Sophie Germain’s favourite problem - Ana Caraiani

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[00:00:18]All right. Thanks very much to Lucas for organizing this event and for inviting me to give a talk and thanks a lot to all of you for coming. Uh so I'm a mathematician working in the area of number theory at Imperial College London and I first learned about Sophie German when I was about 13 years old and I read this amazing book by Simon Singh about FMA's last theorem which had which had just been solved at the time. So FMA's last theorem was one of the greatest mathematical mysteries in history. Um, and it's a problem that, as you heard in Lucas's talk, captivated Sophie. She returned to it over and over. The goal of this talk is to introduce you to Ferma's last theorem and to the mathematics surrounding it. And that means going back thousands of years and then coming to the present day and showing you how this mathematics is still alive today. So my talk is broken up into three parts. First I'm going to discuss ancient mathematics and then we're going to come to talk about actually FMA's last theorem how it was formulated and the first progress on it including Sophie German's work and then at the very end I will talk about modern number theory which led to the solution of this problem. So let's get started with the first thing which is something that we know as Pythagoras's theorem.

[00:01:55]Um this so Pythagoras was a mathematician in ancient Greece who lived about 2500 years ago. And this result tells you that whenever you have a triangle with a right angle and with sides of lengths a, b, and c such that c is the longest side, the diagonal, then you always have the relationship a 2 + b 2 is equal to c^ 2. And this is a result that was known in one form or another to many other cultures, not just to the ancient Greeks, but also Egyptians, Babylonians, Indian mathematicians, Chinese mathematicians. And we don't really know for sure when this result was first discovered or when it was first proved.

[00:02:48]But what I want to do is I want to give you a proof of this theorem. in fact a geometric proof that appears in an Indian manuscript from the 12th century.

[00:02:59]So just somehow look at this figure, compute the area of this figure in two different ways and you will be able to prove Pythagoras's theorem. So this figure consists of one larger square and then one smaller square inscribed inside of it. The smaller square has side lengths equal to C, which is the length of the diagonal in our original triangle.

[00:03:26]And then I guess around the corners you have four copies of our original triangle. And then these make up a larger square. So let's compute this area in one way.

[00:03:40]The area of the smaller square is just C^ squ because C is the length of the side. And then if you look at the corners, you can group the the two triangles uh you can group the four triangles that are the same as our original one into groups of two. As you do this, you make a rectangle with side lengths A and B. So the area of that is equal to A * B. As you have twice of two groups of those, you get 2 * A. So the total area of the figure if we compute it in this way is C ^2 + AB. Okay. There is another way to compute this area which is just look at the larger square.

[00:04:26]The larger square has side length equal to a + b. And that means that the total area is also the square a + b squared.

[00:04:38]Okay. Okay, so we're going to do just a tiny bit of algebra to factor out a + b 2 as a 2 + b 2 + a. And then because we're computing the area of the same figure, we remember this is the same as our initial computation which is c^ 2 + a. Now you just cancel out the term twice a and you're left with Pythagoras's theorem. a^2 + b^2 = c^2.

[00:05:10]Okay. And so this is one of hundreds of proofs of Pythagoras's theorem.

[00:05:16]And I chose this one well because we mathematicians think in different ways and some people prefer algebra and some people prefer to visualize things to think geometrically. And I guess I I tend to get lost my eyes glaze over if I see a complicated equation. But I like to think in terms of pictures and ideas.

[00:05:38]And so that's why I chose this more geometric proof. But if you don't like this one, then you can look up the hundreds of proofs and I'm sure you will find one that you will like.

[00:05:48]Okay. Um, but this is a talk about number theory. And so a question that captivated ancient cultures that were aware of this result was can you make the sides of this triangle to be equal to whole numbers? Is there a right angled triangle whose sides are all three whole numbers? Right? So we want our A, B, and C to all be whole numbers. Can you do this? This was known again to ancient cultures. For example, 3, four, and five are numbers that have this property because 9 + 16 gives you 25.

[00:06:33]And this is a triple that was known to ancient Egyptians. And there are also old Babylonian tables with lots of triples like this. You also have 5, 12, 13, 8, 15, 17. And I could go on forever.

