abc Conjecture - Numberphile

and the proof has been announced of a unsolved so far conjecture called the abc conjecture if this proof is right then it's going to be news on the scale of fermat's last theorem was in the 90s which was this big unsolved problem it was this huge event so it's really exciting we don't know if this proof is right yet so a japanese mathematician called matizuki has released these papers and all together it's 500 pages long he's been working on this for a very long time he's come up with his own theory of maths a whole new body of maths and he's called it inter-universal geometry i know nothing about that very few people do even the experts don't know much about this at the moment so it's going to take a very long time to make sure if the proof is right because they're going to have to learn this whole new theory of mathematics so the abc conjecture involves the most simple formula you can think of it's this a plus b equals c doesn't get much easier than that and that's where it gets its name from the rules are these are whole numbers and they don't share any factors so that means if i can divide a by 2 then i'm not allowed to divide b by two or if i could divide a by three then b is not allowed to be divisible by three then i have to share any factors like that all right let's try an example that works 1024 plus 81 equals 1105 right now let's just check they don't share any factors in fact i picked these on purpose this one is 2 to the power 10 and this one is 3 to the power 4. so they don't share any factors there oh and this one i'll do the same is 5 times 13 times 17. now this is what i want you to notice on the left hand side we've got lots of prime numbers we've got all 10 of them over here and another four loads of them over here on the right hand side you only have three and this is what you tend to see most of the time this is what's normal you get if you get lots of primes on the left you only get a few on the right hand side so this is what the conjecture is about i want to show you one there where it doesn't work three plus 125. that's equal to 128. uh let's just check they don't share any factors then well that's three this is five or cubed and this 128 is 2 to the power 7. now this one is not like the first one i showed you we've only got a few primes on the left but we've got loads more on the right so you've got more on the right than you do on the left that's unusual that's weird so rather than not working like i said earlier it's the unusual example these don't happen so often so the technical way to say this is this times those primes together so i'm going to do that so 2 times 3 times 5 times 13 times 17 and that equals 2 13 260. so it's a big number and it's bigger than the right hand side which was this that's what you that's what normally happens okay so if you do this you get a bigger number right i want to show you this one that i said was unusual if we do the same thing 3 times 5 times 2 that's equal to 30 and that's smaller than 128 so that's the difference so this is unusual this is much smaller than the right hand side this number uh which when you multiply the primes together is called the radical of abc it's called the radical because it is well wicked the abc conjecture is the radical which i told you how to work that out that's this the radical of abc is bigger than the right-hand side i said that was c right that's what you get normally um in fact the conjecture is more than that uh it talks about the powers of that too but there are exceptions and these are the exceptions when k equals one that's the power is one uh there are infinitely many exceptions just like the one i've just shown you there infinitely many even though i said these were the rare ones the unusual ones there are infinitely many of them but if you take a power bigger than one even if it's only a little bit bigger even if it's like a power of one point zero zero zero zero one tiny tiny little bit bigger if it's a bigger than one then you get finite finitely many exceptions and this is a little bit surprising because yes if it's just a little bit bigger than one you get finitely many exceptions you could count them off you could write them down you could say here are all the exceptions for this power and that's unusual that's unexpected now this is the conjecture it's very abstract it's very pure this was made in the 80s this conjecture but if this can be proven what it's going to do is it's going to prove a whole bunch of other stuff had a stroke and that's why it's big news originally they thought that fermat sas theorem which i talked about being solved in the 90s they thought that this was the way to solve it because there is a way that if you can solve this you can solve a version of fermat's theorem it didn't turn out that way because thermostat's theorem was solved first i heard of it um i think before it went around the nerdy blogs and i thought well we could talk about it on number foul but then it's still not been checked so maybe we shouldn't talk about it on numberphile but then when all the blogs started going mad about it i thought so are my askers i mean that's how you probe extra dimensions you that's how you probe the very small