I got three numbers for you, Brady, and I want you to tell me what they got in common. 2025, which is the year, of course, 666, which is very scary, and 1729, which you probably recognize as a >> taxi card. >> Taxi card number. Exactly. But what have these all got in common? >> They're harsh numbers. >> Oh, come on, baby. That's annoying. What's that mean? >> That's the title of the video, probably. >> Yeah. Yeah. Yeah.

They bring joy. These are these are numbers that bring joy. And that is, of course, a harsh ad number. Harshad means a joygiver and these are all examples of of harshad numbers. I'll show you why they are harshad numbers. So so let's take the first one 2025. If I divide it by the sum of its digits. So 2 + 0 + 2 + 5 I get 225 which is a whole number. >> So that's 225 / 9. >> So now I could do it with 666. Okay. Divide by the the sum of the digits which is obviously going to be 18.

That's going to give me 37. another whole number and I could do it as well for for 1729 as well. Any number which is divisible by the sum of its digits is what we call a harshad number. >> So Harshad isn't named after a mathematician called Harshad or something like that. >> No, it's Sanskrit for for the word joygiver. So it was they were introduced originally by Capri who we've come across before. They were also raised into some prominence by a mathematician by the known Ivan Nan in the ' 70s.

So somewhat after Capric think about various numbers. Pick a number ask if it's harsh. So, I thought maybe we can think about birthdays. So, I looked up your birthday, Brady. Am I allowed to talk about your birthday? It's on Wikipedia anyway. So, so it's known. So, do you think your birthday is is that? So, if I write down your birthday, I'm going to write it down now. Okay. Getting on a bit there, lad. Aren't you? Yeah.

Big 5 this year. I know. Next year. Next year. What we talking about? Yeah. Big 5 coming up. 1976. Okay. Do you think this number is hard, Shad? >> I don't think you would have gone for the trouble of bringing it up if it wasn't. Well, I can check. It's So, you're all excited now, aren't you? You're all excited. Well, it's not going to be harsher. Okay, I know it's We'll come back to that later. Uh, so we divide by the sum of its digits.

Okay. So, I need to divide that big number by 38. I mean, we could check it with the American system as well, by the way. Yeah. So, that comes out at So, it's just it's rubbish. Ready? Your birthday's rubbish. Yeah. So, one of the things you can ask is what kind of numbers are going to be harsh, for example. Well, factorials are commonly harsh ad as you can imagine. Why? Because they've got a lot of factors, right? You know factorials you just you do 1 * 2 * 3 * 4 and so on, right?

But not all factorials are harsh. As it turns out, 432 factorial is not a harsh number. Somewhat surprisingly, it's the first factorial that's not harsh. So it's not guaranteed that you're harsh of your factorials, but factorials are often harsh. Every single digit number is kind of obviously harsh because obviously you just divide it by itself and you get one. How many of them do you think they are? How how many harsh numbers do you think we have? >> I'm going to guess that it there are an infinite number of them. >> There are an infinite number.

That's true. Uh they do get the density of them does go down over as you get to larger and larger values. we get fewer and fewer but it is true that there are an infinite number. One thing we can ask about as well is whether we can get blocks of harsh ad numbers. So for example do you get like harsh ad pairs? So it's two two consecutive numbers which which are harsh and and you do for example uh >> all the single digits. >> Yeah.

Okay. Well let's go let's jump away. Yeah of course. Very good. Yeah but even beyond that right. So, so for example, 20 and 21 both these are are harsh numbers and that's actually the smallest sort of non-trivial harshad pair. You can have harshad triples. Uh the smallest one of those uh is 110, 111, and 112. You get harsh ad quadruples. And you can carry on and ask, do you get these blocks? There's a block of 17. The smallest block of 17 is known.

Fay's gonna put it on the screen now. So then you can ask about what about blocks of 18? They exist, but we don't know where the smallest one is. Blocks of 19, they also exist, but we don't know where the smallest one is. Blocks of 20 exist, don't know what the smallest one is. Blocks of 21. None. There's no blocks of 21. >> Proven. >> Proven. No blocks of 21. Harshad numbers. This is This is 100% known. You can never get a block of 21. >> The block of 20 that hasn't been found.

The smallest hasn't been found. Does it definitely exist? >> Yes, >> it does. Right. >> It does exist. Yes. So, the blocks of 20 exist. Blocks 21 do not exist. >> Infinite number of blocks of 20. >> Oh, good question. Uh, I don't know the answer to that top of my head. I'm gonna say I I want to say yes. >> I reckon yes. Thanks. >> I reckon yes. I mean, we know that the actual the harsher numbers themselves there's, you know, there's an infinite number, albeit they get much less dense.