[00:06:53]Uh so these triples that are both whole numbers and the sides of a right angle triangle are called Pythagorean triples, right? Because they have to satisfy Pythagoras's theorem. And so question that I was hinting at is are there infinitely many such triples? Are there infinitely many whole number solutions to Pythagoras's equation? And when I said I went on I could go on forever, I really meant it because it turns out that yes, there are infinitely many Pythagorean triples and really even in an interesting way. So for example, if I started with 3, four, and five, I could easily generate infinitely many triples just by multiplying these numbers 3, four, and five by the same constant. So 6 8 and 10 is also a Pythagorean triple.

[00:07:49]Um 9 12 and 15 is also a Pythagorean triple. But the really interesting way to think about this question is to ask are there infinitely many triples that are not just obtained from a smaller triple by multiplying by the same constant. And it turns out even the answer to that question is yes. And I'm not going to prove this for you in this lecture. But this is the sort of thing that you would be able to do kind of find that there are infinitely many triples and even describe them precisely parameterize them in some sense um in say a first year uh sorry a first course in number theory in university. So that's the sort of thing that is within reach.

[00:08:42]And I want to take a bit of a step a zoom out step back and say that this kind of question fits into a very general pattern of questions.

[00:08:55]Uh that has to do with diaphine equations. So this Pythag Pythagoras equation x^2 + y^2= z ^2 um is a special case of a dapantine equation when you're trying to solve it in whole numbers. Dopantis was another ancient Greek mathematician and he studied these kinds of equations more systematically and what he wanted was he was looking for polomial equations or equations that you could construct in some algebraic way and then try to find solutions in whole numbers and these have been studied throughout our history for example there's Pel's equation x^2 - 17 y^2 = one is an example of an equation due to Pel. This turns out to have infinitely many solutions. There's Mell's equation x cubed = y^2 + 2. This is an equation that really only has one solution in whole numbers. Namely, you take x= 3 and y = 5 and that's it. and somehow proving that Pel's equation has infinitely many solutions and solving Mell's equation are other things that you would do say in a first course on number theory. They're difficult but they're still within reach. But just to give you a sense of kind of the how how vast this area is sort of general dopantine equations are largely unknown.

[00:10:33]We don't know whether they have infinitely many solutions or whether they have you know a few finitely many solutions or some of them may have no solutions at all. And um you know if you follow the news on maths just a couple of weeks ago there was an announcement uh for the Abel prize which is kind of the Nobel Prize equivalent in mathematics. This was given to a German mathematician called Gird Fings. um it was one of one of his kind of greatest achievements was showing that a certain class of Dopantine equations if they're polomial equations and the polomials have high enough degree um they only have finitely many solutions.

[00:11:20]So it it is an active area of research in mathematics today and particularly I want to mention kind of what I work on is is arithmetic geometry kind of starts out with these kinds of equations like either um Pythagoras's equation or Pel's equation or Mell's equation and interprets them as giving rise to some geometric objects and then you kind of can try to characterize whether the equation has infinitely many solutions or only finitely many using the properties of the geometric object.

[00:12:00]So in some sense Pythagoras's equation relates to the unit circle because I could divide by z and then it would turn into x over z ^2 + y over z ^2 = 1 and and so it turns into looking for points on the unit circle.

[00:12:18]All right. So this this was kind of the zoom out the area of Dophantine equations. But this talk is not about kind of general dapophantine equations many of which are still unsolved today.

[00:12:31]But it's about a specific Dopantine equation which fascinated mathematicians for hundreds of years and this is known as FMA's last theorem. So in 1637 a French mathematician called Pierre de Ferma was reading arithmetica by Daphantis and thinking about Pythagorean triples when he formulated the following result. He said that the diopantine equation x to the n + y to the n = z to the n has no non-trivial whole number solutions when n is greater than or equal to 3. So pythagorean triples give you infinitely many whole numbers whose square such that the square of x plus the square of y adds up to the square of z. But Ferma thought that this could not happen if instead of squares you had cubes, you had third powers or fourth powers or any other higher power. And I sort of said he formulated is it as a result. In fact, he he said something he he made a note on the margin of his copy of this book by Dapantis that he's realized that this can never happen when the exponent is greater than or equal to three. And he has a marvelous proof of this fact. But unfortunately, the margin is just too small to contain his marvelous proof. And this is something that really captivated mathematicians. Really pretty much every famous mathematician that came in in the centuries after right after Ferma was captivated by this question. And maybe yeah, just to kind of point to Lucas's talk, maybe what made it so tantalizing is that it didn't seem impossible, right?

