Mathematicians explains Fermat's Last Theorem | Edward Frenkel and Lex Fridman

would it be useful to maybe try to explain from Oz Last Theorem sure it's easy to do [Laughter] I'm an optimist I think I always think that everything can be explained you know even though I say that not everything can be explained but in in mathematics you know in within this particular framework I think that I always feel optimistic when people ask me to explain something I always start with the assumption that they will understand yes you know so let's try ethereum one of the jewels sort of of mathematics of all time a beautiful story also behind it Pierre Verma a great French mathematician who lived in the beginning of mostly worked at the beginning of about 17th century and uh he actually has to his credit a number of important contributions but the most famous is called firma's last ethereum or return was a great ethereum and the reason why it became so famous is in part because he actually claimed to have proved it himself and he did it on a margin of a of a book that he was reading which was a actually important book by diafantos about equations with coefficients and whole numbers and he wrote on the margin literally I this equation you know this problem which I will explain in a moment um I have solved it I have found a proof but this margin is too small to contain it at some point I I give I was given a public talk about this and I made it as a joke I made a tweet in which I wrote that I have proved this theorem but 280 characters are not enough and the kind of customer admit sentences yeah so this was 17th century Twitter style proof okay but a lot of mathematicians took it seriously because he had great credibility he did did made did make some major contributions and the search was on so for 350 years about 350 years it remained unproved with many people trying and failing until in 1994 no in 1993 Andrew Wiles announced a mathematician from Princeton University announced the proof and it was very exciting because he was one of the top number theorists in the world and unfortunately about a year later a gap was found so that's exactly what we were talking about earlier you have 99 of the proof this one little thing does not quite connect and this nullifies the whole thing even though well you could say there's some interesting ideas but it's not the same as actually having a proof so he apparently was really frustrated and he was really a lot of people thought that it's going to be another 100 years or whatever and then luckily he was able to enlist with with the help assistance with of his a former student also great number theorist Richard Taylor they were able to do that one percent so to speak well some people might say it maybe not one but five percent or whatever but it definitely was an important ingredient but it was not he had a sort of like a big new set of ideas and this one thing didn't pan out they were able to close it with Taylor and it finally was published and I think was accepted and refereed in 95 and since in is believed to be correct now uh what he proved actually was not verma's theorem itself but a certain statement which is called shimurtaniam away conjecture named after three mathematicians to Japanese mathematicians and one French porn mathematician who works and also at The Institute for advanced studying in in in Princeton and it was my colleague at UC Berkeley Ken ribbett who in the 80s connected the two problems so this is how it often Works in mathematics you want to prove statement a instead you prove that a is equivalent to B so after that if you can prove B this would automatically imply that a is correct so this is what happened here a was forma's Last Theorem B was simultaneous conjecture and that's what Andrew Wiles and Richard Taylor really proved so it requires to get to firma's Last Theorem it requires that bridge which was established by my colleague Ken Ribbit at UC Burke so now what is the statement of hermoslav's theorem um let me start with Pythagoras since we already talked about it let me start with Pythagoras Theorem which describes uh the right triangles so what is the right triangle is a triangle in which one of the angles is 90 degrees like this so it has three sides the longer side is called hypotenuse and then there's two other sides so if we denote the high points the lengths of hypotenuse by Z and the two other sides X and Y then Z squared is equal to x squared plus y squared so that's the equation or x squared plus y squared equals z squared and it turns out this equation has Solutions in natural numbers main is actually infinitely many solutions in natural numbers for example if x is you take x equals 3 y equals 4 and Z equals 5. then they solve this equation because 3 3 squared is nine four squared is 16. 9 plus 16 25 and that's 5 squared so x squared plus y squared equals z squared is solved by x equals three y equals four is equals five and there are many other Solutions of that nature and we should say that natural numbers are whole numbers that are non-negative that's right one two three four five six and so on now what's sperma's Last Theorem Firma asked what about what will happen if we replace squares by cubes for example so X Cube plus y Cube equals z Cube are there any solutions in what do you call natural numbers it turns out there are none what about fourth powers again none or it seems like none right so that was the that was the statement so the theorem says that the equation x cubed plus y Cube equals e cubed has no Solutions in natural numbers uh remember natural means positive whole numbers so of course there is a trivial solution zero zero zeros so that this works but you need all of them to be positive x to the fourth plus y to the fourth equals z to the fourth also has no Solutions uh x to the fifth plus y to the fifth equals z to the fifth no Solutions so you kind of see the trend x to the N plus y to the N plus equals z to the N if n is greater than two has no Solutions in natural numbers that is a statement of therma's last year deceptively simple as far as famous theorems are concerned you don't need to know anything beyond standard arithmetic addition and multiplication of natural numbers that's why a lot of people uh both um Specialists and amateurs try to prove it so because it's so easy so telling so easy to formulate so in fact I think Verma proved the case of Cubes I think he did actually prove some elsewhere in the case of Cubes but so it remained like force there are infinitely many cases right you have to even if you prove it for cubes and for fourth power and fifth then still there is six sevens and so on the infinitely many cases in which it has to be proved and so you see um the separate simple result took 350 years to prove and you know but in a sense it's like