8128 and Perfect Numbers - Numberphile

BRADY HARAN: So we're going to talk about the number 8,128. And it's a famous type of number called perfect number. Now, this was the largest perfect number known in antiquity, or for very long time. But what is a perfect number? The smallest perfect number is the number six. So let's look at the number six. Now what does it mean? What divides into six? Well one divides into six. That's always true. Two divides into six. And three divides into six. Now, six does divide as well, but for what we're going to do, we're going to ignore the number itself. So ignore six. But if we add up the numbers that divide six-- one, two, and three-- add them together, and we get the number six. We get the number itself. So this is a perfect number because the divisors add up to the number itself. It's not too small, and it's not too big. It's just right. It's perfect. And in antiquity, they were given this rare, perfect property. So the next one, after 6, is the number 28. Let's write that down. 28. So what divides 28? Add these together, and you get the number 28. It's the second perfect number. The third perfect number after that, bit of a gap after that, the third perfect number is 496. Let's look at the divisors of this. And we've got 1. Quite a few of these. 2, 62, 124, 248. We're running off the edge. We add all this together in the same way as before. We get 496 again. It's perfect. Now, there's only one perfect number between 1 and 10. That's the number 6. There's only one perfect number between 10 and 100. That's 28. There's only one perfect number between 100 and 1,000-- 496. And there is only one perfect number between 1,000 and 10,000, and that is 8,128. Let's try it out then, OK? 1, 2, 4, 8, 2,032, and 4,064. Add those together in the same way, and you will get 8,128. Today, even with our computers that can compute massive numbers, we have only found 47 perfect numbers. That is all. Four were known to the ancient Greeks. Still today, 21st century, 47 we've got. There may be more. There may not be. We don't know if there's an infinite many of these numbers. And that's an open question. You'd think that that's something we would know by now. Are there infinitely many of these or aren't there? We don't know the answer to that yet. MALE SPEAKER 2: What's the biggest perfect number? BRADY HARAN: So far? I wouldn't know that. Let's find out the largest perfect number. Well, Wolfram Alpha says, do you mean largest integer? There's no largest integer. So that's helpful. Interestingly, we don't know if there are odd perfect numbers either. All the perfect numbers we have found are even. That's another question we don't know. The largest known perfect number is, apparently, 1,791,000 digits long. Don't ask me to say it. We could be here 'til next week. But that is the full thing. Nearly 2 million digits long. I could be scrolling through this for quite some time.