Primes on the Moon (Lunar Arithmetic) - Numberphile

this started and my friend mark Lebrun who lives in California here's a computer scientist he likes to consider unusual ways of doing arithmetic and he came up with this idea you know that in school children have a lot of trouble when they learn arithmetic all this there's all this trouble with the Carrey's it's very complicated so he said let's let's make it simple for the kiddies we're not gonna have any caries we're gonna abolish all caries we're gonna change the rules of it now when you add two digits to numbers like 7 & 8 the result is not 15 anymore on the moon it's 8 you pick the bigger of the two numbers so 1 plus 1 is 1 sure 1 plus 2 is 2 when you multiply you pick the smaller so 1 times 2 is 1 and that's it those are the rules that's lunar arithmetic so 3 times 8 is 3 3 times 8 is 3 3 plus 8 is 8 you take the bigger as the two actually maybe they didn't maybe they will let's do some realistic let's do an addition for example supposing we wanted to add say 58 and 19 what you do is you say 8 plus 9 is no remember you take the bigger of the two and there are no carries a plus 9 is no 1 plus 5 is 5 you take the bigger the - bigger of 1 & 5 is 5 the biggest of 9 + 8 is 9 so 58 plus 19 is equal to 59 that's lunar addition what about multiplication suppose we want to multiply say 17 by 24 remember digit by digit so four sevens are four very good you pick the smaller all ones are one and now two sevens two sevens are two two ones are one and now we add them up one one plus two is two you pick the bigger of the two and you bring down the four so 17 times 24 is equal to 124 so now you can do a lot of arithmetic in lunar arithmetic and you can produce a lot an interesting number sequences 35 and a half minutes of PLSS time expanded now what are the squares what's 1 times 1 is 1 2 times 2 is equal to pick the smaller it's 2 3 times 3 3 squared is 3 6 squared is 6 7 squared is 7 8 squared is 8 9 squared is 9 what about 10 squared 10 times 10 we have to do it zero times zero you pick the smaller 0 times 1 is 0 1 times 0 is 0 1 times minus 1 you add them up 1 0 plus 0 pick the bigger is it and we get a hundred just like back on earth but what about 11 squared well we have to do it 1 1 times 1 1 we get 1 1 1 1 add them up 1 1 plus one is one what so 11 squared is 1 1 1 111 12 squared 2 times 2 is is 2 2 times 1 is 1 1 times 2 is 1 1 times 1 is 1 always pick the smaller and we add them up 1 1 plus 1 is 1 2 is 2 so 12 squared is 112 and so on 19 squared is 9 and so on so that's the sequence of lunar squares yes we love you don't here we'd like you to aim it a little bit more to the right or if you are interested in the formal details this is an associative commutative distributive law so it works we don't have subtraction we don't have division but apart from that we can do anything what I would like to tell you about next are the lunar primes what are the prime numbers on the moon you're really gonna like this and I need more paper what are the lunar probes well what is a prime first of all we have to decide what it means for a number to be a prime now you remember back on earth 7 was a prime because you couldn't write 7 as 2 times 3 or anything except 1 times 7 if the only way to factorize it is as 1 times itself and it's a prime number on the moon is 7 a prime on the moon well notice 7 is equal to 8 times 7 it because 8 times 7 you pick the smaller of the two which is 7 so 7 is 8 times 7 so 7 is not a prime because it has a factorization but really before we get down to the details of this we have to decide what precisely it means to be a prime so formally and numbers of prime if the only factorization is 1 times itself what does one mean on the moon we have to say what is 1 if I put it in quotes on moon well what is 1 1 has the property that if you multiply anything by 1 you get what you started with 1 times 3 is 3 1 times 111 is 111 right on the moon that's not true 1 times 3 on the moon is not 3 it's 1 1 times I well one time so 111 happens to be 111 by accident that well 1 on the moon 1 times anything is basically all ones and zeros 1 times 30 is ten because you go through a digit by digit and pick the smaller so one on the moon is not one what is one on the moon would you care to make a guess what's the number is that I haven't thought about it at nine it's not very good fantastic brilliant because when you multiply something by nine it just copies what it sees nine times seven two seven