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Collatz Conjecture Explained
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0:00
- This is the most dangerous problem in mathematics, one that young mathematicians are warned not to waste their time on. It's a simple conjecture that not even the world's best mathematicians have been able to solve. Paul Erdos, a famous mathematician, said, "Mathematics is not yet ripe enough for such questions."
0:21
Here's how it works. Pick a number, any number. Seven? Good choice. Okay, we're gonna apply two rules. If the number is odd, we multiply by three and add one. So three times seven is 21, plus one is 22. If the number is even, we divide by two. So 22 divided by two is 11. Now, we keep applying these two rules. 11 is odd, so we multiply by 3, 33, and add 1, 34. Even, divide by two, 17, odd. Multiply by 3, 51, add 1, 52, even. Divide by two, 26, still even. Divide by two, 13, odd.
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So we multiply by 3, 39, add one, and that's 40, which is even, so we divide by two, 20, divide by two, 10, divide by two, five, odd. Multiply by three, 15, add one, 16, divide by two that's eight, and then four, two, and one. Now, one is odd, so we multiply by three and add one, which equals four. But four goes to two, goes to one, so we're in a loop, and the lowest number is one. Now, the conjecture is this: every positive integer, if you apply these rules,
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will eventually end up in the four, two, one loop. This is commonly called the Collatz conjecture after German mathematician, Luther Collatz, who may have come up with it in the 1930s. But the problem has many origin stories and many names. It's also known as the Ulam conjecture, Kakutani's problem, Thwaites conjecture, Hasse's algorithm, the Syracuse problem, and simply 3N+1. Why is 3x+1 so famous? - Among professional mathematicians, maybe it's not famous but infamous,
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in the sense that if someone actually admits in public that they're working on it, then there's something wrong with them. (laughs) - [Narrator] The numbers you get by applying 3x+1 are called hailstone numbers, because they go up and down like hailstones in a thundercloud, but eventually, they all fall down to one, or at least we think they do. You can think of the numbers as representing the height above the ground in meters. So a number like 26 would start 26 meters above the ground.
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And if you apply 3x+1, it rises up as high as 40 meters. And, in total, it takes 10 steps to get to one. So 10 is called its total stopping time. But take the very next number, 27, and it bounces around all over the place. In fact, it climbs all the way up to 9,232.
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As an altitude, that is higher than Mount Everest, before it too falls back to the ground. In total, it takes 111 steps for 27 to get down to one and end up in the four, two, one loop. The paths that different numbers take vary so widely, even numbers right next to each other. So how do you even start to make progress on this problem? Well, honestly, mathematicians struggled. - People just decided that this was something invented by the Soviets to slow down U.S. science,
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and it was doing a good job at it 'cause everybody's sitting there twiddling their thumbs and making no progress on this trivial thing that you can tell a school of children. - [Narrator] Jeffrey Lagarias is the world authority on 3x+1. - The first time I met him I was a senior in college, and he pulled me aside and he said, "Don't do this. Don't work on this problem. If you want to have a career, do not start spending time writing about this or publishing any papers about this.
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Do real math for a while to establish yourself." - [Narrator] Alex Kontorovich didn't listen. He and Yakov Sinai looked at the paths of the hailstone numbers. Were there any patterns? But obviously all of them ended up at one. But what about the paths they take to get there? The pattern is randomness. Here is the sequence of a large number chosen at random. The graph peaks and then drop so low that you can't really see what's happening at this scale. But if you take the logarithm,
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you find this wiggly graph with a downward trend. It looks like the stock market on a bad day. And this is no coincidence. Both are examples of geometric Brownian motion. That means if you take the log and remove the linear trend, the fluctuations are random. It's like flipping a coin each step. If the coin is heads, the line goes up, tails, it goes down. 3x+1 is just like the random wiggles of the stock market. Over long-enough periods, the stock market tends to trend upwards,
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while 3x+1 trends down. Another way to analyze 3x+1 is by looking at the leading digit of each number in a sequence. Here are the hailstone numbers starting with three as the seed. And we can count up how many numbers start with a one, how many start with a two, how many start with a three, and so on to make a histogram. We can do the same thing for the sequence that starts with four, and that's a short one, and for the sequences that start with five, six, and seven. Again, for each sequence,
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we're just counting up how many numbers start with each digit, one through nine, and adding that to our histogram. If you keep doing this for more and more numbers, eventually the histogram settles into a stable pattern. For the first billion sequences, you'll find one is by far the most common leading digit. 30% of all numbers start with one, around 17.5% start with two, 12% start with three, and the frequency decreases for higher digits. Fewer than 5% of all the numbers start with nine.
