The Oldest Unsolved Problem in Math

had predicted. Now, Euler refined the form a little further and showed that an odd perfect number must satisfy this condition, but even Euler could not prove whether they existed or not. He wrote: "The most difficult question is whether odd perfect numbers exist." Over the next 150 years very little progress was made and no new perfect numbers were discovered. The English

math

ematician Peter Barlow wrote that Euler's eighth perfect number "is the largest that has ever been discovered because, as they are merely curious without being useful, it is not likely that any person will attempt to find one beyond it." But Barlow was wrong.

Mathematicians continued to search for these elusive perfect numbers, and most began with the list of primes proposed by Mersenne. Next on his list was 2 to 67 minus 1. So far, Mersenne had done an excellent job. He had included Euler's eighth perfect number and had avoided others such as 29 which turned out not to lead to a perfect number, but 230 years after Mersenne published his list, Edouard Lucas showed that 2 to the power of 67 minus 1 was not prime. could find its factors. 27 years later, Frank Nelson Cole gave a talk to the American Mathematical Society without saying a word, walked to the side of the blackboard and wrote 2 to the power of 67 minus 1 equals 147,573,952,589,676,412,927.

He then walked to the other side of the board and multiplied 193,707,721 by 761,838,257,287 giving the same answer. He sat down without saying a word and the audience burst into applause. He later admitted that it took him three years of working Sundays to figure this out. A modern computer could solve this in less than a second. Since the year 500 B.C. Until 1952, only 12 Mersenne prime numbers had been discovered and, therefore, only 12 perfect numbers. The main difficulty was to check whether the large Mersenne numbers were really prime. But in 1952, the American

math

ematician Raphael Robinson wrote a computer program to perform this task and ran it on the fastest computer of the time, the SWAC.

In 10 months, he found the next five Mersenne primes and, therefore, the 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 2281 minus 1, which has 687 digits. At the end of 1994, the largest Mersenne prime number was 2 to the power of 859,433 minus 1, which has 258,716 digits. Since these numbers were becoming so astronomically large, the task of finding numerous final primes became increasingly difficult even for supercomputers. Then, in 1996, computer scientist George Woltman launched the Great Internet Mersenne Prime Search or GIMPS.

GIMPS spreads the work across many computers, allowing anyone to offer their computing 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 the time. And the best part is that if your computer discovers a new Mersenne prime, you'll be listed as its discoverer, adding you to a list that includes some of the greatest mathematicians of all time. There's even a $250,000 prize for the first billion-digit cousin. In 2017, church deacon John Pace discovered Mersenne Prime number 50 using GIMPS. The number 2 to the power of 77,232,917 minus 1 has more than 23 million digits and was also the largest known prime at the time.

To celebrate this achievement, Japanese publisher Nanairosha published this book, "The Largest Prime Number of 2017." And all it is is that issue spread across 719 glorious pages. It's wild. The size of this font is very small. The book quickly reached the number one spot on Amazon and sold out in four days. A year later, the 51st Mersenne Prime was discovered. It is 2 to the power of 82,589,933 minus 1, and this number has 24,000,860 2048 digits. But there's something I enjoy about the absurd, like there's knowledge here, but it's not the kind of knowledge that anyone will ever read in a book. But in some ways it's a good thing that there is this physical artifact that has the number, if we ever lost all the prime numbers.

You know, someone might find this book and say, here's the most important one. To this day, this remains the largest known prime number. And since numbers in this form grow so quickly, 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 so far we've only found 51. So one might suspect that there are only a finite number of them, which would mean that the Fifth Nicomachean Conjecture would be false, that there are no infinitely many perfect numbers, but that might not be the case.

The Lenstra and Pomerance Wagstaff conjecture predicts how many Mersenne primes should appear as a function of the size of P. This is the real data, the conjecture works remarkably well. But most importantly, it predicts that there are infinitely many Mersenne primes and, therefore, infinitely many perfect even numbers. Mersenne primes are so large and rare that they require a lot of time and computing resources to find them. But a guess is not proof. And to this day, this

problem

shares the title of

oldest

unsolved

problem

in mathematics with the other open problem. Are there odd perfect numbers?

The easiest way to solve this problem is to find an example. Then maybe we could check different odd numbers and see if one of them is perfect. That's exactly what researchers attempted in 1991. Using a clever algorithm called chain factors, they were able to show that if an odd perfect number exists, it must be greater than 10 to the power of 300. 21 years later, Pascal Ochem and Michael Rao raised that lower limit to 10 per 1,500 and recent progress raised that number to 10 per 2,200. With numbers this large, it's unlikely that a computer will find one anytime soon. So we'll have to be smart.

What would a test be like? How could we actually test this? - I think the main idea that people have been trying to approach this problem with is to find more and more conditions that the odd perfect numbers must satisfy, it's called this network of conditions where you have to have 10 prime factors now that we know and such There are thousands of non-distinct prime factors and it has to be greater than 10 to the power of 3000. And it has to do all these different things and we hope that eventually there will be so many conditions that can strain the numbers so much that they can't exist. . - Since Euler, mathematicians have been adding new conditions to this network. - But so far it hasn't worked. - But there could be another way.

