Science & Technology Q&A for Kids (and others) [Part 137]

Okay. Hello, everyone. Welcome to another episode of Science and Technology, Q and A for kids and others. And I see all kinds of questions stored up here. So let's see. Let's start off with the first one from Stefan. Can you tell us some not widely known insights about prime numbers? This is a case where having being able to actually use Mathematica Wolfram language and show you stuff would help a lot. But I'll say a few things. I'm not going to do the share screen thing here today. All right. What is a prime number? Well, You could think about it in the following kind of way. Imagine that you were making a number, let's say 12, and you were making a rectangle whose area was 12. And the sides of the rectangle should be some integer, some whole number. You're making bunch of putting together 12 squares to make a rectangle. Can you do that? Well, yes, you can do it. You can have something where there are three squares on one side and four squares on the other side. Three times four is equal to 12. You make that rectangle. Question is, can you make a rectangle like that for any number? Let's think about the number seven, for example. Well, think about six, that's two times three. That's all good. Think about eight, that's two times four, or four times two or whatever. Think about, well, if we take 12, for example, that's either two times six or three times four or whatever. We can make our rectangle. Try and do it for seven. You can't do it. And that's seven is a number that can't be made by making something times something equals seven. The only thing you can do that with is one and seven. One times seven is seven. But there's no two times anything, three times anything. That makes seven. Okay, so that's for those who don't know it. That's what prime numbers are. And some numbers are prime. 2 is prime, 3 is prime, 5 is prime, 7 is prime, 11 is prime, 13 is prime, et cetera. And so first question is, are there an infinite number of primes? If one wants numbers that have this property that they can't be broken down as multiplying together other numbers, will you always have numbers? Will there always be numbers which are prime? Answer yes, that was something that was known to Euclid, actually in antiquity, a couple of thousand years ago, There are an infinite number of primes, and there's probably some nice ways. I wonder if I could verbally prove that there are an infinite number of primes. Let me Think about that. Let me not try doing that right now. I was thinking of Euclid's proof, which is a little bit tricky to explain, but there's an easier proof that has to do with kind of thinking about the information content of numbers. But let's not worry about it. First statement is there are an infinite number of prime numbers. They keep going forever. Okay, next question. There are some. When you look at the prime numbers, there's like two and three. Those are only one apartment that's never going to repeat. The maximum closeness of primes further on is two apart. So 11 and 13. What's the next? What's the next, what's called twin prime primes, which are just one apart, but two apart, so to speak. Like 11 and 13, 5 and 7, another twin prime pair. Okay, so a question that you could ask is, okay, there are infinite number of primes. What about twin primes? Are there an infinite number of twin primes? That's a question people asked a couple of thousand years ago and nobody knows the answer. People have tried to compute twin primes and yes, you can keep going. There are very big twin primes. But are there an infinite number of twin primes? The answer to that is not known. And it's one of these problems, it's one of the older unsolved problems in mathematics is are there an infinite number of twin primes? So we think about the prime numbers and there are all kinds of obvious questions we might ask like what fraction of numbers are prime? And the answer is that as you get to bigger numbers, the fraction that are prime goes down. There are more ways that the number could be written as a product multiplying together other numbers. So the density of primes goes down. And roughly as you get to size N, prime numbers of size N, the density goes down like the logarithm of N. So it's kind of like if you look at. Your, it sort of slowly goes down, it goes down. The rate at which it goes down is sort of proportional to the number of digits in the number, the number itself. Like when you get, you know, between 100 and 1,000, you've got 900, you've got 900 numbers, but you only are going up by one digit, adding on that, that extra extra digit to represent the thousands column and so on. And the density of prime numbers kind of goes down slowly just by sort of looking at the number of digits and numbers, not the actual sizes of numbers. So one of the things that has been a long running thing that people have tried to do is to prove what's called the prime number theorem, which is the estimate of what the density of primes actually is. And turns out that it roughly goes down like this logarithm, then a little bit more accurately like a log integral function, and then a little bit more accurately than that, it goes down with this formula that depends on this thing called the Riemann zeta function. And the Riemann zeta function, something invented in the 18, I think 1859, if I remember correctly, is a mathematical function, and it has a formula, and there's a pretty straightforward formula for it. Zeta of S is the sum of 1 over n to the S for n going from 1 to infinity. So, for example, zeta of 2 is 1 over 1 squared plus 1 over 2 squared plus 1 over 3 squared plus 1 over 4 squared, and so on up to infinity. That particular case, zeta of 2 happens to be equal to PI squared over 6. Zeta of 3, you can't express it in terms of PI's. Zeta of 4 happens to be PI to the 4th over 90, et cetera, et cetera, et cetera. Okay? So what's related to prime numbers is not Zeta of 2 or 3 or 4, but Zeta of a complex number. So you can take that kind of the function which can be defined by that sum, and you can continue that function outside of just ordinary positive numbers and so on, and you can continue it even to complex numbers, numbers which include things like the square root of minus one and so on. And when you look at that, you're asking that zeta function thing has zeros. There are values of s and zeta of S. There are values of S for which zeta of S is equal to zero. And the prime number theorem, the thing that gives you this estimate of the density of primes has a particular form if and only if the zeros of the zeta function lie on this thing called the critical line. Critical line is a line in the complex plane. It's S equals a half plus I times some quantity t. And I'm getting a bit technical here, but basically, when you make a plot of the zeta function, you'll see it wiggles around like crazy. And the question is, when does it hit zero? And the answer is that, so far as we can tell, for the first, let's see, about 100 billion, I think it's known zeros of the zeta function all lie on this critical line. So they all have this feature that they are in the place that they need to be. So that this prime number theorem that gives you the estimate of the density of primes is correct, but we don't know for sure that the zeros, zeta function all lie on that line. That's a thing called the Riemann Hypothesis. That's been an open problem for more than 150 years now of showing that that's true or not. And the Riemann Hypothesis is kind of one of the big sort of exciting unsolved problems of mathematics. It's an interesting problem because it kind of relates questions about numbers to questions about mathematical functions, to questions about actually things including geometry and other areas. It's kind of a meeting point for a lot of different kinds of areas of mathematics. There's a question even of whether, given the axioms, the assumptions we make about what's true in mathematics, things like that, if you have two numbers X and Y, that X plus Y is the same as y plus X, that's an axiom, that's an assumption assumption about how addition works for any numbers X and Y. Well, there are a collection, it's a fairly small number of standard axioms that one uses to sort of build up mathematics as we know it today. And there's a question of whether those axioms are sufficient to answer the question of whether the Riemann Hypothesis is true. Could it be the case that the Riemann Hypothesis is somehow not provable from those axioms? And that seems like it's a serious possibility that it could be something which, knowing whether it's true is something where you kind of have to do all this sort of infinite computation, and you can't do that. And you can't sort of ever reach the point of saying, yes, we know it's true just from these axioms. And in fact, it turns out that the Riemann Hypothesis is. You can write it as something which doesn't involve zeta functions, things like that. It's just a question about something where you're doing a sort of iteration of integers. So there's a very famous problem called the 3N/1 problem where you're saying, take a number N, let's say it's 100, and if n is even, divide it by 2, so you get 50. In that case, if n is odd, compute 3n plus 1. And so that would be in this case, if it was odd, it would be. Let's say it was. Oh, boy, this is going to tax my mathematical abilities. This is why I build, this is why I've spent my life building systems that do things automatically, is because I'm pretty bad at doing this myself. But let's say it was 99. Okay, that's not an even number. Then we compute 3N plus 1. I think I can manage that. That's 300 minus 3. So 297 plus 1. 