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Now, this pattern is not unique to 3x+1. It actually comes up everywhere, from the populations of countries, to the value of companies, all the physical constants and the Fibonacci numbers, just to name a few. The distribution is known as Benford's law, and it is even used to detect fraud. If all the numbers on your income tax forms obey Benford's law, then, well, you're probably being honest. If not, you may be hiding something. In elections, Benford's law can be used to spot irregularities,
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though you have to apply it correctly. Benford's law works best when the numbers involved span several orders of magnitude as they do for 3x+1. But Benford's law can't tell us whether all numbers will end up in the four, two, one loop or not. For that, we need a different sort of analysis. Now, at first glance, it seems strange that when you apply 3x+1, all numbers should end up at one. I mean, consider that there are the same number of odd and even numbers,
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but odd numbers are more than tripled, while even numbers are only cut in half. Therefore, it seems like every sequence on average should grow, not shrink. But here's the catch. Every time you multiply an odd number by three and then add one, it will always become an even number, and that means the next step is to divide by two. So odd numbers are not actually tripled by 3x+1. They're increased by a factor of about three over two. I'm neglecting the plus one
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because it's insignificant for large numbers. And 3/2 is actually the most an odd number can grow in one step. Think of the path from one odd number in a sequence to the next odd number. After multiplying by three and adding one, you have an even number. And 50% of the time, dividing by two brings you back to an odd number. But a quarter of the time, you can divide by four before you get to the next odd number. So, for a quarter of numbers, the next one in the sequence
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will be 3/4 of its initial value. An 1/8 the time, you can divide by eight before getting to the next odd number, and 1/16 of the time, you can divide by 16 and so on. If you take the geometric mean, you find, on average, to get from one odd number to the next one, you multiply by three over four, which is less than one. So statistically speaking, 3x+1 sequences are more likely to shrink than grow. Take 341 for example, multiply by three and add one, you get 1,024, which you can divide by two
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and then divide by two again, and again, and again, and again, 10 times in total until you're down to one. One way to visualize these paths of numbers in 3x+1 is simply to show how each number connects to the next one in its sequence. This is called a directed graph. It looks like a tree or a series of little streams that flow into each other. If the conjecture is true, it means that every single number is connected to this graph. Every tiny stream all the way out to infinity
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eventually flows into the massive river of four, two, one.
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Some mathematicians have modified this visualization by rotating the graph at each number; anticlockwise if it's an odd number, and clockwise if it's even. And then you end up with a structure that looks like a coral or seaweed. And by adjusting the degree of rotation for odd and even numbers, you can create these beautiful, organic-looking shapes.
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Now, there are two ways the conjecture could be false. There could be a number somewhere, some seed, that starts a sequence of numbers that grows to infinity. For whatever reason, it doesn't obey the same numerical gravity as all of the other numbers. Another possibility is there exists a sequence of numbers that forms a closed loop. All the numbers in this loop would be unconnected to the main graph. But thus far, no loop or sequence that shoots off to infinity has been found.
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And not for lack of trying, mathematicians have tested by brute force every single number up to two to the 68. That's 295,147,905,179,352,825,856 numbers.
