Erdős–Woods Numbers - Numberphile

a positive integer please 24 24 is great and now I'm going to do a sequence until I get bored so 24 25 26 27 28 29 30 31 32 33 34 uh then I say I'm the board 35 yeah, okay then 24 up to 35 uh , you may have noticed that I made the final points in different colors, which will be important. This is the game. I want to cross out any number in my sequence that shares a factor with the endpoints so that it shares a factor with 24. or it will share a factor with 35 24 is even that means I can get rid of everything divisible by two so the even

numbers

are going to disappear so I can cross them out it's also divisible by three yeah so I'm going to get rid of multiples of three, so 24 I can get rid of 27 uh, 30 is gone, but 30 three is no longer divisible by four.

I'm just going through them one at a time, I don't actually need to do that because I've actually already crossed out the even

numbers

, in fact it's a shortcut, you just have to check the prime factors, so okay, let's go through the prime factors, so I made two, I made three, five, this was not divisible by five, but 35 was. so I'm going to use that 30 already gone 25 I can cross it off big six I don't need to make seven uh none of these divisible by seven oh oh I have 29 and 31 left oh I think I'm already stuck because 29 is a prime number, right? and 31 is a prime number and they are not going to share factors with these numbers here, so I have reached a stuck point, so here I cannot cross these two.

My question is: Can we find a sequence where we can cross them all out? What do you think? Where should we start? Should we try larger numbers? Do you think it's possible? Do you think it's impossible? Should we do the sequence? longer should shorten the sequence what do you think? well, you want the sequence to be as short as possible, okay because it makes your job easier, yes, as short as possible, you think you want to go up, you want to go out further, where you are. less likely to be hampered by prime numbers, right, I agree with that, yeah, less hampered by prime numbers coming out further, um, okay, so I have a maybe I have an example here that doesn't have any Prime number.

I'll do an Example of that nice long sequence that has no prime numbers starting at 114 and I'm going to go up to 126. I just want to show you the example of this, so we'll play the same game. It's worth it. knowing what the prime factors are I think that could save us some work 114 even yes, great, let's cross out everything divisible by two, they're gone 114 divisible by three I know my divisibility tests, we've done this before 114 yes, divisible by three let's cross out multiples of three 1 2 3 cross it out 1 2 3 already done 1 2 3 cross it out 1 2 3 big five now I can't cross out any five okay seven if there was a way to solve it multiples of seven but I want to show you this better way to calculate If a number is divisible by seven 126 is divisible by seven so let's cross out the multiples of seven.

Here I'm going to work backwards 1 2 3 4 5 5 six 7 that's gone 2 3 4 5 okay, uh, what about the 11? Can we do 11? Is this divisible by 11? That? No, no, is this divisible by 11? again, yes, but maybe I should have used a different number, maybe if I had spread this out a little more I could have. This is the right question, so where do we start? How long is the sequence? Can we cross it out more? Is it impossible then this question was thought up by a mathematician named Alan Woods? He was actually playing with some serious high-level math. He just realized this while he was playing around with this work he was doing with prime numbers, things like that, and he thought it was impossible, so it's worth a try.

Think about it. I won't give you the answer yet. Think about it. What could it be? Well, I can tell you there is an answer. It is not that big. It's not too small, keep looking, it's in thousands, the answer is in thousands, here's the answer, so if this was another type of number file video and I didn't want to follow the answer right away, I would have said: Hello everyone , this is a number file video about the number 2184 and we're going to do a sequence um and I'm going to go up to uh 2200 let's do it5 206 20789 2090 2091 202 20345 2067 2199 2200 and I'm going to hold on to this it's going to work we can cross out each number those bookends they're going to break the middle to the right let's do the check uh divisible by two that's even let's do the check let's eliminate the even numbers they're gone let's find out if it's something divisible by three uh, this is yes, so this end point is divisible by three, so let's get rid of the multiples of big three divisible by five, yeah, 2200 is divisible by 5, let's get rid of the ones that were gone, uh, seven, this one is divisible by seven, okay?

We already finished 11, that one is divisible by 11 2,200 so 1 2 3 4 5 6 7 8 9 10 11 we can get rid of that it's already done and we have one left, what about 13? Can we divide by 13? If this is divisible by 13 1 2 3 4 5 6 7 8 9 10 11 12 13 that disappeared and that is our first sequence that we can cancel completely using those endpoints, then the length of that sequence goes from 2,184 to 2,200 or in others words, it's 2.184 up to 2.184 + 16, so 16 is the length of that sequence, not quite because you have to add the endpoints, it's actually 17 long, but it doesn't matter, so we call that kind of length of that sequence 16 is called the English wood number, so it is the first and its corresponding starting point is 2184.