298. So 99 would go to 298 and so on. You just iterate keep and so 298. Then you just divide that by 2 and you keep iterating this. So the 3M plus 1 problem, the big question is, does it always end in. Does it always get down to. Well, it ends up with a little cycle of numbers 1 and 4 and so on at the end. Does it always get down to that? Or could it be the case that you're computing 3N +1 and somehow you end up getting bigger and bigger and bigger numbers because you keep on hitting the 3N plus one thing rather than the N over two thing and it sort of escapes to infinity. Nobody knows the answer to that. Turns out that there is a version of the Riemann Hypothesis that looks a very lot like that. It's more complicated than just 3N +1. It's a formula. I actually sort of tried to compress the formula for this a couple of years ago. And you can get it down to something that's I think about one line of Wolfram language code of just sort of simple mathematical functions on whole numbers. And then the Riemann hypothesis becomes this question of if you do this iteration, will it always terminate? So those are a few things. Now you ask about sort of other things about primes. Another question about primes that's surprisingly unanswered is what the I mentioned twin primes, or they're primes that are just two apart. There's a question of how far apart typically are primes. And that turns out to be something that's been investigated empirically. There's not a lot known about that. If you look at the early primes, the most common distance apart to find primes is 6. If you keep looking at bigger and bigger, bigger primes, you'll find that the distance apart, the average distance apart, goes up a bit. But that distribution, there's a definite kind of distribution of how far apart typically are primes. And it's not well understood how that works. So it's sort of a remarkable thing because primes are pretty easy to define, yet there are already kind of questions about them that are surprisingly hard to answer and may not even be answerable from the kind of assumptions we've so far made in math. Okay, here's one more thing about primes. How Easy, is it, to figure out if a number is prime? Well, when it comes to factoring numbers, taking some number, I don't know, 102, and saying, what are the prime factors of 102, how do you break 102 into. Well, let's take one. I can actually do in my head, 33. Okay, 33 is equal to three times 11. Okay. How do you know? How do you take a number and figure out what its factors are? So one way you can do it is if you guess the factors and you guessed one of them right, then the number's going to start unraveling because you can just divide by that factor. You can keep going and you get the factors. But knowing whether a number is sort of guessing the factors of a number, if you guess them right, you can sort of unravel it. But systematically going from the number to its factors seems to be pretty hard. And the type of problem where you can sort of, once you guess the answer, it's easy to solve the problem. These are so called NP problems. It stands for non deterministic polynomial time. What does that mean? That means, as you look at, so you take a problem like, I don't know, multiplying numbers together, okay, you have an end digit number and you ask the question, how long does it take to multiply one end digit number by another N digit number? The way they teach you in school, for doing long multiplication, it takes about N times N steps, N squared steps. Turns out there are methods for multiplying numbers that do it in N log log N type steps very, very close to N steps. Those are the things we use in Wolfram language, for example, to do large number multiplication. It's very not like you do it in the sort of schoolbook multiplication method. It's a very different kind of method that just for sort of the fancy statement is it uses Fourier transforms to do these computations. That's at least an approach to doing it. But any case, and by the way, the reason that sort of schoolbook multiplication is it doesn't need to take that long is that in sort of the schoolbook multiplication method, you're multiplying every digit by every other digit. But it turns out as you do that, you're kind of doing many digits, you're doing many multiplications more times than you need to. There's ways to sort of compress the number of multiplications, and that's how those kind of methods work. But so multiplying numbers together is an operation which, as you increase the length of the numbers. At best, it goes up like N squared, where N is the length of the numbers. It's a polynomial in N. And so that. So if you are given the factors of a number, multiplying them all together to check that, yes, indeed, those factors multiply together to make the number. That's something that you can do with computational effort that scales just with roughly with the length of the number of digits in the number. Okay, but let's take the other question of taking a number and figuring out what its factors are. Well, one way you can figure that out is just say, let me guess that it might have two as a factor. Let me guess it might have three as a factor, four as a factor, five as a factor, and so on. And just keep on trying to divide the number by successive integers. Okay? That's a way you can guarantee to find out what the factors of the number are. The only problem is if your number is if you have some number. It has 100 digits. For example, the amount of effort it takes is about the size of the number. So it's 10 to the 100. Let's say you have a number that's in the millions, you have to sort of try dividing by every number less than that number to see whether anything divides. It's very different from multiplying the number where all you have to look at is the number of digits, which is six, as opposed to the size of the number, which is a million. And so these problems that have this feature that if you can guess the answer well, in this particular case, the only sort of. The very systematic way to work out what are the factors of number is you do trying. You try to divide it by numbers up to the actual size of that number. And that will take. If the number has n digits, that could take like 10 to the power n steps, because there are 10 to the n numbers that are up to the size of the number, so to speak. So, for example, for the millions, a million is 10 to the power 6. So there are 6 digits for 1 million 6 zeros, but the actual number is a million. Okay? So this question of factoring, it's a problem where if you could just say, I know the answer, it's this. You could check it easily. But to systematically find the answer, you have to go through lots and lots of steps to find that answer. And one of the big questions is this P versus NP question. Are all problems where if you guess the answer, you can check it quickly. Is it possible that every time you have a problem where you could Guess the answer and check it quickly, that in fact there is an algorithm for solving that problem that can systematically do it quickly. It is strongly believed that there are plenty of problems which have the property that just because you can check them quickly doesn't mean you can compute them quickly. And it's believed that factoring is an example of that. It's believed that to work out the factors of a number in general is in some sense is hard. I was going to say exponentially hard, but it's known not to be quite exponentially hard. Exponentially, if you've got a number of length N, does It take like 2 to the N or 10 to the N or something steps to do it? It's not quite as bad as that, but it's not something which is a polynomial in N, so far as one could tell. So in other