What was Fermat’s “Marvelous" Proof? | Infinite Series

you know the tale about firma's last theorem in 1637 pierre du furma claimed to have the proof of his famous conjecture but as the story goes it was too large to write in the margin of his book yet even after andrew wiles proof more than 300 years later we are still left wondering what proof did firma have in mind the mystery surrounding firma's last theorem may have to do with the way we understand prime numbers you all know what prime numbers are an integer greater than one is called prime if it has exactly two factors one in itself in other words p is prime if whenever you write p as a product of two integers then one of those integers turns out to be one in fact this definition works for negative integers too we simply incorporate negative one but the prime numbers satisfy another definition that maybe you haven't thought about an integer p is prime if whenever p divides a product of two integers then p divides exactly one of those two integers let's call this definition b and let's think about it does it sound plausible here's an example suppose our prime is 3 and notice that 3 divides 12 for instance now look at the different ways 12 can be factored as a product of two numbers what do you see no matter how we write 12 3 always divides one of the two factors you may think that's a silly observation but it does not hold for composite or non-prime numbers for example 4 also divides 12 but 4 does not divide nor does it divide 6. and the idea is that this observation will hold for all multiples of three for example 3 also divides 30 and no matter how you write 30 as a product of two numbers three will always divide one of the factors now 6 also divides 30 but it does not have this property in particular 6 is not prime so this new definition of prime is perfectly valid even though it's not the one that we're so used to so you might wonder why don't we ever hear about definition b is it because these two definitions are actually conveying the same concept in other words is every integer that's prime in the sense of a also b and conversely is every integer that's prime in the sense of b also prime in the sense of a it turns out the no not always that is the answer is yes if you're working with the integers in fact i encourage you to get out pen and paper pause the video and prove that an integer satisfies definition a if and only if it satisfies definition b however and here's where it gets interesting if we replace the integers by a different number system a system where we can still add and multiply and factor things just like we do with integers but where those things aren't necessarily integers then it is not always true that these two definitions coincide to see why let's look at an example let's replace the integers by a different number system what exactly well in gabe's episode beyond the golden ratio he explained how fee the golden ratio is just one of a family of metallic means but the golden ratio also lives in a different family fee is the number one half plus one-half times the square root of five but what about other numbers of the form plus a fraction times the square root of five there are infinitely many numbers of this form and the golden ratio is just one of them the set of these numbers form what's called a quadratic field which plays an important role in algebraic number theory but for the rest of the episode let's just focus on the case when a and b are integers collectively we'll denote these numbers by z adjoin square root of 5. now the nice thing is that we can add and multiply these numbers together for example to add 1 plus 2 root 5 and negative 4 plus 3 root 5 just add the integer parts and the square root parts together so their sum is negative 3 plus 5 root 5 and we can also multiply them together we'll just use the familiar distributive law which some folks like to call foil so their product is 26 minus 5 root 5. moreover zia joined root 5 also has prime numbers given by definitions a and but because we replace the integers with z adjoin root 5 we need to modify definition a little the reason is that z adjoined root 5 may contain numbers

that behave like the number 1 even though they aren't the number 1. i'll explain here's the new definition a number p and z adjoin root 5 is prime if whenever you write p as a product of two numbers then one of them is a unit is a word that means has a multiplicative inverse that is a number u is a unit if there exists some other number v so that u times v is 1. for example in the usual integers 3 is not a unit it does not have a multiplicative inverse okay yes 3 times a third is equal to 1 but 1 3 is not an integer so that doesn't count in fact the only units in z are 1 and negative 1 and that's why unit is a good generalization of the number one okay so we have two definitions a and b if we work with the integers then these two definitions coincide but now i claim that because we are working in zia joined root 5 they do not coincide in particular the number 2 is prime by definition a but not prime by definition b first let's see why two is not prime according to definition b notice that four can be written as two times two but it can also be written as one plus times negative one plus root five this means that 2 divides the product but 2 does not divide either factor 1 plus or a negative 1 plus root 5. in other words and you can verify there are no integers a and b so that one plus root five equals two times a plus b root five similarly if you replace one by negative one this shows that two is not prime according to definition b however it is prime by definition a why i'll let you work that one out it's a little trickier but not too much i recommend using a proof by contradiction along with something called a norm i won't go into the computations now but if you're interested check out the references below alright let's summarize we have two definitions a and b when working with the integers these definitions imply each other but in z join root 5 they do not why the reason is because the integers possess a very special property that z adjoined root 5 does not have before i tell you what that property is let me just say that this overall discussion is a part of something called ring theory the study of rings but not this kind of ring a ring is a mathematical object a set of elements that behave a lot like integers even though they may not be and z adjoined root 5 is one such example the neat thing is that once you have a ring you have enough mathematical structure to talk about primality in particular our two definitions a and b have technical names in ring theory an element in a ring is called irreducible if it satisfies definition a and it's called prime b so earlier we saw that 2 is irreducible and z joined root 5 but it is not prime now here's the punch line primality and irreducibility will coincide if and only if your ring has a very special property and the integers have that property what is it the fundamental theorem of arithmetic namely that every integer has a unique factorization into a product of primes more generally if you're looking for a buzzword the integers form a unique factorization domain or ufd and according to abstract ring theory irreducible and prime are equivalent concepts if and only if your ring is a ufd specifically if each element can be uniquely written as a product of irreducible elements what's interesting is that not all rings are ufd's and this brings us back to fermat's last theorem the absence of unique factorization is precisely why one of the many attempts to prove fermat's last theorem wasn't successful in 1847 french mathematician gabrielle lemay thought he had proved fermos conjecture by factoring an expression like this which occurred in the ring z joint alpha which is not a unique factorization domain and so his technique didn't work fortunately having a faulty proof isn't always a bad thing in fact the lack of unique factorization was spotted a few years earlier in a different setting by german

