31 and Mersenne Primes - Numberphile

okay so today we're going to talk about meren numbers uh so we have to pick one of our meren numbers to exemplify the category now Brady always makes me do this so I've decided to pick 31 Because by the time you watch this video I will be 31 so it's a nice personal number for me so 31 is our example of a meren prime so a meren prime is a prime number to begin with like 31 31 is a prime number a prime number is as you probably know a number that can only be divided by one and itself U the first few prime numbers are 2 3 5 7 31 is a prime number but it's a special type of prime number because it's one less than a power of two what are the powers of two two you got two uh you got four you've got 8 16 32 and so on okay so that's just 2 * 2 * 2 * 2 * 2 that's the sort of thing none of these are prime numbers but let's have a look at one less well one now if I'm honest we don't say one is prime there's a special reason for that let's have a look at the others three is a prime number and it's one less than the power of two so that counts it's a m Prime Seven is a prime number and it's one less than the power of two fantastic 15 now that's not a prime number so that does not count and 31 there you go that was our example of a meren prime now there aren't many of these meren primes in fact there's only 47 meren primes uh now they start off as three 7 31 the next one after that is 127 so let's have a look at the next couple uh powers of two 64 128 and well this is 63 and that's not prime but this is 127 that is prime so it's one of our meren primes they're called meren primes CU they're named after a French mathematician called Marin meren who was a monk a mathematician and a musician and a fan of alliteration uh his favorite cartoon was Mighty Mouse his favorite film was Mad Max so he was in contact with a lot of other mathematicians around the world and he was trying to make a list of this special type of prime number and that's why they're named after him are they used for are they used for are they used by code Breakers are they used to make better iPads or or are they just a game for smart mathematicians like you uh well prime numbers are obviously very useful to mathematicians they're famous and they're famous for a reason meren primes are a special type of prime number something very special about meren primes they are related to the perfect numbers we've talked about perfect numbers before they were six uh 28 496 8,128 they were known by the ancient Greeks they were given this idea that they were perfect unique M and primes and the perfect numbers are two sides of the same coin let me show you why if I pick a m n Prime let's call it M for M N Prime okay so m is our m n Prime Times it by n + 1 divide by two and that will give you a perfect number let me show you 3 * 4 IDE 2 that's 12/ 2 that's 6 that was the first perfect number okay let's try the next one seven 7 * 8 IDE 2 that's 56 2 that's 28 that's the second perfect number 31 same thing 31 * 32 divide 2 and that will give you the third perfect number 4 9 6 and let's try 127 1 27 * 1 28 and have it it's 8,128 two sides of the same coin right so you can help to find the next meren prime it's called the Great internet meren Prime search uh a great website with an unfortunate acronym it's at mer.org and you can try it yourself download the program it uses collaborative computer power defined the next meren prime why why should I do that why should anyone do that for the glory for the glory because mathematicians will thank you for it uh like the perfect numbers we don't know if there's infinitely many perfect numbers we don't know if there are infinitely many meren primes or not so that's another open question something that maybe hey you could work out what's it going to take to work that out how how will that question ever be answered it's not going to be easy it's not going to be obvious it's going to take a different way of looking at things coming at it from a different angle something that other people haven't tried yet maybe not your usual School maths but something else whatever it is something we haven't tried yet the man or woman who does it what will what will what will it mean for them if a man or woman comes up with that proof what what will happen if you do come up with that proof you will have Glory thrust upon you you will be famous for coming up with such a proof something that has eluded mathematicians for hundreds of years is is it something you can have a hunch about yeah I think technically we would have to be M and Prime agnostic but it's definitely something you have a hunch for mathematics is done that way ma people think mathematics is a dry subject that it's logical but people have intuition they have to pursue their intuition so if you ask me I would say yeah infinitely number of meren primes yeah I would think so I might be wrong