words, factoring seems to be a hard problem. To take a number and grind it down to its factors seems to be a hard problem. It is not. There are a class of problems that are in this class, NP problems where the answer is easy to check, but it's not obvious how you systematically get to that answer. There are a class of so called NP complete problems which have the feature that if you could only solve one of them, you can translate it into all the other problems. There are problems about graphs, there are problems about numbers, et cetera, et cetera, et cetera. If you could solve just one NP complete problem, there's a translation to all other NP complete problems. And so there's kind of a strength in numbers in the sense that people have tried picking away at different ones of these NP complete problems. Nobody's ever managed to find a fast way to solve any of them. And so it's believed that they're all hard to solve. Factoring isn't known to be NP complete. It's what's called NP hard. So it doesn't get the benefit of kind of that herd of NP complete problems, but it still probably has this property that it can't be solved quickly. Okay. Surprising fact is that testing whether a number is prime can be solved quickly. So taking a number and breaking it down into primes seems to be hard to do. But if you give me a number, testing whether that number is prime turns out to be something you can do rather quickly. And actually the thing that allows you to do that, it starts from a thing called Fermat's little theorem. People may have heard of Fermat's last theorem, which is this result in number theory that says if you have X, Y and Z are three integers that, for example, we know that X squared plus Y squared equals Z squared, there are values of X, Y and z that make that true. For example, 3, 4, 5, 3 squared plus 4 squared equals 5 squared, 9 plus 16 equals 25. Similarly, 5, 12, 13 is another one of these so called Pythagorean triples. So they were known to Pythagoras. That solves the problem. X squared plus Y squared equals Z squared for whole numbers X, Y and Z. Okay, so then the question is, what if you, instead of doing squared, you say it's X cubed plus Y cubed equals Z cubed. There are no known solutions to that for positive numbers, positive whole numbers X, Y and Z. In fact, in that particular case, it's been known for a long time, you can prove that it's not possible to find numbers X, Y and Z that solve that. So Fermat's last theorem is this question of whether X to the n plus Y to the n equals Z to the N for n greater than 2, whether there are solutions to that. And that was something that was, was not. It was for many, many years that was a big sort of unsolved problem, is that people had looked for solutions and never found them. Finally, in the early 1990s, it was proved with a surprisingly elaborate proof that, no, there really aren't any possible solutions to that. That's Fermat's last theorem. Fermat, Monsieur Pierre de fermat from the 1600s, had had a thing called his little theorem, which is a theorem that says that, well, a power of a number mod another number has to have a certain result. And that formula says it has to have a certain result. When the thing that appears in it is a prime number. It says if the thing is a prime number, then a number, some integer, pick an integer, raise it to a power, and that power is a prime. And then take that mod, I think I'm remembering this correctly, take that mod another number. Then if that number that appears there is a prime, the result has to be one. If it isn't a prime, then. But it still could be the case that some number other than a prime could give result one. But if it's a prime, it always has to give a result one. So if the result was two, then you know, that number isn't a prime. So the idea is you can probabilistically test for primality by saying, let's just pick different numbers for that base there. And try taking the number that we have, that's our candidate prime. And raise that base to the power of that prime and compute this mod thing and see what the result is. And if we say for any particular base, we say the number's one. Okay, that's evidence that this thing that we fed in could be a prime. If the number comes out to be two, the game is over. We know our number isn't prime, but. So you can try feeding in a bunch of different numbers and see a bunch of different bases and see whether for each of those, the thing comes out with one and says, okay, it could be a prime. So people had known for a long time. It's sort of an interesting case. Well, actually had known since the 1970s, I think that there was a sort of probabilistic test for primality where you can just try different numbers against your candidate prime and say, could this be a prime? Could this be a prime? And if it's yes, yes, yes, yes, yes, this could be a prime. If there are enough times you say, yes, yes, yes, it could be a prime, then you might get more and more convinced, yes, it really is a prime. Okay, sometime in. When was this? Probably 15 years ago. Now, finally, some people showed that if you do that enough times, more or less, and you do a few extra things as well, then you can nail down and say, yes, this number I have is a prime. And the effort to do that is vastly less than the effort to try and factor a number. So primality testing, testing whether a number is prime is something to be done quite quickly. So there are actually the. For example, you can ask the question. Another type of question you can ask is, what's the millionth prime? What's the 10 millionth prime? What's the 100 millionth prime? You might have thought that if you say, what's the billionth prime? Or whatever, that that would be a lot of effort to work out because you have to. You know, you might think you'd have to go through all the primes up to the billionth prime and see which one is going to be that billionth prime of the billionth and first prime and so on. Turns out that's not necessary. It turns out there's a trick that relates to the Riemann zeta function. Remember I mentioned that the Riemann zeta function is related to the density of primes. Well, if you can get an accurate enough estimate of the density of primes, you can work out up to this size. How many primes will there be up to this size? If you estimated the density how many primes there are per number, so to speak. You know that well enough. Then eventually you can kind of nail down how many primes must there be up to this size. And so it turns out it actually can get easier to work out the, you know what? Well, almost easier at least to work out what the 10 billionth prime is than some smaller prime using that method of doing estimates. Okay, my last, I'm doing sort of free association about things to do with primes. Another big question, it's been a long running question about primes is, is there a formula for primes? That is, is there a mathematical formula that you can write down where you know that every number that you can get out of this formula is a prime? So, for example, Fermat himself had a candidate for that. Well, there's a chap called Mersenne who is kind of a contemporary of Fermat's who has a more famous version of this, which is a number of the form 2 to the N minus 1. I think. Am I remembering this right? I think the fermat version is 2 to the 2 to the N plus 1 minus 1. I'm not sure this is the kind of thing. Wolfram language knows that immediately. I, my brain doesn't know that immediately. But in any case, they're numbers of the form, roughly things like 2 to the N minus 1, 2 to the N is obviously not a prime because it's just divisible by 2. But 2 to the n minus 1 might be a prime, and sometimes it is a prime and sometimes it's not a prime. And people had wondered, was it always going to be a prime? And people had wondered, could there be a formula, is a finite formula we can write down that is just like every has some parameter, let's say n in it, and we just compute this whole thing with, you know, two to the n minus three to the whatever plus whatever plus whatever, that that formula will always give primes. So far as it's known, there is no such formula. I don't think there's any proof that no such formula can exist, although I think, I think it's. Let me think. Yeah, no, I don't think, I don't think one knows. I don't think one has really a way. Well, let me think. I mean, if you say it can only contain arithmetic operations, I suspect that is provable, but I'm not sure that that's actually proved. You can certainly make up equations like that. When I was mentioning X squared plus Y squared equals Z squared, that are equations involving integers. You can make an equation involving integers where the only solution