achieved his second breakthrough and did what Descartes couldn't.

He proved that every even perfect number has Euclid's form.

This Euclid-Euler theorem solved a 1600-year-old problem and proved Nicomachus's fourth conjecture.

Math historian William Dunham called it the greatest mathematical collaboration in history.

But Euler wasn't finished yet.

He also wanted to solve the problem of odd perfect numbers.

So for his third breakthrough, he set out to prove Descartes other statement that every odd perfect number must have a specific form.

Because if an odd perfect number does exist, you know two things first n is odd.

And second sigma of n equals 2n.

Now any number n, you can write as a product of different prime numbers and each prime can be to some power.

So let's take that and put it into Euler sigma function.

So you get sigma of n equals sigma of all of those primes to their powers, which equals 2n.

But since all of these factors are primes, you can actually split up the sigma function into the sigmas of the individual prime powers.

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, then the sigma function will be even

because 1 plus 7 equals 8, you'll always get an even number because odd plus odd is even

if the prime number is instead raised to an even power like seven squared, then the sigma function returns an odd number.

Sigma of 7 squared equals 1 plus 7 plus 7 squared, which equals 57.

Because odd plus odd plus odd equals odd.

So if you have the sigma function of an odd prime raised to an odd power, it will give an even number.

If instead it's raised to an even power, you get an odd number.

And this is where Euler's genius insight comes in because here on the right side you've got 2 times n where n is an odd perfect number, and 2 is even.

Well, what that means is that on the left side there must only be one even number because if there were two even numbers, 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 because n is odd and there's only a single 2 here.

So only one of these sigmas here can give an even number, which means that there is exactly one prime that is to an odd power and all the others must be to an even power just as Descartes had predicted.

Now, Euler refined the form a bit more and showed that an odd perfect number must satisfy this condition, but even Euler couldn't prove whether they existed or not.

He wrote "Whether there are any odd perfect numbers is a most difficult question."

For the next 150 years very little progress was made and no new perfect numbers were discovered.

English mathematician Peter Barlow wrote that Euler eighth perfect number "Is the greatest that ever will be discovered for as they are merely curious without being useful, it is not likely that any person will ever attempt to find one beyond it."

But Barlow was wrong.

Mathematicians kept pursuing these elusive perfect numbers and most started with Mersenne's list of proposed primes.

The next on his list was 2 to the 67 minus 1.

So far, Mersenne had done an excellent job.

He had included Euler's eighth perfect number while avoiding others like 29 that turned out not to lead to a perfect number,

but 230 years after Mersenne published his list, Edouard Lucas proved that 2 to the 67 minus 1 was not prime, although he was unable to find its factors.

27 years later, Frank Nelson Cole gave a talk to the American mathematical society without saying a word,

he walked to one side of the blackboard and wrote down 2 to the 67 minus 1 equals 147,573,952,589,676,412,927.

He then walked to the other side of the blackboard and multiplied 193,707,721 times 761,838,257,287 giving the same answer.

He sat down without saying a word and the audience erupted in applause.

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.

From 500 BC until 1952 people had discovered just 12 Mersenne primes and therefore only 12 perfect numbers.

The main difficulty was checking whether large Mersenne numbers were actually prime.

But in 1952, American mathematician Raphael Robinson wrote a computer program to perform this task

and he ran it on the fastest computer at the time, the SWAC.

Within 10 months, he found the next five Mersenne primes and so corresponding perfect numbers.

And over the next 50 years, new Mersenne primes were discovered in rapid succession, all using computers.

The largest Mersenne prime at the end of 1952 was 2 to the power of 2,281 minus 1, which is 687 digits long.

By the end of 1994, the largest Mersenne prime was 2 to the power of 859,433 minus 1, which is 258,716 digits long.

Since these numbers were getting so astronomically large, the task of finding numerous end primes became more and more difficult even for supercomputers.

So in 1996, computer scientist George Woltman launched the Great Internet Mersenne Prime Search or GIMPS.

GIMPS distributes the work over many computers allowing anyone to volunteer their computer power to help search for Mersenne primes.

The project has been highly successful so far, having discovered 17 new Mersenne primes, 15 of which were the largest known primes at that time.

And the best part, if your computer discovers a new Mersenne prime, you'll be listed as its discoverer, adding yourself to a list that includes some of the best mathematicians of all time.

There's even a $250,000 prize for the first billion-digit prime.

In 2017 Church Deacon John Pace discovered the 50th Mersenne Prime by using GIMPS.

The number 2 to the 77,232,917 minus 1 is more than 23 million digits long, and it was also the largest known prime at the time.

To celebrate this achievement the Japanese publishing house, Nanairosha published this book, "The Largest Prime number of 2017."

And all it is is that number spread over 719 glorious pages.

It's wild.

The size of this font is so tiny.

The book quickly rose to the number one spot on Amazon and sold out in four days.

A year later, the 51st Mersenne Prime was discovered.

It's 2 to the 82,589,933 minus 1, and this number has 24,000,860 2048 digits.

But there's something I enjoy about the absurdity, like there is knowledge in here, 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 that like has the number, if ever we lost all the prime numbers.

You know, someone could find this book be like, here's the big one.

As of today, this is still the largest known prime.

And since numbers of this form grow so rapidly, the largest Mersenne Prime is almost always the largest known prime.

Computers have been incredibly successful at finding new Mersenne primes and their corresponding perfect numbers, but we've still only found 51 so far.