nine times one is one just copies what it sees so on the moon nine is the one so on the moon a prime number has the property that the only way you can factorize it is nine times itself so seven is obviously not a prime because seven is equal to 8 times 7 what about 8 8 is not a prime because an 8 is equal to 8 times 8 so start a prime it has a factorization in which one of the fact neither factor is 9 any number if for it to be a prime the biggest digit has to be a 9 so 25 is not a prime because we could multiply it by 5 and get what we started with so we thought I have a 9 in the number so 19 is the smallest we must have a 9 there's no way to build 19 except what 19 times 9 right you can't factor is because it's got a 9 in it so if you know if it was 18 we can multiply 18 by an 8 and get 18 we'd have a factorization so you can always multiply by the biggest digit and get what you started with so the big biggest digit had better be 9 these Prime's are going to be rare than yes but not that all that rare so yeah how rare are they well let's look at the prime numbers so what about 9 itself so back on earth we don't say 1 as a prime because it's we wouldn't have unique factorization 2 is the smallest prime but on the moon just as one isn't a prime on on earth 9 isn't a prime but apart from that we get 19 so the primes the lunar primes 1929 39 49 59 69 79 89 90 is a prime 91 as long as there's a nine and there's no obvious way to factorize it you have to check that 90 is a prime we can't get 90 by multiplying two numbers you have to check it but it's true it's easy to check 99 we get 109 yes that's a prime but 119 is not because as we saw a moment ago 19 times 19 yes 119 so 1 1 9 is not a prime but roughly speaking most members that have a 9 in them the primes so those are the lunar primes if we were to go to the oais and we're in the search window we type lunar primes we used to call them dismal primes because teaching of arithmetic in schools were so dismal it's a dismal definition I think we like lunar primes much better so well I can show you the graph or two graphs one of the first 200 terms technically it's called a pimp lot because we draw little pins you don't see the pins there because they're quite dense then you get a scatterplot justice plane graph of however many terms we have and it looks like we're getting about what is it two thousand five hundred terms there's one more thing I would like to show you I'd like to prove that they're infinitely many primes you remember infinitely many lunar primes first of all our prove that 109 is Breyer and the proof of that is suppose it wasn't it would have to be the product of two two-digit numbers they would have to be two digit numbers because of the rules so it would have to be a B times C D but we've got to get a one at the beginning so one of a and C remember how it works the table as the multiplication the tableau you get one at the beginning either a or C has one-and-two get a nine at the end both B and D have to be nines that's the other only possibility because when we do the multiplication 9 times 9 is 9 9 times 1 well that's that's 1 pick the smaller of the two okay now we do the C C times 9 is well it's C whatever C is multiplying it by and 9 doesn't change it and C times 1 well C can't be 0 so it's gonna be 1 now we add them up and we get 1 1 plus C I don't know what it is it's the bigger of 1 and C well it's gonna be C it could be 1 anyway what we get is the number 1 c9 but C can't be 0 because if C was 0 we wouldn't get the 1 in front so this can't happen this is a contradiction if C was equal to 0 we wouldn't get the 1 in front we cannot factorize 109 as a product of two primes and then likewise 1 0 0 0 0 9 is prime by the same argument there are infinitely many Prime's yeah no matter how many you can put as many zeros as many zeros as you want and you get a prime nice proof yeah yeah nice no Luna Prime's coming down the ladder now are the ones that the mathematicians like these are the primes or numbers that are related to primes and they are they are elevated above the ordinary numbers 0 1 5 27 one nine four one eight six five and so on

Channel: Numberphile

Views: 267,730

Rating: 4.511672 out of 5

Keywords: numberphile, moon, lunar arithmetic, squares, prime numbers

Id: cZkGeR9CWbk

Length: 12min 58sec (778 seconds)

Published: Wed Nov 07 2018

Reddit Comments

[u/Gusfoo]

I have to say I vigourously enjoyed the proof at the end of infinite lunar primes.

Nice proof, prof.