to that equation is primes. And you don't have to have any fancy things about mods and anything like that. You can just have an arithmetic formula that is a complicated polynomial formula equals zero, let's say. And the only numbers that will solve that are primes. That was something that was established in the 1960s. There are also other weird things. For example, if you have even just polynomials and you allow numbers that aren't necessarily just positive numbers, you allow positive and negative numbers, and you ask for what numbers is this polynomial you have positive. There are cases there that at least I've found where at least seems to be the case that you get only that the only numbers that satisfy that are prime numbers. And I'm not sure I know why. All right, a few comments about prime numbers. Let's see, It seems to be some questions about. About prime numbers. Let me just. JJH asks, Are the. Is the distance between twin primes now quite well known? Yeah, you can certainly compute, you know, with Wolfram language, I'm sure we can compute up to I don't know how many digit primes, 30, 40 digits? I don't know, maybe bigger than that. And we can certainly look at experimentally, what is the distance between twin primes. And yes, the distance between prime numbers, like the distance between twin primes, has some distribution which gradually gets broader, gradually further and further between numbers between prime numbers. And that's reflected in this prime number theorem, which says that the density of primes gradually goes down. And there are similar, much more elaborate versions of that. Involve twin primes. Let's see. Okay, Seb asks, what does the kind of factoring that you learn in an algebra 2 class actually have to do with real life? All right, let's talk about that. So let me give some examples about, well, prime numbers, where they show up, so on. Here's an example. So let's see how to make this work. Well, it has to do with Saturn. Saturn has these rings you see through a telescope. They look really cool. It's kind of like the sun has a bunch of planets going around which all lie in one plane. When the solar system was formed, there was an accretion disk of just stuff that was all sort of spinning around the central mass, which was the sun. And gradually that accretion disk formed into it, sort of formed into lanes, and those kind of eventually condensed into planets. Okay, so planets have moons going around them that are kind of like little tiny solar systems where There are usually in a particular plane that's usually aligned with the plane of the orbit of the planets. There are moons going around. And of course, then you can ask, are there moon moons where there are moons going around? Moons going around planets, and the planets are then going around the sun. And moon moons are still a controversial kind of issue because their orbits don't tend to be stable. But any case, the. So around Saturn, there are not only are there actual moons, a whole bunch of them, but they're also just random little pieces of ice and rock and so on, which are all like an accretion disk. They're all forming this disk in the plane of the planet. Okay. But if you look at Saturn's rings, they have little gaps in them. And the question is, what are those gaps? And this is a story about numbers. And let's see. Roughly the point is this, that, Well, okay, you have a moon going around a planet and it is exerting a gravitational force on other things. If you have. How am I going to explain this? Essentially, what happens is if you have two things going around the planet and one of them goes around, takes two days to go around, and the other takes three days to go around, then most of the time they won't sort of be in the same place around the planet at the same time. But every so often, if one takes two days and one takes three days every so often, sort of the. Let's say in that example, let me think. I'm thinking it's six days, is that right? Yes, yes. After six days, one of the planets will have gone around three times, the other planet will have gone around two times, and they'll both wind up in the same place. So that leads to what's called a resonance, where you kind of get these two things that are sort of going around and they end up in the same place. And how do you avoid that? How do you make it be the case that they don't end up in the same place? Well, those numbers need to be relatively prime. It needs to be the case that there's no multiple of one of the numbers that's equal to the other number. So, for example, if we have. Let me see if. Well, it's. Again, I'm sorry, this is a. You're asking me things with numbers. You're asking you math things which are very easy to compute with our fine technology, but not so easy to just talk about. But let me just sort of. So one example, you're asking where do things like primes and so on show Up One example is in things like the gaps in the rings of Saturn having to do with these particular kind of numerical coincidences of where things line up with periods, of how long it takes moons to go around. That's one example. Let me pick a different example. Well, there's a kind of bizarrely related example also about. Let me think. Yeah, I mean, this is. So I think I shouldn't go in this direction because it's so much easier to do. Maybe sometime I'll ask me again about these things when I'm actually doing something where we're actually typing on a computer, because I think it's going to get difficult to describe these things. But let me go in a different direction with prime numbers and factoring. The big application area for factoring that people talk about a lot is public key cryptography. So let me try to explain that. So what is cryptography? Cryptography is take your message like, you know, sort of the invasion is going to be by sea or something. Take that message and you want to grind it up into some form that looks random to somebody who just finds the message lying on the. On the sidewalk somewhere, but where the correct recipient of that message has a key that allows them to decode the message and find out the invasion will be by C or something. And so the earliest kinds of forms of cryptography that people know about, there's a thing called the Caesar cipher that was probably used by Julius Caesar. And what was done there is you take a piece of text like. Like the. Well, C, sea. Okay, you take those letters. Obviously, for Caesar, it will be mare, because it'd be in Latin. But the. So, but let's take it in English sea. And our code is just going to be, let's add two to the letter. So in the Alphabet we go S, T, U. So we'd say add two. We'd get U. The E would be G, and then the A would be C. So we'd get ugc. And so you pick up this message and it says ugc. And you don't know what that means unless you know the number two, in which case you just say, well, I'm going to subtract two from each of those letters and then I get sea. And that was the kind of cipher that it said Julius Caesar used. And I think it was a time when not that many people could read to begin with, let alone could kind of figure out, oh, you can subtract from these letters and so on. Very simple cipher. The key is the offset, and that's sort of how you need to know that key in order to be able to do the decoding. Okay. Years later, people started using so called running key ciphers, where instead of just adding two to each letter, so to speak, going two steps in the Alphabet, you would go first two steps, then four steps, then six steps, then eight steps and so on. You would increase the number of steps, you would run the number of steps up. And that was a type of cipher that Even in the 1600s, people had a hard time breaking. People didn't know kind of how to think through sort of how you would take that message and figure out what the key was and break it. Okay. So then we get to much later in, for example, the time of the Second World War and so on. And one had sort of elaborated versions of running key ciphers that involved rotor wheels where you would have all the letters of the Alphabet on some wheel and you would click it up and you would. And these wheels would have letters in different order. And you would have this way of kind of having an electrical kind of plug board where you would kind of randomly permute things and things. That made, for example, this thing called the Enigma machine, which was used by Germany in World War II and was famously broken by the British. People like Alan Turing were involved in that. And what did it mean to break it? Well, what it meant was there was a key that was set every day. You would have to everybody, you know, at every, every field commander in the army or