So you might suspect that there are only a finite number of them, which would mean that Nicomachus's fifth conjecture would be false, that there aren't infinitely many perfect numbers,

but that might not be the case.

The Lenstra and Pomerance Wagstaff conjecture predicts how many Mersenne primes should appear based on how large P is.

Now this is the actual data the conjecture performs remarkably well.

But more importantly, it predicts that there are infinitely many Mersenne primes and so infinitely many even perfect numbers.

The Mersenne primes are just so large and rare that they take a lot of time and computer resources to find.

But a conjecture is not a proof.

And up until this day, this problem shares the title of oldest unsolved problem in math with the other open problem. Do any odd perfect numbers exist?

The easiest way to solve this problem is by finding an example.

So maybe we could just check different odd numbers and see if one of them is perfect.

That's exactly what researchers tried in 1991.

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 to the power of 300.

21 years later, Pascal Ochem and Michael Rao raised that lower bound to 10 to the 1,500

with recent progress pushing that number up to 10 to the 2,200.

With numbers that large, it's unlikely that a computer will find one anytime soon.

So we'll need to get smart.

What would a proof look like?

Like how could we actually prove this?

I think the main idea that people have been trying to approach this problem with is coming up with more and more conditions odd perfect numbers have to satisfy, it's called this web of conditions where it has to have 10 prime factors now that we know and maybe thousands of non distinct prime factors and has to be bigger than 10 to the 3000.

And it has to do all these different things and we hope that eventually there's just so many conditions that can strain the numbers so much that they can't exist.

Since Euler, mathematicians have kept adding new conditions to this web.

But so far it hasn't worked.

But there might be another path.

When Descartes was looking for odd perfect numbers, he came across 198,585,576,189, which you can factor as 3 squared times 7 squared times 11 squared times 13 squared times 22021.

Put this into Euler sigma function and you find it is equal to two times the original number.

In other words, it is perfect.

That is if 22021 were prime, but it's not because it is equal to 19 squared times 61.

And filling that in shows that it is not perfect.

Numbers like this that are very close to being odd perfect numbers are called spoofs.

Spoofs are a larger group of numbers.

So odd perfect numbers share all properties of spoofs and then a few extra ones.

And the goal is to find properties of spoofs that ultimately prevent them from being odd perfect numbers.

For example, one condition of odd perfect numbers is that they can't be divided by 105.

So if you find that spoofs must be divisible by 105, then this would prove that odd perfect numbers can't exist.

In 2022 Pace Nielsen and a team at BYU found 21 spoof numbers including Descartes number,

and while they discovered some new properties of spoofs, they didn't find any that rule out odd, perfect numbers.

So how large would an odd perfect number have to be?

They don't exist.

You don't think odd perfect numbers exist?

No, they don't exist.

I wish they did.

That'd be really cool if if there was just this one gigantic odd, perfect number out in the universe.

They don't exist.

No.

How are you convinced that they don't exist?

There is something called a heuristic argument where it's not a proof.

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.

And you put that those pieces of information together and you think, okay, on average how many numbers should be perfect.

This argument, which was made by Carl Pomerance predicts that between 10 to the 2,200 and infinity, there are no more than 10 to the negative 540 perfect numbers of the form N equals pm squared

With odd perfect numbers the heuristic says we shouldn't expect any.

We've searched high enough now that we think we have enough evidence they shouldn't exist anymore.

My understanding is this heuristic argument. It also predicts that there are no large perfect numbers even or odd.

So...

That's true.

So there's a downside.

Yeah, there's a downside because it says there shouldn't be large, even perfect numbers and we actually expect there to be infinitely many.

And so, okay, so...

why do I believe the heuristic in this case and not this case?

You're right.

Am I being hypocritical about that?

There are other aspects you can add on 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.

Euler was right when he said whether there are any odd perfect numbers is a most difficult question.

So are there any applications of this problem?

I can say no.

Now, many people may think that if there are no applications to the real world, then there's no point studying it.

Why should anyone care about some old unsolved problem?

But I think that's the wrong approach.

For more than 2000 years, number theory had no real world applications.

It was just mathematicians following their curiosity and solving problems they found interesting proving one result after another and building a foundation of useless mathematics.

But then in the 20th century, we realized that we could take this foundation and base our cryptography on it.

This is what protects everything from text messages to government secrets.

Whenever you have a group of people put their minds towards a problem, something good's gonna come out of it.

If it's only, if it's only at the beginning, this doesn't work.

Okay, well, as Edison said, I learned 999 ways of not making a light bulb.

Eventually I got a good way to do it.

It's the same with math.

You have a problem and you throw your mind at it and others do too.

And you come up with new ideas and eventually something good comes from that process.

Einstein's general relativity was built on non-Euclidean geometries, geometries that were developed as intellectual curiosities without foresight of how they would one day change the way we understand the universe.

How many people do you think are working on the problem of perfect numbers right now?

I'd guess around 10 people currently have papers in the area, 10 to 15.

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.

Hundreds of people have thought about this problem for thousands of years.

What can I do?

You can do something.

Why should you do math if you don't know that it will lead anywhere?

Well, because doing the math is the only way to know for sure.

You can't tell in advance what the outcome will be.

Like this problem might turn out to be a dud.

We might solve it and it might not mean anything to anyone, or it could turn out to be remarkably helpful.

The only way to know for sure is to try

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