We're going to talk about triperfect numbers. Triperfect numbers‽ We've heard of perfect numbers, haven't we? We've done that on Numberphile before. It was one of our first ones, wasn't it? I think we need to do a quick . . . recap of what a perfect number is because that's going to be relevant to what a triperfect number is. So a perfect number is a number like six. Six is a perfect number and that's because factors of 6 are: 1, 2, 3, and 6 itself. In fact, what we're going to do: we're going to ignore the number itself. If we add up the other factors: So we'll have 1 + 2 + 3. We're going to end up with the original number. We're going to have 6. The next perfect number is 28, and it works the same way. If you look at the factors of 28 and add them up you'll get the original number. The next one is going to be 496: that's a perfect number. The next one after that is 8128, and they keep going, right. So they are quite rare. We think there's infinitely many of them, but we've found 50 so far. The largest one we've found is the most recent, which is a perfect number that is 46 million digits long. So we do think there's infinitely many of them. Although we haven't proven that to be true, and all the perfect numbers we've found are even. We haven't found an odd perfect number yet. So that's something that hasn't been proven yet. But this definition I gave you was kind of weird because I ignored the number itself. So I said out of all the factors except the number itself You could just add all the factors including the number itself. That's perfectly fine, and then you could say a perfect number is: you add up all the factors and you get double the original number. That's absolutely fine. What happens if we add up all the factors and get triple the number? That is a triperfect number. Well, the smallest triperfect number is 120. If we look at the factors of 120, you have: and we'll include 120 itself. So those are the factors of 120 and if you add them up you will get 360. You will get 3 times the original number. So that is the first try perfect number. Let me write out the next few. Let's start a list here with 120 and the next one is 672. The next triperfect number after that is 523 776. The next one after that is 459 818 240. The next one after that is 1 476 304 896. and the next one after that—I might put this one in—I'll put it here: So I've got six there, written out and that's all of them. That's all we know and we also think that's a complete list. We think there's finitely many of them. We think there's six of them only. That is a complete list. There are six triperfect numbers. Now, that's not proven, but we really do believe that this is a complete list. Couple of reasons for this. We have searched for them, and we've searched quite a long way. We've searched huge numbers, like numbers that are 350 digits long, like really big numbers, but they just stop. So we have a triperfect number in the hundreds. We have triperfect numbers in the thousands. We have them in the millions. We have them in the billions, and then they just stop and it would be really weird if there was a some . . . some . . . some sort of massive triperfect number so there's this massive, massive gap and then another triperfect number suddenly appears. That would be really weird if that's true. Now we don't know that this is a complete list, but it would be really weird if it wasn't a complete list. And there is other reasons why we think this is complete list because it's also related to perfect numbers. In fact it's related to odd perfect numbers. If you remember I said we haven't found any odd perfect numbers yet. And there is a relation between odd perfect numbers and triperfect numbers. If you had a number. Let's just call it n, and if that is odd perfect, then 2n is triperfect. So that means that if we do find an odd perfect number one day, and it's going to be massive if we ever find it. If we find it one day, then automatically we're going to find a new triperfect number. So either we're going to find this odd perfect number and this list is not complete after all, or this list is complete which means that aren't any odd perfect numbers, which we also believe—not proven—but we strongly believe that's true. BRADY: James, are there quadperfect numbers? I'm glad you asked, because yes, absolutely. So there are 4-perfect numbers, 5-perfect numbers. Shall I show you them. We can look at them. I've got them. Yeah. BRADY: That wasn't set up! I thought that was a real question. The smallest 4-perfect number that we know is 30 240. So if you look at the factors of that number and add them up you'll get four times the original number. There are 36 known 4-perfect numbers, and then the list just stops again, and then we can't find any more, so we think, again, we've got a finite list here, and we think we have all the 4-perfect numbers. You can take it a step further, of course. Look at 5-perfect numbers. The smallest 5-perfect number is 14 182 439 040. There's sixty five of those, that we know of, that we've found, and again, it's the same thing. It looks like it's a finite list because we found 65 of them, and then it just stops and we can't find any more. So we think that's a finite list as well. Same is true for 6-perfect numbers and 7-perfect numbers. We think we found a complete list then they just stop. >>BRADY: If you love numbers, and I think maybe you do because you're here watching numberphile Then you really should check out brilliant a site full of quizzes Puzzles courses all dedicated to mathematics and science among them is this course here on number theory? It's one of my favorites and you can see here a few problems specifically about perfect numbers Brilliant is a treasure trove of interactive content. That's gonna make you really love learning It's good fun and makes you a bit smarter along the way now There's a whole bunch of stuff on the site for free, but you can also get 20% off brilliance Premium Membership Which gives you even more stuff by going to brilliant? Org slash numberphile that also tells brilliant you came from here. So that's good for numberphile, too Our thanks to them for supporting this episode and I'll include in the description Some links to some excellent stuff on brilliant. You might wanna have a look at