something, there would be some courier who would tell them, this is going to be the key of the day. And so then they would set that key on their, on their encryption machine and decryption machine. And then everything they would send out will be encrypted with that key. The things they would get in, they would use that key to decrypt things. And so what had to be done was to figure out, well, what was the key. Now, there were lots of possible keys, so it wasn't easy to just go through and try all the possible keys. It's kind of like passwords in modern times. If you have a long password, you'd have to try a lot of passwords before you get to the one that you actually used. That is, unless you pick a password which is like your date of birth or your. Or the name of your favorite names of animals or something, which is a much smaller set of possibilities than a password that has, oh, you know, uppercase letters and lowercase letters, and it's all a jumbled mess, so to speak. Then the only way to Find the password is basically just to try all the possibilities until you get there. And the longer the password, the more possibilities they are. So, okay, so one way, so kind of the idea was to break the Enigma machine. To break the code. You have to try the keys and find a key that works. Okay, so what happened at Bletchley park in the code breaking effort in World War II was first of all they used some technology which was sort of early pre electronic computers. Actually it came from telephone switching circuits and things where they would have machines that would just go through and systematically try click, click, click, click, trying different combinations. That was first thing. The second thing was figuring out how to prune down the number of things you have to try how to know, oh, it couldn't be one. Once you've checked this and this and this, you know, you can exclude this and this and this and this and this and reducing the size of the search space to be able to do the decryption. Okay, but so then that whole idea of, of what amounts to encryption by substitution and permutation, substitution as in you've got this letter and it's going to be replaced by that permutation, as in you mix up blocks of letters. That idea of having blocks of substitution and permutation, that's been a long running idea in cryptography. And in fact even in methods that are used today are basically big messy collections of substitution and permutation blocks. So that was sort of the state of the art in cryptography. Now one important feature about those methods is they're typically set up so that whatever key you use to encrypt is the same key that you use to decrypt. If you take the example of old Julius Caesar, let's say your key is add two to the letter. Well, that's your encryption key says add two. Your decryption key is essentially the same. It's just subtract two. And in the case of the substitution permutation ciphers, it was actually the same key. It's actually the same key that is used that allows you to go both forwards and go backwards. Okay, so that was kind of the state of the art of cryptography until sometime in the 1970s when it was that you would make a key that would be used for encryption and also for decryption. And obviously the issue of how you distribute the keys is non trivial because if somebody knew what the key was, if somebody intercepted your courier who was delivering the key and found the key, then they don't have to do any of this. Cryptanalysis and fancy things with computers and so on, they just say, oh, I know the key. So that's. And they know. Okay, so that was that story. Okay. Then came the idea of so called public key cryptography. And the big idea there is that the key to encrypt is different from the key to decrypt. And the thing this comes from is factoring numbers. And so sort of the basic way this works is that roughly the, well, let's see, you're roughly the, the key to encrypt is a number. That's all you need to do the encryption. But if you want to do decryption, you need the factors of the number. And because it's hard to go from the number to its factors, you can't go from the public key to the private key. Okay, so what's the point of this? Well, it means that you can have something where you say, this is my public key. You want to send me a message, just encrypt it with this public key. And you can publish your public key. You can put it, put it on your website. This is my public key. But if you want to, if you, if you want to, if you want to get your message, everybody can send you a message. If you want to know what that message was, you have to use your private key to decrypt the message and figure out what, what the message was. And the, the most popular way to do this uses so called RSA cryptography, which is based on the difficulty of factoring numbers. It's based on this idea that you can have a public key that's the number and private keys that are essentially the prime factors of the number. It gets a little bit more elaborate in its actual implementation, but that's basically the idea and it actually relates to Fermat's little theorem, the thing I mentioned before, the actual way this encryption and decryption is done. But in any case, the, so public key cryptography is used all over the web and all sorts of places for. And there are all sorts of protocols that you can use in public key cryptography. For example, you can make digital signatures. You can say, here's this thing, I signed it. And to prove that I signed it, I have to be able to show you my, I have to do things that only I can do because I have my private key and I can do stuff that I can do with my private key. But you can see on the outside that I did it by looking at a public key. And also in cryptocurrencies and blockchains this whole idea of public key cryptography is extensively used. And there are many protocols, there are many sort of ways to factor this idea of. There's a different key to encrypt and to decrypt. So that's a big place where factoring numbers is used. Well, actually, it's the inability to factor numbers that's used. And people will often wonder, oh, if we had a faster, better, stronger computer, maybe a quantum computer or something, could we take all these things which people have assumed can't be factored and just crack them all? My guess is no. And my guess is that's a quite separate story about why quantum computers won't work to do that. One of the things that was sort of a big break in the, in the kind of idea of quantum computers is happened in the 1990s. A thing called Shor's algorithm was, was developed that is a way of factoring numbers using a quantum computer. Well, the problem is that what more or less happens in a quantum computer, the big idea is that instead of there being a particular sequence of instructions that are performed, quantum mechanics allows you to sort of have many threads of history that you can sort of have running in parallel, sort of all at the same time. And then at the end of it, you just have to say, well, what was the result from that? Knowing the result is a hard thing. And that's something we figured out with our physics project. Some estimates of the difficulty of sort of untangling a quantum computing process like that. But the idea for factoring is roughly that, well, you use all these different quantum threads, and each quantum thread is trying a different number to divide the number you started with by. And so remember, if you could just go through, if you have the number a million, and you could just work out for every number up to a million, you could just say, does it divide that? Is it divisible by that number? Can I divide it in as whole numbers with that number, et cetera, et cetera, et cetera? With 2, 3, 4, 5, 6 and so on, I do all those trial divisions. If I can do all those trial divisions, then I can know what the factors of my number are. So the idea for a quantum computer is let's get all those parallel threads of history. Each thread is doing one trial division, more or less. So that sounds really good. It's like, oh, we'll be able to do this. The problem is that when you want to measure the result, when you want to figure out, well, what actually happened, which thread, you know, yes, one of these threads successfully was a factor for the number. But it's like, well, what actually happened? Which thread was it? Where did it go? That turns out to be really hard because we humans aren't sensitive to all these separate threads of history. We sort of knit all those threads of history together to believe that definite things happen in the world. And so that's a piece that isn't part of the kind of mathematical formalism of quantum mechanics. So it's