Do you want to see the others? I would love to know some more. Yes, they are obviously rare. They feel strange. They're hard to find yeah let's take a look at them so these are the lengths of the sequences first and then I'll show you the starting number as well so 16 which starts at 2.184 the next one is 22 starts at 3.521 210 the next one uh it's 34 that starts at 47 B 563 m752 566 wow next next I think I want to do a couple more uh next 36 that starts at 12 billion 993 million 165.00 320 oh that's a little, that's a little strange because that number actually disappeared, you would think they would get bigger and bigger, that number has actually decreased a bit so maybe it's not as simple as that.

I can't write these massive numbers anymore. However, I'll show you a couple more lengths, so we have 34 36 46 56 64 and let's say they don't repeat, although couldn't there be another length of 16? You are absolutely right, so there are other sequences of length 16, in fact there are infinitely many of them, so the smallest starting point is 2.184 and in fact there are infinitely many starting points that have length 16. You can do the same trick. There are a couple more things we know. that there are infinite numbers of wood erish um like in what are these sequences like in these the smallest numbers I noticed that they are all even they have to be even that's a great question they also thought they said they are all even are they?

It has to be even that's exactly what they said no, no, but the first odd is 93 uh and they're you, they're pretty regular, it's not easy to find, but they're pretty regular, but yeah, 93 is the first odd. Ireland wooden number, so what we have here is a sequence of Irish wooden numbers and that means some numbers are missing, not all numbers will have a sequence that you can play this game with. I have many questions. comes to mind, wow Dr. Gri, like, do chains ever overlap? Am I ever in one of those Oswood swings and then and but I'm also in another one?

Oh, what a good question, questions, bra. thinking like a mathematician I don't know what those answers are I don't have all the answers for those U's but I don't see any reason why they can't overlap but I don't know so these are things that you could ask questions about I mean it's fun to play with them, isn't it? The way I presented it, I just presented it as a fun numbers archive, kind of like playing with numbers, having fun with it, just finding it as a special. property of a number like that Alan Wood, however, was working on some real high level mathematics to do this, uh, just to give you a little idea of ​​what he was looking at, he was considering taking maybe two sequences like this .

Let's say we start at a and then we do A+ 1 and A+ 2 and so on until we get to a right plus K, which will be our other end point. Let's say you take another sequence, let's call it starting at B, which is a. completely different number then you have you know B+ 1 B plus 2 so like the game that I have shown you and we are going to go up to B plus K also so these two sequences have the same length what he wanted to know If you look at the prime divisors of each term in these two sequences, there will be some point where they differ, so they might be the same for a while, but at some point they will be different prime factors.

Just do an example of what I mean by that, let's say I have 14 15 16, so there's a very short sequence there and another sequence that starts at 224 225 226, so what we're looking at are the prime divisors pr of each term, so in these first terms this is what 2 * 7 is and the prime divisors of this number are also 2 and 7, it is 2 a ^ 5 * 7, so they have the same prime divisors, although this is larger because I'm using two more often, they have the same prime divisors. 15 is 3*5 225 is 3^2*5 square so they have the same prime devices 16 well that's going to be what 2 to the power of ^4 and 2 26 is 2*113 and that's a prime so yeah , see, they matched they matched and then they failed, so what Erish and Wood wanted to know was if there was a k value, something like a finite number, a number that could even tell them what it was and if I use it as the length of my sequence , fail to. so it's going to fail for every two possible sequences you try to put together uh for all the infinite numbers out there, that's cool, this drives high level mathematics uh, if certain conjectures are true, we can say that most of the time it will be k equals 3, which is what we have here most of the time and by most of the time I mean most of the time with a finite number of exceptions, which actually in the Infinity of numbers a finite number In fact, you could just enumerate your finite number of exceptions, then find the one with the longest length and then use that as the value of K and job done, that's the K you can use and then that would break all the exceptions. possible sequences for what would be a finite K, but that's if we solved these high-level math problems that still remain unsolved, so even though this was a silly math problem with numbers I gave you, we're really pushing the boundaries of mathematical knowledge here , so I'm saying that so far the two longest number sequences we know of that satisfy this criterion are two, yes, there are no three numbers.

No, I don't know of any sequence that has three numbers and the match works and fails on the fourth. I only know examples that match. for two and then failing on the third amazing amazing factorial 77 plus one which turns out to be 71 squared. I've shown you three, yeah, that's it, that's it, there are people who publish under pseudonyms, okay, so here's an arish numbers story from a pseudonym, so there was an article they were working on, Arish and my wife, Fan Chong.