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We know, for certain, that every single one of those numbers eventually comes back down to one. We have tested nearly 300 quintillion numbers, and none of them disproves the conjecture. In fact, given this information, mathematicians calculate that any loop other than four, two, one must be at least 186 billion numbers long. So, it seems pretty likely that the conjecture is true, but this doesn't prove it. One way mathematicians have attempted to prove it is by making a scatterplot
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with all the seed numbers on the x-axis and a number from each seeds sequence on the y-axis. Now, if you can show that in every 3x+1 sequence there is a number that is smaller than the original seed, well then you have proven the Collatz conjecture, because whatever number you pick, you know it will at some point get smaller, and that smaller number as a seed also gets smaller, and so on down to one, meaning the only way any sequence can end is in the four, two, one loop.
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This has not yet been shown. But, in 1976, Riho Terras was able to show that almost all Collatz sequences reach a point below their initial value. In 1979, this limit was reduced with almost all numbers going to less than X to the power of 0.869. And then in 1994, it was further lowered to less than X to the power of 0.7925. In this case, the term almost all numbers has a technical mathematical definition. It means that as the numbers you're looking at go to infinity,
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the fraction that end up under the curve goes to one.
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Then in 2019, one of the world's greatest living mathematicians, Terry Tao, was able to show 3x+1 obeys even stricter criteria. He showed almost all numbers will end up smaller than any arbitrary function f of x so long as that function goes to infinity as x goes to infinity. But the function can rise as slowly as you like. So, log x is an example, or log, log, log x works too, or log, log, log, log x. What this means is, for almost all numbers,
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you can guarantee that there is an arbitrarily small number somewhere in its sequence. In a public talk he gave in 2020, Terry Tao said, "This is about as close as one can get to the Collatz conjecture without actually solving it." This is an impressive result, but it's still not a proof. So why can't we prove the conjecture true? Could it be because it's not true? I mean, everyone is trying to prove it true, which means almost no one is looking for counterexamples.
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- It had happened to me just two years ago, where there was something I was trying to prove, that I was trying for three years to prove, and I couldn't get it to work. And then I found a counterexample, and then I realized what the correct statement should have been. And then a month later, I proved the correct statement. Maybe we should be spending more energy looking for counterexamples than we're currently spending. - [Narrator] I mean, remember how the number 27 grows all the way to 9,232?
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Here is a plot of seed numbers up to 10,000, with the largest number reached for each seed plotted on the y-axis. The y-axis stops at a 100,000, but not all numbers can be shown at this scale. The seed 9,663, for example, climbs as high as 27 million.
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And as yet, no one has proven why a number couldn't just shoot off to infinity. And it would take only one to disprove the conjecture, or some set of numbers could be part of a loop not connected to the main graph. As far as we know, there is only one loop: four, two, one. But something strange happens if you include negative numbers. Applying the same 3x+1 rules as before, there is not one loop, not two loops, but three independent loops of numbers. And they start at low values like -17 and -5.
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Why should there be disconnected loops on the negative side of the number line, but not on the positive side? Now, one of the most convincing pieces of evidence supporting the conjecture is Terry Tao's proof that almost all numbers have a number in their sequence that is arbitrarily small. But proving that almost all numbers obey this criteria isn't the same thing as proving that all numbers do.
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How many numbers between one and 100 are perfect squares? The answer is 10. So, 10% of numbers up to 100 are perfect squares. How many numbers between one and 1,000 are perfect squares? The answer is 31. So only 3.1% of the numbers up to 1,000 are perfect squares. And the higher you go, the smaller this percentage becomes, such that in the limit, you could say almost all numbers are not perfect squares. The fraction of numbers that are not perfect squares goes to one as X goes to infinity.
