4937775 - Numberphile

JAMES GRIME: Today, Brady, we're going to look at Smith numbers. And Smith numbers were born in an unusual place. They were born in a telephone directory when somebody was looking up the phone number of their brother-in-law and wrote down the number, and then realized it had some unusual properties. 493-7775-- BRADY HARAN: And that's a Smith number? JAMES GRIME: That's a Smith number. BRADY HARAN: And whose number was it? JAMES GRIME: It was, in fact, Harold Smith. And he was, I think, the brother-in-law of Albert Wilansky, who was trying to contact him and found him in the phone directory. So Wilansky made the breakthrough in realizing that there was some nice property about this particular number. So this is, perhaps, a bit big to start off with in defining a Smith number. But a Smith number is a special number. The sum of the integers that make up the number is the same as the sum of the prime factors of the number. 27 is a Smith number. Well, first of all, let me just write out the prime factors of 27. So 27 can be written in the following way-- 3 times 3 times 3. OK. Now, let me do the following. Add these two numbers together that make 27-- 2 plus 7. And you'll notice that that's equal to the sum of the prime factors-- 3 plus 3 plus 3, which equals 9. And that's what makes up a Smith number. A famous one, the one that you've already covered in Numberphiles, which is the beast number-- 666. So that breaks into 2 times 3 times 3 times 37. OK? If you multiply these together, you'll get back 666. Now, let's add these digits together. So 6 plus 6 plus 6-- and that is the same as 2 plus 3 plus 3-- and now comes the subtle bit. You don't add 37. You break 37 up into its digits. So let me put 3 plus 7. And hopefully, this works. 6 plus 6 plus 6 is 18. 2 plus 3 is 5 plus 3 is 8 plus 3 is 11 plus 7 is 18. And so that's why 666 is also a Smith number. Let's go to the original Smith number. And let's have a go at that one. What gets me is how he realized that this was going to work. First of all, get the prime factors. So it's 3 times 5 times 5 times 65,837. It's that bit that I really think is really neat. What we can do is add up the digits that make up the number. 4 plus 9 plus 3-- plus 5. And this should equal 3 plus 5 plus 5 plus-- and then we do this trick, right? We break this up into its individual digits-- plus 3 plus 7. I hope this works, Brady, otherwise I'm looking a right idiot. 4 plus 9 is 13 plus 3 is 16-- plus 5 is 42. BRADY HARAN: 42! JAMES GRIME: Yay. OK. Let me confidently write 42 there. And now, let's check if it works here. 5 plus 5 is 10 plus 3 is 13-- plus 10 is 42. It worked. BRADY HARAN: There we go. It's a Smith number. JAMES GRIME: And that's a Smith number. BRADY HARAN: Do you like those ones? JAMES GRIME: I like that number. I think it's really nice. BRADY HARAN: What do you like about it? JAMES GRIME: I like the fact that, first of all, it's a number where you've broken it down into its-- I'm a physicist, right? I believe in building blocks, quarks, the building blocks of matter. These are the building blocks of mathematics, right? The prime numbers are the building blocks. So you first of all, can break it down into those prime numbers. And then, somehow, by just adding together these prime numbers, you get a property of the original number over and above the original property, which was you multiply the primes together. It just has a nice additional feature to it. 4937775-- am I speaking to this person? BRADY HARAN: Just to see what happens. AUTOMATED VOICE: --recognized. Please check the number. If you need help, call the operator on 100 from your mobile. JAMES GRIME: Oh, well. He's probably had it disconnected he's so cheesed off with people ringing him. What really blew me away was when I discovered what the biggest known Smith number is. And I thought I'd write that one down for you, because I'm not sure how they were able to check it all. OK. It's a big one. So 10 to the power 1,031 minus 1-- that 1's very important. OK. That's all multiplying the following-- 10 to the power of 4,594 plus 3 times 10 to the 2,297 plus 1 again. And all of that bracket is raised to the power 1,476. And that's not the end of it. That, then, all gets multiplied by 10 to the power 3,913,210. And this is the largest known Smith number. In fact, in terms of the little mass, it's given as 16 times 100 to the power n, where I'm going to let this number n vary. So if n is 0--