Interesting 666-digit Numbers - Numberphile

Key Takeaways

this is the third and final video in our little Trio about really big numbers and the famed 666 links to all of them can be found in the video description apocalypse apocalypse yes apocalypse so apocalypse numbers are just numbers with 666 digits okay so that's pretty basic that's not particularly exciting but you can ask are there any that have interesting features so you could ask what about the ones that are prime numbers so the prime apocalypses okay so again you can just write nice simple computer code that can find these for you and it's not too difficult so let me just tell you about the about them and we'll see if we can read anything into them we'll play our silly game again okay so what's the first apocalypse Prime so imagine the first number that is an apocalypse number will be one with 665 zeros after it I'm going to guess I'm going to hope that the first one is one 664 zeros and then a one no a it's not Su no no it's not it's not it's that's composite yeah yeah yeah what is it it's this one so it's one with lots and lots of zeros yeah and then it ends with one two three the first one yeah you like the one two three structure there that's nice happy with that I was only out by 122 yeah you were already out by a few that's true so this is the first okay whatever I don't know I just don't know what to make of that don't know what what I'm going to do with that but let's do the next one the next one bit of a jump to the next one not not a massive jump the next one ends with 1837 okay so 1837 what's that got to do that sounds like a date should we have maybe that's a date right yeah okay what Daye 1837 what happened in 1837 well I looked it off a few things happened there was a big earthquake in Syria lots of people died devastating earthquake in Syria Queen Victoria ascended to the throne that's a significant world event New York was kicking off left right and Center there were loads of riots in New York City basically driven by poverty the banks were failing as well so that happens I'm not sure this is a good candidate for the apocalypse to be honest on balance so the next one after this is well this okay that's the next prime Apocalypse Now that's the future right so maybe maybe maybe we should worried about that date who knows right okay so that's when Everton will next win the league that wouldn't be I think that's much later that's much l that's not going to happen yeah there was 19 85 was the last time they want anything and then nothing nothing for a long time after okay so how many apocalypse primes are there in this in this span yeah so there's probably around so the way you can you can ask this is you can ask um you know about the prime counting function which is which is basically how many so the prime counting function Pi of n basically tells you how many prime numbers are smaller than n essentially how need to go up to n and you know how this behaves for large n because it's known that this goes like n over log n so you can use this Prime counting function to work out how manyy are between uh 667 digits and 6 and and 6 and the last 665 digit if you like so if you do that estimate um you find there's going to be about by my count estimate about 5.9 * 10 662 prime numbers that have 666 digits so there should be about this many ocalypse primes that's a lot I would have thought when numbers got that big primes were going to be more sparse than that well and they do PR numbers they do but but the problem is there's so many of you're talking a huge range there's a lot of 666 digit numbers there's an awful lot of them and so yeah actually whilst it does get you know density does go down there's still a lot of them because you're covering a large range it's all a bit unsatisfactory I think this I think because there are a lot agree don't like there's a lot so actually what pickover is looking for he looks for Fibonacci numbers that are apocalypse so Fibonacci apocalypse don't this now that's a slightly different question so if you remember what the Fibonacci sequence is so Fibonacci sequence is is something like the following so you start off with zero and one take the two two numbers before and add them together to get the next one so I take the two numbers before zero and one add them together one right now I take the two numbers before which is one and one add them together I get two two numbers before it's one and two add them together I get three and you can carry on right build a Fibonacci Sequence at some point you would expect maybe you would or maybe you wouldn't that I'm going to start getting 666 digit numbers in the Fibonacci sequence now where are they okay so again you write a computer code you find them okay so that's why I did um although some of these were already known and it turns out I would have thought there weren't many no there's not there's not there's only a handful right but they do exist so the if we call sort of the you know the nth Fibonacci number FN then it turns out the this one is apocalypse so this this is the 3,184 Fibonacci number that has 666 digits in fact the next one does as well that one does as well in fact you if you go up to 31 88 that one does as well so all those do but the next one won't so there are five yeah yeah yeah so there's five of them yeah exactly that's it and also maybe these are years I can't resist it I can't resist it and none of these are prime no so um so that's that's another feature to it so there are five 666 digit long Fibonacci numbers exactly I did not know that before today yeah and of course the Fibonacci numbers you know they're kind of whilst primes are interesting Fibonacci seems to have something to do with nature it seems you know the Fibonacci does show up a lot in nature you know we know that that the the growth of of sort of biological systems seems to recognize the Fibonacci sequence we know that you see the the golden ratio in nature um and of course the Fibonacci sequence the the ratio the numbers tend to the golden ratio so who knows maybe there's something in this ridiculous love it love it man that's brilliant I love that I love that stuff I could talk about this all day ever write down plus one okay you couldn't do something like that because obviously you know you're going to get caught into logical Loops it's it's going to be a nightmare and it's considered ungentlemanly okay so you couldn't do that the room was packed and R went first now what R did was he wrote down a series of about 30 or 40 Wes so it seems like a large number