hard to kind of estimate how difficult it is to do that knitting together, but it seems to be pretty hard. So it seems like this idea that you'll be able to use a sort of quantum parallelism to always work out, solve these factoring problems, I don't think it's going to pan out. So anyway, that's a big sort of use case for factoring in the world. Actually it's for not factoring is the really the big use case. There are, gosh, there are so many places where in practice one wants to do these things. Let me give you just a very everyday example. You got a bunch of pictures and you're going to make an array of pictures and you're going to put it on your page. And I've been in the situation many times. How many pictures should I have? Well, If I have 18 pictures, I can make a, you know, I can make three times six or I can make two times nine. I can lay out my pictures in those arrays. If I have a prime number of pictures, well, I'm, I'm out of luck. You know, there's no way to lay them out in a rectangle, so to speak. But that's a very practical place where. Oh, yep, you need to know the factors of a number to know. How do you get to lay out your pictures in a nice, in a nice array? Let's see. Okay, there's a question here. Perpetual asks. I'm not sure I quite understand this question, but it has to do with so called one way functions. That's what happens in the case of factoring and multiplying things together. You can multiply stuff together one way, but going backwards, that is breaking the number up into the thing that you can make the number out of by multiplying is hard. So that's a so called one way function where you can go one way but you can't go back. One place in physics where, which I've studied a lot, well for 50 years actually, but I think I finally have a decent resolution and wrote a book about this recently, is in the Second law of thermodynamics, the law of entropy increase, the law that says that things tend to get more random. You put gas molecules in a box, they bounce around, they'll tend to look more random in their configurations than when you started. That phenomenon is a phenomenon very much like encryption. And that phenomenon is this kind of one way type thing where through this process, through this phenomenon I call computational irreducibility, you know, the rules by which these molecules are bouncing around and those individual rules are completely reversible. Like you can say, oh, the molecules bounced this way and out that way, or they could have started out that way and bounced back this way. So each individual collision is reversible. But together you've got a sufficiently complicated computation that's going on that you can't figure out the reversal of those things. Or a typical observer like us looking at these gas molecules won't figure that out. So that's a kind of a big, big use case of the whole story of sort of one way functions is this idea of the randomness of things tends to increase. And that's something very similar to encryption, that the randomness tends to increase. You can't go backwards. I've tried to find public key cryptography systems that work on different kinds of mathematical bases or computational bases. This one about factoring is a real good example where you multiply the numbers together, you get something which is hard to go backwards on. There's a question of are there other kind of structures where it works that way. So for example, in the case of this encryption done by the gas molecules bouncing around, you can idealize that by something where you just have lines of black and white cells and you have some rule that says how blocks of those black and white cells can be replaced by new cells that makes these things called cellular automata that I've studied a whole bunch of. And I wondered a lot, particularly in the 1980s, was there a way of using cellular automata to make a public key cryptosystem? I never succeeded in doing that and nor has anybody else since I don't think one can. There's no way to know that. It's not possible. We couldn't figure out how to do it. The same kind of idea with cellular automata. You can have sort of, oh, this cellular automaton can be applied, and then that one that's kind of like multiplying numbers together. It's like multiplying cellular automata. And is there a way of taking the combined cellular automaton and finding out that the pieces, the sort of prime cellular automata from which it's built, are hard to get to know that that isn't a hard problem. So there wasn't the same kind of thing as with numbers. The only common other sort of method for making public key cryptosystems is so called elliptic curve cryptography, which is another thing that's based on kind of number theory. I can roughly explain how that works. Maybe. So, remember I mentioned X squared plus Y squared equals Z squared? And what are the integer values of X, Y and z that make that Satisfy that equation? So 3, 4, 5 is an example that satisfies that equation. Okay, let's take the equation X squared plus Y cubed equals something a little bit funkier, a little bit, you know, a little bit more complicated that that equation. If you kind of plotted out what the, what the possible values of X and Y are if they were arbitrary real numbers, you could figure out what those numbers could be, and you would get this thing called an elliptic curve. And then the question is, are there integer values or actually rational values, rational numbers that satisfy that equation? What are the, what are sort of the rational points on an elliptic curve? And it turns out there's a whole elaborate theory, relates to group theory and so on, of what possible rational points there can be on elliptic curves. And that sort of story of going from one set of points to another set of points and so on is something that's kind of like factoring versus multiplying. And you can build other. Another kind of problem that's like the factoring problem that is sort of easy to do one way and hard to do it the other way. And so there's a whole bunch of public key crypto systems that are based on elliptic curves. And it's kind of like they come in little species where there's a particular elliptic curve. One elliptic curve. It's like, oh, we're using the EC184, whatever. No, it's, that's not right. I forget the designations of the elliptic curves. But we're using this named branded elliptic curve. And then you can work out all sorts of properties of it. And things get kind of complicated because in the practicalities of the world, something like the National Security Agency in the US that's responsible for government cryptography and so on will say, we've got an elliptic curve. Here it is, here are its properties. Use this elliptic curve. And then people get all suspicious, oh, why are you giving us this elliptic curve? It must be an elliptic curve that you've Somehow cracked. So there's a whole sort of skullduggery of elliptic curves that people sometimes bring up. Let's see. Oh, by the way, I should have mentioned when I was talking about cellular automata, I never figured out how to make a public key cryptosystem. But cellular automata are great for private key cryptography because they're very good at making randomness. And the key is just what configuration you start the thing off from. Once you started off from that configuration, just run the rules, you get this pattern. The pattern has lots of random features. You can use that randomness to encrypt your message. So roughly, here's how it works. Let's say your message is a bunch of ones and zeros, and Your message is 111000. Okay, now let's say that you have generated a sequence that is for all practical purposes random. You did it using a little key, but that one little key is enough. Once you sort of run the cellular automaton, you get this long sequence, 1, 0, 0, 1, 1, 0, whatever, a random looking sequence. All you do is you take that random looking sequence and every time there's a one, you flip the bit in your message. Every time it's a zero, you don't flip the bit. So if we've got 111000 and we say the first pieces of our sort of ciphering stream are 1 0, 1, then that 111 at the beginning will turn into 0 because we flipped that first bit 1 because we didn't flip the next bit 0 because we flipped the next bit. And so what you need, so you encrypt by just sort of combining, you're doing an exclusive or operation combining or some mod2 operation. For each bit, you combine it with one bit in the sequence that you got out, often called the key sequence. That's a little bit confusing because the key is really the thing you did to start the whole thing off. But you're generating this whole long sequence of seemingly random bits. You combine