[00:14:40]Ferma claimed he had a proof. So there should be a proof out there. you should be able to to show that this is indeed true. Um perhaps if he had just speculated, I wonder if this can happen, this would have just been lost and forgotten by history. But instead, it it really became one of the greatest mathematical mysteries.

[00:15:02]All right? So he didn't provide a proof except in FMA's notes. You can find some proof at least in the case of exponent four. And then later on Oiler used the idea from FMA's proof and refined it and allowed it to apply to exponent three as well. And then a few other mathematicians came later on and did something for for exponent five and exponent 7. So there was a little bit of progress. And just to mention what ideas show up um in this area, FMA's original proof was based on infinite descent.

[00:15:45]Um so this is a property of whole numbers that they're bounded from below, right? You you can't keep finding smaller and smaller whole numbers because you reach zero or you reach one depending on how you how you think about whole numbers and and you can't go lower than this.

[00:16:03]And so the idea is that if you have a non-trivial solution to Ferma's equation given by whole numbers, then in the case of exponent 4, you can produce a smaller a solution that's smaller in some sense that's still a non-trivial solution to FMA's equation with the same exponent four. And so if you do this every time then you get a contradiction um because you can see that as you go to very small numbers you can't find anything but a trivial solution right so one um so you're allow you're not allowed to take one of these numbers to be equal to zero because that would be a trivial solution to FMA's equation if you ask all of x y and z to be whole numbers greater than zero But you can always go one step lower then at that point you find a contradiction. So this is this idea of infinite descent.

[00:17:05]Um that was used by FML in the case n= 4 and by Oiler in the case n= 3. But I just want to reiterate that this was something really special about the exponent. And even the generalizations for five and seven didn't look like they could attack the case of a general exponent n. That's what makes FMA's last theorem so difficult.

[00:17:32]Okay. So to move forward a little bit, let me just say that there is a way to simplify form last theorem a little bit.

[00:17:42]The first step that you can take once the case n= 4 is done is to observe that the exponent n factors as a product of prime numbers. Right? These are numbers that don't have any divisor other than themselves or one. So 2 3 5 7 13 are primes but 4 6 9 8 10 are not prime numbers. Anything that's not prime can be factored uniquely as a product of prime numbers. This is a fact known as the fundamental theorem of arithmetic.

[00:18:22]And using this fact and using the fact that you know how to so that form's last theorem in the case n= 4 was proved you're reduced to considering the case of p's powers where p is an odd prime.

[00:18:39]So it's enough to look at FMA's equation x to the p + y to the p = z to the p in the case of an odd prime. If you can't have any solutions to that, you're not going to have any solutions for any integer that is a multiple of that prime. You're also not going to have any solutions for any integer that's a multiple of four. And that takes care of all the integers greater than or equal to three.

[00:19:08]And then further on mathematicians broke up the problem into two cases. The case where the product of the numbers xyz is a multiple of the prime p and the case where it isn't. Somehow the first case where p doesn't divide the product xyz was considered a bit simpler. This is the case where kind of the the first significant progress on FMA's last theorem for general exponent was made and that was due to Sophie German. So she was the first mathematician who made progress for general exponent and this you will see more about her work in James Maynard's talk but I just want to mention the following theorem that is due to Sophie German.

[00:20:08]This says that so let's say p is our exponent for which we're trying to prove form's last theorem and let's say that p is prime but also 2 p + 1 is prime. So for example this could apply to five where 11 is also prime or it could apply to p= 11 where 23 is also a prime number.

[00:20:35]And if this condition is satisfied, Sophie German proved the first case of FMA's last theorem. There are no solutions to FMA's equation that are relatively prime that do not um have P as a factor. And that's really a stunning achievement for you know a large class of primes. So using this theorem and using a stronger version that Sophie German actually proved as you heard in Lucas's talk one could get um the first one could get FMA's last theorem for exponents up to 100 in fact with with more more work but still inspired by Sophie's ideas one could go up to exponent prime odd prime exponent up to 280 and that's so much better than what Ferma and Oiler and and the few people who came after them were able to do and just for for very limited specific exponents.

[00:21:40]So the primes that satisfy this theorem are called Sophie primes.