mathematician edward coomer who introduced what he called ideal numbers precisely to get around the issue in short the discovery that not all number systems or rings have an analog to the fundamental theorem of arithmetic set the stage for more than a century's worth of brand new mathematics which then led to andrew weil's proof of ferma's last theorem in 1993. so what proof did fermat actually have in mind when he wrote in his margin well i'm not a historian but it's very possible that he assumed that properties of the integers like unique factorization and the equivalence between prime and irreducible will always hold just like lemay thought but as we saw today things aren't always what they seem if you'd like to learn more about the ideas discussed in today's episode be sure to check out the links below see you next time i'd like to start by thanking everyone who supports us on patreon amounts big and small help keep the lights on here at pbs infinite series we genuinely really rely on your support to help the program keep going and we'd particularly like to thank roman pinchuk who is our first converse level supporter extremely generous i don't know what to say but thanks and now let me get to some of your comments from our earlier episodes on the piano axioms and the construction of the natural numbers using sets first off i want to shout out robert lowe from the uk who we talked on twitter i believe he's a viewer but he wrote a blog post about very similar information recently the piano axioms and the formulation of the natural numbers in sets and so forth he and i had a chat about whether you should start the natural numbers at zero which is a big controversy anyway his blog is very good and the post is excellent it's a good supplementary take on the material that we did here the link to it is in the description you should go check it out gerard 10 made the highly upvoted comment that for someone who was saying let's not talk about numbers i was numbering the axioms one two three and so forth as many people pointed out in their replies to gerard those labels are arbitrary i could have called them cheese cocoa puffs and frankenstein who cares but point taken touche john lang or lang pointed out the similarity between the piano axiom formulation of the naturals and the lambda calculus and other people also mentioned explicitly the church numerals the relationship between all this stuff is no coincidence this is how the numbers can be formulated from a abstract computer science perspective as well i encourage you guys to google that stuff lambda calculus church numeral on a somewhat related note jesse myez or mace pointed us toward the book software foundations by pierce i checked it out looks like it's a good resource i added a link to the author's website down in the description of those earlier videos okay a lot of you including john lucabasso nar91 and joey bove feistauer hope that's pronounced right brought up that there are non-standard models of the natural numbers that can be extracted from the piano axioms at least in the standard first order logic formulation of the axioms but as andrew keppert pointed out the version of the axiom of induction that i gave is the second order logic formulation where you're quantifying over uh sets and subsets and so forth so for people who don't know the distinctions between first and second order and higher order logic i assume that's the majority of the audience don't worry about that conversation but for the rest of you the reason that i tried to sidestep this and why i made the disclaimer that i made in the second episode which maybe i should have made earlier in the first episode was because i wanted to make this as accessible as possible just to give people an introduction to these ideas without having to get hung up on the nuances of first order versus second order logic formulations those are the reasons i made those choices and tried my best mibam or mabame asked whether it's a good analogy to think of the von neumann or zermelo set theoretic constructions as an implementation this is a computer analogy and to think of the piano axioms themselves as just an interface to the natural numbers i think that's a great analogy craig tyle pointed out that in my little philosophical aside at the end of the second episode that i probably should have referenced paul banasarath the philosopher of math from princeton who championed the structuralist view of mathematics uh you're right and this just goes to show how weak my philosophy foo has become that i'd forgotten about bonastrap i haven't read him in over a decade but you're right i mean these are not original ideas on my part they've been championed elsewhere in the literature good call finally david giles said that gabe should be given a million bucks to do this youtube stuff full time i agree now i already get a huge pile of cash which is why my clothes look so nice but another million wouldn't hurt so elon musk you watching no i'm saying