that with your plain text, your original message, then you get your ciphertext, which is every bit has been essentially randomly flipped. And to go back, all you need is that key sequence and you do the same operation again. And if you flipped, if it was a one in the key sequence caused you to flip, then if you apply this key sequence again, you'll flip again and you'll get back to what you started from again, again. So that's a way to make a non public key cipher where it's the same Sort of the same key, original key, that you use to get what you need to encipher, to encrypt, as, to decrypt. And so that's something you can do very nicely, very well with cellular automata, something I made up in the 1980s, and I guess there's been quite a lot used since then. So that's a. So let's see, My questions here about. How can we prove that there's randomness on a big scale? Oh, that's a big can of worms. What does it mean that something is random? What it means is one can't see a pattern in it. Well, how does one. Sort of a question is, how does one look for that pattern? Let me give you an example. The digits of PI, so far as we know, those digits don't have an obvious pattern to them. But of course, at some level they have a pattern because they are the digits of PI. And that's something we can say about them that we can't say about other digit sequences. But on the other hand, if we say, well, do the same number of ones and twos occur in the decimal digits of PI? So far as we know, the same number of ones and twos occur on average. So there's nothing special about that aspect of the sequence. But this question about when sort of what counts as random, and you can start asking questions like, if you bash the sequence with sort of all possible computations up to some level of difficulty of the computations, will you guarantee that you can't find a pattern, you can't find a regularity, you can't compress the sequence with any computation that's below some level of difficulty of the computation. And that's something which you can potentially, with certain footnotes show in certain cases. So that's sort of a conditional definition of randomness. By the way, the whole issue of doing cryptanalysis, the whole issue of kind of can you figure out what key was used to make this encrypted message? That's all about finding regularities, breaking out of the pure randomness. So, for example, key fact is, if you're encrypting a piece of English text, then when assuming that every letter is encrypted, the encrypted version of every letter is just another letter, like this T turned into a Z, this H turned into a Q, et cetera, et cetera, et cetera. Well, the point is that the original text has certain regularities. So, for example, there are more E's than Z's in the original text. A T is often followed by an H. Et cetera, et cetera, et cetera. A Q is usually followed by a U. Those kinds of things, those kinds of regularities are what allow you to sort of tease out regularities from the apparent randomness of an encrypted message. And that's sort of the essence of cryptanalysis, is to find those kinds of regularities. And the question of whether cryptanalysis is even possible depends on whether. Well, it depends on the fact that your original message had certain regularities in it. And in particular, one of the things this was discovered, probably discovered during World War II. It wasn't written up until Claude Shannon did it in the late 1940s, but it was probably discovered earlier than that that there was this notion of what turned into information theory. But this idea of, well, in that particular instance of this, it's when you do, you do all letters occur with equal frequency? Okay, no, they don't. If we're going to predict the next letter, if we're going to say, what's the most likely next letter, it's going to be an E. If we know nothing else, predict E, you'll be right more of the time than if you predict Z, for example, if you start making a longer block, like you've seen that it says th. Okay, what's next? Well, actually, quite possibly E is next. Certainly Z is very unlikely to be next if you had th. So you can start sort of increasing, you can say, well, what are the probabilities for next letters to be this or that thing and that? The question of sort of how far you can get by just saying how far you can reproduce a message by just saying what are the probabilities? Is sort of something which turns into the information content of the message insofar as it's. Yeah. And so, by the way, zooming to the current time, LLMs, large language models like ChatGPT, they are all about kind of based on the probabilities of what came before, what is the most likely thing to come next? And they're doing that not for letters, but for words or tokens, pieces of words. And they're asking the question, given that you saw the cat, sat on the, what is the most likely next word? And they're basically saying, given the things that we see, the examples that we see in a few billion web pages, what is the most likely next word there? And the real trick with large language models, which wasn't obvious it was going to work at all, is that, well, yes, in that particular case, there are probably enough instances on the web of the cat's head on the mat or whatever, that we could just look at the statistics of the web to figure out what the next word was. But most of the time, most of the long prompts you might give to ChatGPT, they don't exist anywhere on the web. So it's kind of got to. Got to extrapolate from what it's seen on the web. It's going to make a model that allows it to predict what the next word is likely to be. And the fact that that model can be made using neural nets, that's the big surprise. And the fact that when the model once made, seems to make same kinds of predictions that humans would make, that's the big surprise. That's the thing that really made, has made large language models take off. Presumably the reason it makes predictions that are about the same as what humans would make is because the architecture of these neural nets is at least close enough to the architecture of the neurons in our brains that sort of the same kinds of things happen. Eli is asking, could there be an inverse to the law of entropy increase, something like that? Under certain conditions, structural organization always increases, which maybe gives rise to something like biological evolution. Yeah, that's a very interesting question. Darwin, Charles Darwin believed that sort of the increasing complexity of biology was inevitable. He has a nice quote. The very last line of his Origin of Species book was something like, as the Earth goes circling around the sun according to the fixed law of gravity, so ever more complex organisms are being evolved. I'm paraphrasing a bit. He thought that there was a law as inexorable as the law of gravity that gets planets to keep orbiting things, that there would be some inexorable consequence of natural selection that would cause progressively more complex organisms to be produced. That seems not to be true often. And we see it not only in natural selection, in biology, but in technology and other places. Often there's a simple solution that's really good and it blows everything else away so it doesn't force the increase of complexity. Now, I have to say that for myself. The more that I understand biology, the more surprising it seems at some level, in the sense that one might have said, well, biology is just like chemistry. They're a bunch of we have the right kinds of molecules and they're bouncing around and they have chemical reactions and they do the right thing. That seems to not be what's happening in biology. What seems to be happening is that right down at the scale of individual molecules, molecules are being carefully pushed in the right direction. They're being transported down little fibers. They're being oriented by fitting into little gaps in the shape of a protein molecule or something. Everything is kind of being organized. And that process of kind of that sort of orchestrated thing that is biology is a really remarkable thing. Now, we've seen other examples. We see it in technology, we see it in software systems. It isn't the case that the electrons in a microprocessor that are carrying the electrical signals are bouncing around randomly. No, the. They're bouncing around. They're being organized by the details of a program that's causing them to have a one here, a zero there. These things are added to that. This gate is open, that one's closed, and so on. It's all orchestrated. But now the question is, is there a bulk theory of orchestration that is we can say about when we have our molecules and they're just bouncing