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Circular Prime Numbers
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0:01
19,95 can you imagine why that special 19,95 any guesses I saw the heading on your cheat yeah why why what what what's so special about it is it a circular Prime oh yeah it is a circular Prime but what is a circular Prime 19937 is a prime number but so is
0:27
9,371 and so is 9
0:33
3,719 where I've rotated the number again and so is 37,1 199 that is a prime number as well and so is
0:50
71,9 93 all those rotations the all prime numbers that is a circular Prime
1:03
yeah it depends on the base we're just doing it for fun but uh let's see if we can find some more circular primes are there an infinite number of circular primes who knows who knows I know let's find out these are the boring circular primes two three five seven boring cuz they're one digit you rotate them it's the same number right fine here there are some other boring ones uh 11 is quite boring too rotate it it's 11 again oh right you would get the same effect
1:32
if we had 22 33 44 but those aren't primes are they cuz 22 divisible by 2 33 divisible by three right so 11's quite boring 111 what about that it would be boring but it's not a prime yeah so 111 is 3 * 37 so okay so not a prime fine so what are the boring primes that it's just ones I guess uh the next boring one it has 19 ones it's one quintilian 111 quadrillion 111 trillion 111 billion 111 million 111,111 that's not a boring number but it is a boring circular Prime exactly
2:20
it's oh it's not a boring number I totally agree that's actually really interesting but for what we're doing today which is circular primes it's a boring circular Prime so that's called a rep unit cuz you're repeating one which is another word for unit right rep unit the next rep unit Prime after that has 23 ones the next rep unit Prime after that has 317 ones the next rep unit Prime after that has 1,031 ones and so on we've only found 11 rep unit PR primes it's possible there's
3:01
infinitely many of them we don't know that actually that's not a proven thing we found 11 of them a rip unit I know coincidentally until we find the next one but for circular primes boring let's look at some less boring circular primes we want some more digits please some other digits well we can't have a circular Prime with a digit two in it because at some point that number is going to get rotated and that two is going to get rot rotated to the end and that will be an even number for the same
3:36
reason you can't have a four in it because at some point it will get rotated to the end 6 80 will also be the same because it will get rotated to the end at some point five is also not allowed because it'll get rotated to the end it will be divisible by five which means circular primes can only have 1 3 7 and nine in it so let's try and find some twood digigit circular primes I mean we have the 11 then there's 13 which is a prime 31 though is a prime as well we also have H 17 which is prime
4:15
rotate it 71 is a prime as well 37 is Prime and so is 73 and 79 is a prime if you rotate it 97 is a prime as well those are the twod digigit circular primes I thought 19 was going to be on the list is 91 composite must be must be it's divisible by seven if only we had a test to work out if 91 was divisible by seven but there isn't one so we'll move on all right three digigit I mean where does this end yeah so let's look at the three-digit circular primes uh I'll just
4:47
write them out I'll write them out in their lowest terms 113 which means those rotations are going to be prime
4:58
197 19
5:03
9 337 that's all the three digits they're getting rarer I thought they'd be loads more yeah so not only does your three-digit number have to be prime you have to think of those rotations as well they have to be prime as well it turns out these numbers are going to get rarer ah so if you think of bigger numbers not only has it to be prime which is a rare thing in itself then all those rotations have to be prime so and you're living in a world where primes are becoming more
5:35
sparse exactly exactly so the bigger these numbers are getting the harder it is to satisfy this condition they are getting rarer we think there's an end to this list oh 1,193 and its rotations
5:55
3,779 we've got 11,900 39 19,99 37 that was our one at the start we've got
6:11
193,194
6:22
so how big is our list of circular primes I've got 1 2 3 4 5 9 10 13 15 2 3 4 19 plus the Reps plus those rep units yes which possibly infinitely many but boring I'm amazed I thought I thought you were going to be pulling all sorts of huge numbers at me like phone numbers and things you know so the biggest one would have to be our last one here but rotated because that was written in its lowest terms 99,354
7:01
so we haven't found any bigger than this uh yeah I picked 19,95 CU I thought it was the biggest interesting one because it uses all the digits the one the three the nine the seven CU if you look at the bigger ones more repeated digits in those so that's why I picked my 19,95 so it appears to be a a finite list and again just like we was saying it just it just gets hard harder to find the circular Prime so with bigger numbers it just becomes a harder condition to satisfy so these are all
7:36