mathematicians you know you would think mathematics is such a sterile profession everybody's so serious you know almost like we're all wearing like lab coats and like take an elevator to the to the every Tower and however look at all this drama look at all these dramas like we all select drama we also have narratives we also have our myths here is a guy is the 16th century mathematician or 17th century mathematician who lives a note on the margin and motivates others to find the proof then how many cards were broken that they believed that they found the proof and then later it was realized that the proof was incorrect and so on and brings us to modern day and one last attempt and reviles who is very serious and respected and esteemed mathematician announces the proof only to be faced with the same reality of his hopes dashed seemingly dashed and like there is a mistake it doesn't work and then to be able to recover a year later how much drama in this one story it's amazing but from what you understand for what you know what was the process for him that uh that is similar perhaps to your own life of of of of walking along with the problem for months not years yes so he worked he has given interviews about it afterwards so we know that he described his process that number one he did not want to tell anybody because he was afraid that people find out that he's working on it because he's was such a top level mathematician people would guess that he has some idea that there is some idea so you know if you just know that somebody's has an idea this already gives you a great boost of confidence right so he didn't want people to have that information so he didn't tell anybody that he was working on it number one be lonely number two he worked on it for seven years if I remember correctly by himself and then then he he thought he had it and he was elated obviously he was you know very happy and he announced it at the conference I think it was in Cambridge University or Oxford University in the UK in 1993 I believe so you know this is really interesting because all of us we can really all matters can relate to this because I I remember very well my first problem how I solved my first problem uh I described it in in love and math in my book so uh it was I was how old was I I was 18 years old I was a student in Moscow and I was just I just lucked out that I was introduced to this great mathematician since I you know I was not studying at Moscow University because I find semitism in the Soviet Union so I was in this technical school but I was lucky that I had a mathematician who took me under his wing and Dmitry Fox who actually later came to the to the US and he's still a professor at UC Davis not so far from from me uh so he gave me this problem and it was rather technical so I will not try to describe it but I do remember how much uh effort you know the that excitement but also kind of a fear what if I don't have what it takes you know I lost sleep so this was one one consequence of this I for the first time in my life I had trouble falling asleep and this guy actually stayed for a couple years afterwards so then it was kind of like a wake-up call that I should be to take care of myself not work too late and so on so that was sort of like that experience and I was lucky that I was able to find a solution number one within two months Maybe and it was very it was surprising and it was beautiful like it the answer was was in terms of some something which seemed to be from A Different World from a different area of mathematics so I was very happy but I do remember this this moment when suddenly you see that just like you in this case it was literally I had to compile these diagrams with what mathematicians call Co homology groups and spectral sequences and manually calculate some some numbers and trying to discern some system in it and suddenly I saw that how they all were governed by this one uh one one force so to speak one pattern and that was absolutely wow so it's like I mean what was it so you're sitting there at a desk actually you know I lived in a town outside of Moscow so I used to take I would take a train to Moscow so it's what we call in Russia electric you know like this electric train which was super slow it took more than two hours to cover that distance and I think that the crucial Insight came when I was in this and I just I was I had to contain myself so I don't start screaming you know because there were other passengers in the car so I was sitting there and staring at this paper so you know what I remember that's what came to me I have something now which nobody else in the world has I have a proof of first of all I did not it was not just the proof like in the case of Irma the statement is already made that's why it's called conjecture you know you make a statement you don't have a proof yet then you try to prove it in my case I did not know what the answer would be there was a type of question where the answer was unknown so I had to find the answer and prove it and the answer was very nice so nobody knew as far as I could tell nobody knew because my teacher told me that he explored all the literature and this was not known so this was suddenly I felt that I was in possession of this now it was a little thing it was not cured for cancer you know it was not a large language model you know but it was something undeniably real meaningful and it was mine kind of you know like I had it nobody else I cannot published it I didn't even tell I hadn't even told anybody and it is a very strange feeling you know to have to have that were you worried that this treasure could be stolen not at the time not at the time so later on there were situations where I was exposed to that those type of experiences but at that time I didn't think of that I was still this starry-eyed kid you know who was just obsessed with mathematics with his Beauty and Discovery discovering those beautiful facts beautiful results so I didn't think about I didn't even think that it could be possible that somebody could steal it or whatever uh I just wanted to share it with my teacher as soon as possible you know and and he he understood quickly and he's like yeah good job you know

Channel: Lex Clips

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Keywords: ai, ai clips, ai podcast, ai podcast clips, artificial intelligence, artificial intelligence podcast, computer science, consciousness, deep learning, edward frenkel, einstein, elon musk, engineering, friedman, joe rogan, lex ai, lex clips, lex fridman, lex fridman podcast, lex friedman, lex mit, lex podcast, machine learning, math, math podcast, mathematics, mit ai, philosophy, physics, physics podcast, science, tech, tech podcast, technology, turing

Id: KW20tDDImP4

Length: 15min 14sec (914 seconds)

Published: Wed Apr 12 2023