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And yet we know there are an infinite number of perfect squares and we know exactly where they are. Now, we've tested by brute force all numbers up to two to the 68, and they all obey the Collatz conjecture. And you might be thinking that, well, we should have found a counterexample by this point. But on the scale of all numbers, two to the 68 is nothing. I mean, the Polya conjecture proposed in 1919 by George Polya asserted that the majority of natural numbers
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up to any given number have an odd number of prime factors. The conjecture was eventually proven false by C. Brian Haselgrove in 1958 when he identified a counterexample. What's remarkable is the value of this counterexample was 1.845 times 10 to the 361. That's some 10 to 340 times bigger than all the numbers checked for 3x+1. One way to think about 3x+1 is as though it's a simple program run on a turing machine. The seed number is the input to the machine. So in this picture, two to the 68
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is simply an input tape 68 squares long. You can think of them as a string of ones and zeros or black and white squares. Saying the machine has transformed every input up to this 68 square taped down to one should not give you a lot of confidence that it will do so for all inputs. In fact, it's fairly simple to calculate a number that shows any arbitrary behavior you like, so long as it is finite. But if you want a number that increases by three over two
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five consecutive times, you can calculate that number. If you want a number that climbs by three over two 10 times in a row, or 100 times or 1,000 times, you can easily calculate those numbers. But after the finite section you specify, you have no more control. And every that has ever been tested, always falls to one. If there is a counterexample, it's virtually impossible that someone's gonna guess it. And the space of all possibilities is too big to search exhaustively by brute force.
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- Two to the 1,000 is not a searchable space. So, if we're gonna find it, we have to find it by some intelligent process and not by guess and check. I had been on team 3x+1 for 20 years. And then this point of view, I realized that, what do we really know?
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It's very hard to prove a theorem that's false.
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And so, could it be that everyone is struggling to prove this thing because it's not actually true? And two to the 60 is not a lot of evidence. And even the statistical version is maybe true and not evidence for the non-existence of a divergent trajectory somewhere in the 3x+1 sequence. - Of course there is another option, and that is that we'll never know, that the problem is undecidable. In 1987, John Conway created a generalization of 3x+1.
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It was a mathematical machine that he called FRACTRAN. And he was able to show this machine is turing-complete, which means it can do anything a modern computer can do, but it also means that it's subject to the halting problem, a chance that the machine never stops running and so doesn't give you an output. And this does not prove that 3x+1 is also subject to the halting problem, but it is possible that given what we know, we may never prove the Collatz conjecture true or false.
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- You're gonna be taught in school that we know a bunch of stuff, and it's a lie. They're all lies. Here's this stupid little problem. Come on, really we can't solve this? Really? You know, it just shows that math is hard. If anything, it shows that all of the things that we can solve are miracles. We have no right to have solutions to all of these other problems. - For my whole life, I've thought of numbers as these incredibly regular things full of patterns, symmetry, and repetition.
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But what I'm realizing only now is just how peculiar numbers really are. You can see this most clearly in the coral representation. From a simple mathematical operation comes something intricate, organic-looking, and, thus far, intractable to us. Do all numbers connect to the structure, or is there some unique filament, a spindly little thread, that doesn't connect to any of this, that runs off to infinity? And why is it so hard to tell? I think that's why Paul Erdos said,
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"Mathematics is not yet ripe enough for such questions." (gentle music)
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What I love about 3x+1 is it's a problem almost anyone can understand and play around with. And actually trying to figure things out for yourself is the best way to learn, which is why I subscribed to Brilliant, the sponsor of this video. Now, recently, Brilliant have upped their interactivity. For example, here is a great lesson on the Pythagorean theorem. So, you don't just remember the formula, but you really understand what it means. Now, Brilliant is a website and an app
21:10
designed to get you thinking deeply by engaging you in problem-solving. It's one thing to read through a textbook and think that you get it, but it's quite another to play with interactives and actually test yourself as you go, and Brilliant curates the experiences so they get more and more challenging over time. There's always a helpful tip or explanation that takes your understanding to the next level. I'd highly recommend their course on mathematical fundamentals,
21:36
which now has even more interactivity and it has topics that are relevant to all areas of STEM and algorithm fundamentals for anyone interested in coding. It's great to have a resource like this to inspire you to learn something new every single day. I try to start off my day by challenging myself with Brilliant. And so if you'd like to join me and a community of 8 million other learners, go to brilliant.org/veritasium. I will put that link down in the description.
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So I wanna thank Brilliant for sponsoring this video, and I wanna thank you for watching.
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