I mean, the density does drop, but I can't imagine it sufficiently a big effect to eliminate the blocks, but I don't want to say that. I don't know for sure. Okay, I'm going to be honest there. I don't know. Another place that there you can sort of often find them. they tend to cluster around the number around multiples of nine. That's another weird thing about I can show you a little plot of that that I've prepared.

So here's a plot of the density of the harshad numbers. Okay, so it sort of gives you a rough estimate the local density and that's what the the vertical axis is. And as we go along the sort of number line, the red dots show multiples of of nine. And what you can see here is that most there's not actually that many where the density is very low around multiples of nine. The density doesn't tend to be low. It tends to be sort of amongst the higher ones.

And it's because it's simple. There's a there's a nice fact that if you take any number then it is equal to the sum of its digits mod 9. So let me show you. I could do example for example. Should I get another piece of paper or should I get if you want? [Music] [Applause] [Music] >> Okay. So let's take our 1729 our taxi card number that is 192 * 9 + 1. So modulo 9 this is just one. If I look at the sum of its digits, that's 19, which is 2 * 9 + 1.

Okay, so this is also 1 mod 9. And it's true of any number. Any number is equal to the sum of its digits mod 9. And so this is the reason why you often get harsher numbers clustering around nine because it links the number to the sum of its digits in quite a natural way. You see, >> obviously, we're talking base 10 here. This is this is a point where this becomes very evident. >> Yes, that's true. And we are going to come back to different bases in a minute.

But I want to do show you something else about harshad numbers. You can have hashad chains. Okay. So this is like where you get multiple harshad numbers. I'll give you an example. 6804. If I take the sum of its digits, I get 18. And that's going to give me 378. So this this is a harsh head number. But now I can look at this guy that I've sort of got from it, right? And I can ask, is this a harsh number? Well, 378. If I divide by the sum of its digits, that's also 18.

That gives me 21. Okay. So, I've got another harsh number. Okay. This is a harsh number. What about 21? Well, 21. Divide by the sum of its digits. That's three. Okay. Oh, there you go. Another whole number. So, this must mean 21's a harsh number. What about seven? That's definitely a harsh number. Okay. And I get down to one. So from this original seed I you know I've gone through a chain of um of sort of 1 2 3 four non-trivial harshad numbers.

Okay plus one if you want not counting the one there. So this is called like a multiple harshad number of of length four. And you can obviously have longer uh chains of harshad numbers. Here's a pretty cool one. 2016502 88 84 466176 this bad boy right this guy and I'm not going to do it but you can try at home this will give you a chain of 12 harsh numbers that's pretty impressive you can have infinitely long chains in principle So this number time 10 the n this guy is guaranteed to have be a chain of length n + 2.

So it's an example. So that just shows as long as I can just make it n bigger and bigger and bigger I can get chains of any length. Other bases Brady you wanted to do other bases. Right. Okay. Obviously you can think about harshad numbers in different bases. Same sort of idea basically. >> Spread the joy. >> Spread the joy. Let's spread the joy. So let's think about your birthday again. >> Okay. Let's do it. All right. >> Should we see if it's a harsh shaded in in a different base? >> Are even or odd numbers more likely to be harsh? >> Oh, good question.

Well, we know that. So, even is obviously good because it means you guarantee that two is a fact divide. So, I think even I'm going to I don't know this, but I'm going to claim even is more >> because you're less likely to hit a prime. >> Exactly. Exactly. Okay, we're going to do your birthday again, Brady. So, my little Python code, which I'm very proud of. Okay, let's see what it tells us. So, it says it's a harsh head number in bass 16 and in bass 20.

I've only gone up to 21. I can go higher if you want. Do you want to see? You got any other ones? >> Okay, go high. >> Let's do it. Let's do it. Let's do it. Let's do it. Oh, there we go. It's done it. Oh, there's loads. >> So, I guess the question popping into my head now is are there numbers that are harsh numbers in lots of bases? Like, is there a special property? >> Are you talking about your universal joy givers?

You're sort of Yeah. >> Yeah. The most joyful of numbers. The most joyful of yes there are really there are numbers which are universally joyful which are harshad in every base there not very many but they do exist >> I guess you can only go up to the number itself in bass can't >> exactly exactly >> they are there are four of them >> numbers one numbers two numbers four and number six those are what's called trans harshad they are harshad in every base okay come may so they are indeed universal joygivers here's a funny one the Number 12.