When Descartes was looking for odd perfect numbers, he came across 198,585,576,189, which can be factored as 3 squared times 7 squared times 11 squared times 13 squared times 22021. Put this into Euler's sigma function and you will find that it is equal twice the original. number. In other words, it's perfect. That's if 22021 were prime, but it's not because it's equal to 19 squared times 61. And filling that in shows it's not perfect. Numbers like this that are very close to being odd perfect numbers are called parodies. Parodies are a larger group of numbers. So odd perfect numbers share all the properties of parodies and then some additional ones.

And the goal is to find properties of the parodies that ultimately prevent them from being odd perfect numbers. For example, a condition of odd perfect numbers is that they cannot be divided by 105. So if you discovered that fakes must be divisible by 105, this would prove that odd perfect numbers cannot exist. In 2022, Pace Nielsen and a team at BYU found 21 fake numbers, including Descartes' number, and while they discovered some new properties of the fakes, they didn't find any that ruled out odd and perfect numbers. So how big would an odd perfect number have to be? - They do not exist. - Don't you believe that odd perfect numbers exist? - No, they don't exist.

I wish they did. It would be really cool if this gigantic, strange, perfect number existed in the universe. They do not exist. No. - How are you convinced that they don't exist? - There is something called a heuristic argument where it is not a proof. So if we had proof, we'd be done for. It's just an argument of, okay, we think prime numbers like this occur so frequently. And you put that data 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 power of 2200 and infinity, there are no more than 10 to the power of 540 perfect numbers of the form N equals pm squared.

With odd perfect numbers, the heuristic says we shouldn't expect anything. We've searched high enough now that we think we have enough evidence that they should no longer exist. - What I understand is this heuristic argument. It also predicts that there are no large perfect numbers, even or odd. So... That's true. So there is a disadvantage. Yes, there is a disadvantage because it says that there should not be large numbers, not even perfect ones, and we actually expect there to be infinities. And so, okay, so...why do I believe in heuristics in this case and not in this case?

You're right. Am I being hypocritical about that? There are other aspects that you can add to the heuristic and strengthen it. Let me put it that way. But you're right, it's not a test. - For now, this remains the

oldest

unsolved

problem in mathematics. Euler was right when he said that whether odd perfect numbers exist is a very difficult question. So is there any app for this problem? - I can say no. - Now, many people may think that if there are no applications to the real world, then there is no point in studying it. Why should anyone care about some old unresolved problem?

But I think that's the wrong approach. For more than 2,000 years, number theory had no real-world applications. They were simply mathematicians who followed their curiosity and solved 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 a group of people focus on a problem, something good will come out of it. If it's just, if it's just at the beginning, this doesn't work.

Okay, well, like Edison said, I learned 999 ways not to make a light bulb. I finally found a good way to do it. The same goes for mathematics. You have a problem and you focus on it and others do too. And you come up with new ideas and eventually something good comes out of that process. - Einstein's general relativity was built on non-Euclidean geometries, geometries that developed as intellectual curiosities with no anticipation of how they would one day change the way we understand the universe. How many people do you think are working on the perfect numbers problem right now? - I guess currently about 10 people have jobs in the area, 10 to 15.

If you are a high school student and you love math and you think, I want a problem to think about, this is a great problem to think about . about. And you can progress. You can discover new things. Yes, don't be afraid. Hundreds of people have thought about this problem for thousands of years. What I can do? You can do something. - Why should you do math if you don't know it will get you anywhere? Well, because doing the math is the only way to know for sure. You cannot know in advance what the result will be.

As if this problem could turn out to be a failure. We could figure it out and it could mean nothing to anyone, or it could turn out to be very useful. The only way to know for sure is to try. In today's world, it often seems like you have to choose between following your curiosity and developing real skills you can apply. But the truth is that it is essential to do both. Fortunately, there is a learning platform that allows you to do just that. And it is the sponsor of this video, Brilliant. Brilliant will make you a better thinker and problem solver by helping you develop real-world skills and everything from math and data science to programming technology.

You name it, Brilliant has thousands of lessons you can explore so you can follow your curiosity wherever it takes you, and you'll not only learn key concepts. In fact, you'll apply that knowledge to real-world situations, giving you real practical insight. With Brilliant you can learn by trying things yourself. That's what makes learning brilliantly so powerful. And along the way, you can set goals, track your progress, and level up with guided learning paths that let you delve deeper into specific topics. I think that learning something new every day is one of the best gifts you can give yourself.

And Brilliant is the perfect way to do it with bite-sized lessons you can take right from your phone.So even if you only have a few minutes, you can feed your curiosity, sharpen your mind, and develop new skills. To try everything Brilliant has to offer for free, for a full 30 days, visit shiny.org/veritasium, scan this QR code or click the link in the description and the first 200 of you will get 20% off in Brilliant's annual premium subscription. So I want to thank Brilliant for sponsoring this video and I want to thank you for watching it.