[00:21:47]So a prime number P such that 2 P + 1 is also prime is called a Sophie prime.

[00:21:56]Five and 11 are Sophisure primes. It's not currently known whether there are infinitely many such primes or not. But this will come up in James Maynard's talk later today. It's just I want to highlight that it's remarkable that she connected one of the greatest mathematical mysteries form's last theorem to another really important question in maths which is about the distribution of prime numbers. We know there are infinitely many primes but their distribution is something mysterious and it's again something that is being studied actively in research today.

[00:22:37]Okay. And one more thing that I want to mention about Sophie's work is that the idea, one of the first ideas for proving this theorem has to do with congruences modulo this auxiliary prime number Q.

[00:22:55]Right? So we're assuming P is prime.

[00:22:58]We're also assuming 2 P + 1 is prime.

[00:23:01]We're calling that Q. And that's kind of an auxiliary prime for us.

[00:23:06]And an idea that's very powerful in number theory is that if you have a diaphantine equation, an equation such as x to the p + y to the p= z to the p, you could try to first weaken it and solve an approximation of it. Instead of trying to find whole numbers x and y such that x to the p + y to the p add up exactly to some z to the p. What you could try to do is find x y and z such that x to the p + y differs from z to the p only by a multiple of q. Zero is a multiple of anything. It's a multiple of Q in particular, but this is still weaker to ask X to the P + Y to the P to only agree to Z to the P up to a multiple of Q. And doing this means that you're solving a congruence equation modulo Q.

[00:24:14]Instead of solving your original equation on the nose, asking for equality, you're only asking for an approximation. You're asking for equality, but you're allowed to add or subtract multiples of Q. And this was one of the one of the first ideas coming in this theorem of Sophie German. And the idea of also kind of trying to solve equations by looking at congruences, modular primes comes back hundreds of years later in the actual proof of Fermas theorem. So it's it's one of these very powerful ideas that persists over centuries. And so that's why I wanted to highlight it. It seems first like you're solving something else, this weaker problem, but in fact it takes you a little bit towards it. And I guess as a as a further step in what Sophie German did, I should say that these powers modulo Q modulo Q equals 2P plus one tend to be rather rigid and that relies on something called FMA's little theorem. This is a theorem that Ferma actually provided a proof for. So so he he did in fact prove FMA's little theorem even though he did not provide a proof for his last theorem.

[00:25:33]All right. So that was the progress due to Sophie German.

[00:25:38]I want to mention one more idea in the centuries long quest to prove Ferma's last theorem uh which is somehow something that was tried by a French mathematician called Lame and then it was refined by a German mathematician called Kumar.

[00:25:58]So one thing that makes Ferma's last theorem hard is that it involves a mixture. Let me go back to it. It involves a mixture of addition and multiplication.

[00:26:11]If this was only about multiplication, it would be easy because you could just factor your integers into primes and look at that. Or if it was only about addition, it would be easy. But the fact that it it combines addition and multiplication makes it tricky.

[00:26:28]But then the idea that lame menkumar had was to kind of try to turn the addition into some kind of multiplication and for that one had to build bigger number systems. So the rational numbers are numbers that we're familiar with, right? Fractions, but one can build bigger number systems out of the rational numbers. And one can study equations within them. And one can ask whether this property that I mentioned which is unique factorization you know taking an integer such as say 10 and expressing it as a product of prime numbers 2 * 5 uniquely does something like this hold in bigger number systems. So for example, if you allow yourself kind of a flight of imagination and you add to the rational numbers the square root of minus1, this is called an imaginary number. We're denoting it by i. Then you can factor x^2 + y^2 as a product x + i y * x - i y and then if that's equal to z ^2 then you can start to kind of make progress on the problem of finding pythagorean triples. So this i is an imaginary number. It's just kind of a formal algebraic construct that has the property that it square is minus1. But it is very useful for studying diopantine equations. And if you look at all the numbers that you could generate from rational numbers from fractions by adding in this one additional imaginary number called I and then you're still allowed the usual rules addition and multiplication with their usual properties then you get a new number system that is called the number system of gausian numbers.

[00:28:36]So what lame tried to do was he tried to study FMA's equation for a prime exponent p by factoring x to the p + y to the p and you can do that if you build an even bigger number system and you add in roots of unity. So I is a fourth root of unity but you can add in peace roots of unity and then you can ask whether this kind of unique factorization that integers have persists in the bigger number system.