It's It's weird. It's almost trans harsh. It almost wants to give joy to every single uh number except for bass eight. It does not like bass eight. So, it's harsh in every base except for bass. Do they bring you joy? They do bring me joy, but especially when I start thinking about these cool facts. This this one I think is is is a crazy fact, right? So, and it's proven. So, take any number under a billion, right? Any number under a billion then that number is guaranteed to be either harshad or the sum of two harsh ads just two right any number under a billion is either harsh the sum of two harsh which I think is wild does it fail after a billion or has it just not been tested past a billion >> it will fail eventually after a billion I think because I'm going to tell you the next bit every single number literally every number every positive integer it don't matter how big it is is going to be the sum of at post let's say K harsh add numbers where K is some finite number right this is proven K is some finite number but K is not known >> right >> okay >> so it could be three >> it could be three it could be well it probably isn't three but it it could be tree three but it's it's a wild fact any number is the sum of the most k harsher numbers but k is some finite number currently unknown so this this is a proven fact this is this is this is just this is just true and The proof of it relies on I think some related to dedic functions and all this stuff, right? >> Is the number likely to be what you and I would consider a big number or could it be like you know seven or something? >> I reckon it's probably quite a big number.

Yeah, I don't think it's seven. I think it's probably bigger. >> I don't think it's necessarily tree three. I don't know because we're dealing with something which goes arbitrarily forever. Yeah. >> So yeah, I'm not sure. Now now I'm not sure. Maybe it is big. It's interesting. I'd love to know. I mean it's not known. I mean the answer is is is currently unknown right? So so this is important. Yeah. And we we haven't stopped.

There's more to know. [Applause] So there are super harsh ad numbers. Okay. So these are harshad numbers where when you divide them by the sums of their digits you get a prime number. All right. These are called super harshad. Sometimes called moran numbers. So this is a 18 for example is super harsher because if I divide by some of its digits which is nine I get two which is obviously a prime number. This is an example of something that's super harsh.

There's also something called harshadorphic numbers. It's probably best illustrated with again an example. So if I take the the number 16,218 this is a harsh number. You'll notice that it ends in 18. If I take the sum of its digits, what do I get? I'll get 18. Okay, so this makes 18 harsh admorphic. So it's something where there is a harsh add number which ends in the number you're interested in and for which the sum is the number you're interested in.

That's that's called a harsh admorphic number. I'll give you another example. >> That seems trivial even by our standards, but >> well, wait, there's there's something about it. Okay. So I'll give you another example just so so we're clear what it is. So if I take the harsh number 2715 okay ends in 15. Okay. The sum of these guys is of course also 15. Right? So that makes 15 harsh adorphic. Right now I'm going to ask you a question.

How many numbers do you think are harsh admorphic? >> Well you've set me up now. It's either infinite or some ridiculously low number that's going to surprise me. Do do you think there are any numbers that aren't or any numbers that how many are I mean we know that 18 is we know that 15 is >> Oh you mean what number can make the morphy part? >> Yeah. Yeah. >> Okay. Uh okay. Uh yeah. So what what are we calling that part?

The >> this is this is called the harsh these are you would say that 18 is harshorphic and and 15 is harshorphic. If there's a harsh number which ends in it and who as it is digits >> okay. So, are there some numbers that aren't accounted for? >> Yeah. Or or is every number one or is it just a subset of them? What do you reckon? >> I feel like I feel like every number should be one eventually in an infinite world of harsh. >> So, there are not there is not.

No, it's not every number. There's one number and only one number which is not harsh and that is in base 10 I'm talking about here. >> Right. Oh, please let it be something really arbitrary like you know 209 or something. Was it? >> It's 11 >> 11 >> 11 is the only number which is not which is not a harsh ever. I I think I think it's related to the fact it's base 10 and it's one more than bass 10 basically. It's connected to that fact.

That's it. I mean I don't know if I mean I think they bring him they brought me some joy at least. >> Do you have a head for numbers or would you like a head for numbers? Why not train your brain on the magnificent bounty of courses and lessons available on Brilliant? There's no better way to learn than to flex those neurons than this kind of visual interactive content. It's designed by smart people who care about learning.

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Our thanks to them for supporting this episode of Number File. Is there a point to this? This this does seem highly recreational. Or is there something deeper lurking here? >> I mean, it is it is pretty recreational, I think. I mean, so the root of them was recreation, right? This Capricorn, that was his thing. He he did recreational mathematics. Okay, here's where I I would say this there's there's some