around and so on, we can make global statements about the fact that there is this tendency for the molecules to seem like they're configured in a more and more random way. But is there a situation in biology, for example, where there is sort of maybe inevitable, maybe just it has been set up that way, or maybe as we build the software system, we're gradually adding more and more to the software system. What can we say in global terms about sort of the orchestration of all those pieces? I think it's a very interesting question, and I don't know. It's something where I think that there are certain quantitative laws that are pretty common. Like it's very, very common to find power laws show up. It's very common, like a common one. Not really. Not in biology, although versions of this show up in biology. If you look at the words in the English language and you rank them from the most common word to. To the least common word, and you'll have some ambiguities because it depends on exactly which text you looked at and so on. But roughly, the word rhinoceros is less common than the word the. And so we get some rough ranking. Now, the nth most common word, if we rank them all from the most common to the least common and so on, seems to have frequency about 1n. That's called Zipf's law. And nobody knows quite why it's true. There are some simple ways you can guess why it might be true, why languages might have evolved to have that form, why you can get that kind of law just by saying you put down letters at random with spaces in between, and you can kind of. And then you sort of burn in the results and so on. You can get that. There are a variety of ways to get something like that. But why that actually occurs, and power laws like that, that's just power to the minus one. But. But other kinds of power laws are pretty common to see in lots of kinds of complicated systems, particularly complicated, more orchestrated systems. And to have this theory of why that happens, to have sort of a theory of orchestrated systems is an interesting thing. I've been thinking about it. I don't know how to do it. And what kinds of questions you might be able to ask about those systems is another interesting issue, because if you say, well, what in detail does it do? Well, you have to go right down into all of the details of the system, and you're really not going to be able to answer that other than by essentially running the system and seeing what happens. So what kinds of global statements can you expect to make? Could you expect to make a statement like, oh, biological organisms will always eventually die because the orchestration will eventually fail somehow? Is that a general statement you can make? I don't know if that one is even true. It's not true for some cell lines and things like this. So it's not. But are there. What kinds of global statements might you be able to make? And I think that's something very much an open question that I at least have been interested in about sort of, you're getting something where there isn't randomness in the sense that you just say, oh, my best approximation to this is we pick these things with certain probabilities and that's all we can say. It's clear that there's organization, but you can't make statements about that organization just by. Other than by saying, oh, well, look, I can see that it's organized type thing. All right, well, we should probably wrap up in a moment here. Let me see if there's anything else that relates to the topics we've been talking about here. Memes is commenting here. Are there really an infinite amount of primes? Yes, there absolutely are. Euclid showed that if you start asking for. Well, I mentioned the twin primes case. There are lots of other sort of special kinds of primes and asking whether there are infinite number of those special kinds of primes. Those are typically unanswered questions. For example, one example is primes of the form 4n +1, where n is an integer, and of the form 4n +1. Right. Those you can kind of say, well, how many primes are there One form versus how many of the other form. And you hit these kind of two curves and they're going up and down and chasing each other. And it's not known whether what sort of the outcome of that will be. Actually, it reminds me of another thing about primes you guys were asking me about. Kind of surprising thing about primes. Here's one. So this prime number theorem that gives you an estimate of the number of primes. There's a formula in the prime number theorem that gives you that estimate. And it seems for a long time that the number of actual number of primes is, depending on how you set it up, either less than or greater than what this estimate says. It seems like this estimate is always overestimating or always underestimating, depending on how you set it up, the actual number of primes. And that seems to be true for a long, long way. In fact, Ramanujan, famous Indian mathematician from the early part of the 20th century, who came up with many surprising results that I think he derived by doing sort of calculation and experimental mathematics and then said, oh, by the way, this weird, weird formula with square roots of PIs and square roots of 17s and things, and it happens to be true. He found that by doing calculation and then is like, this is true. He actually thought that he had found a proof of the prime number theorem and therefore a proof of the Rumian hypothesis using the. And that you would always have. But he actually didn't say he had a proof. He said this thing will always be, this estimate will always be greater than the number of primes. But he did that by sort of evidence for the fact that he used this kind of experimental method, because that result is true up to very, very, very big numbers. There's a thing called the skews number, which is like 10 to the power 10 to the power 10. Ish. Don't remember exactly what it is, but it's a very big number. It's a number so big that you couldn't write out its digits on atoms in the universe and have it fit. So it's known that this estimate will flip over at least by that number. Could it flip over much, much sooner than that? But it's known that you go far enough out, the number of primes will. Will end up being, let's say, greater than instead of less than this estimate. So that's another sort of surprising thing about the primes. I mean, I think the, the big picture of the primes is they're always, they've always got surprises. And it is very surprising that something whose definition is as simple as the definition of the primes could have an infinite collection of surprises. And in a sense, that's deeply related, I think, to this idea of computational irreducibility, the idea that you can set up rules for a system that can be quite simple, and yet as you actually run the system, you will always have surprises. And if you keep going, you're going to bigger and bigger and bigger and bigger primes. You'll always have surprises. And that it's also related to Godel's theorem about the fact that there are proofs of infinite length in axiomatic systems like the one that's used in studying properties of numbers and so on. So it's kind of the fact that the primes always bring in surprises. Is itself a very interesting fact about the world and about mathematics and so on. I think that point that sort of the primes bring in surprises is something I suppose people have really concentrated in mathematics on the unsurprising aspects, the smooth, the predictable aspects of primes. Something that's more been something I've been interested in is these cases where you have sort of the unpredictable, the necessarily unpredictable happening. And it's something where even back in antiquity, people knew a certain number of primes. I mean, Plato, for example, was very big on the fact that. What is the number? Is it 4040? I can't remember. There's some number that Plato describes as being an optimal number for the size of a democratic society or something, because it's a number that has many factors. Talking about factoring, there's another application of factoring for you. What is that number? It's around 4,000 something, I think. And so the fact that they knew how to factor that number means they must have known primes up to on the order of that number. And yet this observation that the distribution of primes seems kind of random, that's not an observation you see anywhere. You see people discussing, well, can we find regularities, but not, oh, it seems kind of random? What is the significance of that? All right, well, thanks for asking. Bunch of interesting questions there. And I should go back to my day job and look forward to chatting with you again another time. Bye for now.