[00:29:08]So Kumer showed that this is actually not true. This is a very special property of integers and it persists for the gausian numbers but it does not persist in general if you add in an arbitrary peace power root of unity.

[00:29:25]um he was able to quantify in a precise way how this property fails and by doing that he was able to prove for last theorem for another large class of primes called regular primes.

[00:29:40]So, Kumar proved kind of provided the next very important progress on FMA's last theorem and that was in the second half of the 19th century by exploring unique factorization and how it breaks down in larger number systems.

[00:30:02]And the other reason why I wanted to mention this is that this work really gave birth to modern number theory in some precise sense at least to modern algebraic number theory where we combine techniques from algebra to study the properties of whole numbers. So it was another important milestone. The milestones so far have been FMA kind of formulating this as a result but without a proof that's what we call a conjecture we mathematicians call this then uh Sophie German proving the first case for these Sophie German primes then Kumar and his work on unique factorization which proved it for regular primes and then this brings us to the end of the 20th century where we're really doing modern number theory. And so here's some some developments.

[00:31:01]In uh 1983, this uh German mathematician that I mentioned to you, Fings, who got the prize two weeks ago, proved that Ferma's equation um has only finitely many solutions if you fix the exponent. So if you fix your exponent p um to be this odd prime, maybe at least five, then there are only finitely many solutions. This is something that followed from fall things's work.

[00:31:35]Around the same time, completely independent development, a German mathematician called Frey suggested that maybe one could provide kind of a a way to disprove so to a proof of FMA's last theorem by relating it to some kind of mysterious relationship called the modularity of elliptic curves and kind of Frey made a suggestion and then this was later made more rigorous by a French mathematician called sir and ultimately by ribbit and by 1986 it was clear that fermma's last theorem would imply if only one could prove this very mysterious very advanced very cuttingedge maths known as the modularity of elliptic curves but still in 1986 this was considered very far out of reach by mathematicians working in the area of number theory.

[00:32:36]Well, by all of them except for one, which is Sir Andrew Wilds, now a professor at Oxford University. He proved in 1995 for Ma's last theorem by proving this result on the modularity of elliptic curves. And this also built on work that he did with his PhD student at the time, Richard Taylor. Richard Taylor was my PhD supervisor. So I'm kind of that's my personal connection one one connection to this story.

[00:33:08]All right. So I've brought you to the last part of my talk which is about modern number theory and I want to zoom out again and say rather than telling you directly what is this modularity of elliptic curves I just want to say it it fits within a very large framework that is today known as the Langland's program and this is thought of as somehow a grand unified theory of maths. It aims to connect number theory to other areas of maths. For example, to analysis and I think this is something that Sophie German would have liked very much because her work was in number theory but also in analysis. And it turns out that there is there are mysterious connections between these different areas. And also there are connections to arithmetic geometry which I mentioned to you before. the idea of viewing equations as geometric objects and studying their properties that way and to representation theory which is really kind of the study of symmetry in mathematics and so the way that I like to think about the Langland's program is as a bridge and so that's why I drew this bridge and this bridge you can think of it as having an interesting geometry itself and then on the bridge is somehow the community of people who work on extending ing it and on one side of it I want you to think there's number theory there's FMA's last theorem and on the other side of it there's analysis and now I want to give you a very concrete example of kind of what this bridge might give you in practice.

[00:34:59]So, number theory and analysis.

[00:35:02]I claim that I can give you an object that lives within the world of number theory and an object that lives within the world of analysis and show to you that these objects somehow secretly communicate.

[00:35:19]So on the side of number theory I've written down a cubic equation and you can think of it as a diapopantine equation in two variables x and y and from the more geometric point of view this is something called an elliptic curve kind of geometrically if you were to picture it it might look like a donut. So a surface with one hole.

[00:35:45]Um on the side of analysis I've written an infinite product in a variable Q. This variable Q could be anything. And it's just this infinite product that you can expand out and you get a power series in this variable Q like Q + Q ^2 + Q cubed with some coefficients. And these coefficients are going to be whole numbers. I'm just expanding out this infinite product.