the ones we found so far what have you got to say that all those people who are in the comments right now saying base 10 yeah it is base 10 oh it's just for fun so you could do this in any base you could do it in binary you can do it in other things it would be just for fun it is not a property of the number itself but also that's what we do love it we can go one step further so these were circular primes I was just rotating the digits what about if I just mix up the
8:04
digits in any order and still be prime can I do that so what are they like permutational primes or so they they sometimes called permutation primes and they're sometimes called absolute primes which I prefer oh yeah yeah absolute prime so whatever you do to it however you mix up those digits it's still a prime what about jumble primes or jumble primes is not yeah I like it but you know I know I know you love naming things cuz absolute doesn't quite do what it says on the 10
8:37
yeah yeah I know you like naming things Brady jumble Prime I I'll expect I'll expect Wikipedia to be updated very soon but absolute primes have to be circular primes because we're alled to Jumble them up anyway which means they have to be circular PES the rotations have to work so the only come from our circular Prime list H which means that not going to be very many of them no is there is there other than other than the Reps yeah other than the rep unit other
9:10
than the rep units oh it's exhausting can we just can we just put an asteris up and just say other than the rep unit primes and the twood digit primes obviously exactly so they again the boring ones single digit ones would count as absolute primes boring rep unit Primes count boring the two digit primes are kind of boring too because the only jumble as you said is the circular Prime it's this rotation um so it's only the bigger ones that are interesting and here they are 113
9:52
199 and 337 [Music] that is it which I guess shows you how hard a condition it is to satisfy so that's it so the biggest one would be 337 or if we jumble it up 733 and any order of that would be a prime although because of that repetition the because of the two threes in it there aren't really that many there's there's 337 373 and 733 and none of those none of those three that are interesting don't have a double digit in them yeah and that's part of it as well so it turns out that
10:37
we're thinking again we think this is all there is but let's say there's a big one somewhere out there that we we just don't know we found out we have worked out that if it does exist it will have the form it will be nearly a rep unit number it's going to be a number of this form a a a where a some digit a a lots of A's and a b at the end a can't be anywhere other the way so we've proven that to be true where A and B are either 1 3 7 or nine so it's nearly a rep unit
11:16
Prime because that's how hard this condition is to satisfy how did they prove that that's amazing yeah I mean this might not exist I mean we think it doesn't exist if it does exist it would be of that form and we know that it would have to be very big this number whatever it is would have 6 * 10^ 175 digits that's not the number itself that's how many digits it would have that's the length of the number or bigger right that's how many A's it would have and then a b yeah that's how
11:48
many A's all those A's then a b at the end so the number itself is like the B could be anywhere cuz it yeah and the B basically it's getting sort of rotated through so it's a circular Prime I guess cuz it's just effectively getting rotated through and so uh yeah the number itself would be like a Google Plex kind of number right just a massive number and it just seems very unlikely if it happens wow amazing but it just seems really unlikely we need a bounty or a prize to the person who finds the
12:18
biggest ever jumble Prime well you can offer a bounty and tell you what I am so confident let's put this on video yeah why what could go wrong I am so conf confident I reckon our 337 is going to be our biggest one so uh I will offer a prize what shall my prize be I'm not a rich man um a personal one-on-one dinner with Dr James gr yeah exactly yeah brilliant yeah personal and you and you'll pay sure sure why not so that's the prize on off of that is the Bounty find a bigger absolute prime jumbo
12:59
Prime find a bigger absolute prime a dinner for two on me that's your prize wow what's this I see on brilliant a course of advanced number puzzles well yes please no matter what you're into or how advanced you are brilliant's always going to have something to stimulate educate and entertain it could be math computer science chemistry physics you name it but I do think number five viewers are going to particularly love these ones and here's some other stuff from
13:32
across the site it's fun it's clever it's always really interactive great way to learn could spend ages doing this you can try everything on brilliant for free for a full 30 Days by going to brilliant.org snum file there's also a link in the description and of course the QR code there on the screen there's also a superb offer of 20% off an annual premium subscription by using that link check it out people and now thanks to brilliant for their support of number
14:04
five deal with the categories one at a time I was so pleased when I got to the end this I was like yep I knew it had something to do with being one or more or one less than a multiple of six I did some Algebra I worked it out I showed some friends of mine and one of them said why didn't you do it the easy way and I was like ah you know me I like to do some what easy ways
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