[00:36:16]Okay. Um the reason that this is uh in some sense an object that lives in the world of analysis is because this power series is the fier expansion of some function that is very symmetric and that is called the modular form. And I want to spend a minute thinking about this this name Furier.

[00:36:42]Furier was a contemporary of Sophie.

[00:36:45]According to Wikipedia, he helped get her tickets to the meetings of the academy, right, from which women were generally excluded. And he worked in analysis. Um and FIA analysis is a kind of analysis that aims to understand functions that have lots of symmetries in terms of standard periodic functions such as the sign function or the cosine function. So trigonometric functions the idea is to break up a complicated function and decompose it in terms of as sums of various ss and cosiness. And this kind of theory was developed in the 19th century um and originally applied to say studying like vibrating strings or planetary orbits.

[00:37:36]And it turns out so so this one infinite product that I've written down on the right hand side of my slide is the expansion somehow in terms of exponentials that are sums of cosiness and signs complex exponentials.

[00:37:54]um it's the expansion of a certain very special function called the modular form and the modularity of elliptic curves which is what wilds proved to establish fairma's last theorem is the connection the bridge between these two worlds so let me convince you that these two words worlds really talk to each other I'm going to extract some whole numbers from the left hand side and from the right hand side and then you're going to see how these whole numbers match up.

[00:38:25]From the left hand side, I'm going to look at my cubic equation, but I'm not going to look at it as an equation on the nose, but rather as a congruence equation, modulo different prime numbers, and then I'm going to count solutions. So, for example, I want to look for solutions that differ up to multiples of three and count the number of solutions, modulo three, and that's going to give me a number. And similarly modulo 5, modulo 7 and so on. On the left hand side I just want to expand this product into a sum of powers of Q with coefficients.

[00:39:06]And so the blue is for analysis, the red is for number theory.

[00:39:11]If I look for the coefficient of Q to the L when I expand the infinite product and I let L run over prime numbers then these are the values that I get. So at two the coefficient is minus2 at three the coefficient is minus1 and so on.

[00:39:30]If I look for the number of solutions to the congruence equation modulo L and I let L run over prime numbers, then these are the numbers that I get. So at two I get four solutions. At three I get four solutions. At five I get four solutions.

[00:39:47]At seven I get nine solutions. And so I claim these two very different objects that seem to have nothing to do with each other secretly communicate.

[00:39:58]So, I'll give you a second. Can you see any relationship between the strings of numbers that are in blue and the strings of numbers that are in red?

[00:40:13]>> Sorry, >> if you add them up, right, somebody said, somebody from the audience said this, if you add them up, you recover the prime number L, right? minus2 + 4 recovers two minus1 + 4 recovers three.

[00:40:32]Um if I take L to be 29, 29 plus 0 gives me back 29. So they do secretly know about each other. They have nothing to do with each other, but I promise you this will persist for the infinitely many primes and and so they really do communicate. And so this is a very concrete way of imagining this result u that I mentioned which is the modularity of elliptic curve. And so this is a theorem proved by wilds and Taylor walls in 95 um in a special case and then completed in 2001 the work of Conrad Diamond and Taylor and that tells you every elliptic curve is modular. So roughly every cubic equation that corresponds to this kind of donut um has secretly some generating function that is a modular form. This kind of relationship doesn't just hold true for the particular objects I mentioned here but really somehow much much more generally.

[00:41:35]And this was the key to the proof of FMA's last theorem. And so to end my talk, I just want to mention that you know this is an active area of research today. And so for example in 2023 in joint work with James Newton um we proved the modularity of elliptic curves over other number systems such as the number system that you get from looking at Gausian numbers. Q adjoined this imaginary square root of minus1.

[00:42:06]And on the other hand, instead of just looking at cubic equations, you could be looking at higher degree equations. And for degrees five and six, there are recent results uh due to Boxer, Caligari, G, and Piloni from just last year. And George Boxer and Toby G are my colleagues at Imperial. And then finally another sense in which this area is very much alive today is another one of my colleagues from Imperial who is Kevin Buzzard and who is in the audience today is working on formalizing the proof of FMA's last theorem and that means somehow inputting a large part of the proof of FMA's last theorem into a theorem prover and having it be automatically verified by a computer. So he has a a long-term project, five-year long project to do this. Uh and so yeah, really at the cutting edge of what is going on in mathematics today. So thank you very much for your attention and