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  • Joseph Nebus 6:00 pm on Wednesday, 9 August, 2017 Permalink | Reply
    Tags: , , , , , Fermat's Last Theorem, , ,   

    The Summer 2017 Mathematics A To Z: Elliptic Curves 


    Gaurish, of the For The Love Of Mathematics gives me another subject today. It’s one that isn’t about ellipses. Sad to say it’s also not about elliptic integrals. This is sad to me because I have a cute little anecdote about a time I accidentally gave my class an impossible problem. I did apologize. No, nobody solved it anyway.

    Elliptic Curves.

    Elliptic Curves start, of course, with polynomials. Particularly, they’re polynomials with two variables. We call the ‘x’ and ‘y’ because we have no reason to be difficult. They’re of at most third degree. That is, we can have terms like ‘x’ and ‘y2‘ and ‘x2y’ and ‘y3‘. Something with higher powers, like, ‘x4‘ or ‘x2y2‘ — a fourth power, all together — is right out. Doesn’t matter. Start from this and we can do some slick changes of variables so that we can rewrite it to look like this:

    y^2 = x^3 + Ax + B

    Here, ‘A’ and ‘B’ are some numbers that don’t change for this particular curve. Also, we need it to be true that 4A^3 + 27B^2 doesn’t equal zero. It avoids problems. What we’ll be looking at are coordinates, values of ‘x’ and ‘y’ together which make this equation true. That is, it’s points on the curve. If you pick some real numbers ‘A’ and ‘B’ and draw all the values of ‘x’ and ‘y’ that make the equation true you get … well, there’s different shapes. They all look like those microscope photos of a water drop emerging and falling from a tap, only rotated clockwise ninety degrees.

    So. Pick any of these curves that you like. Pick a point. I’m going to name your point ‘P’. Now pick a point once more. I’m going to name that point ‘Q’. Now draw a line from P through Q. Keep drawing it. It’ll cross the original elliptic curve again. And that point is … not actually special. What is special is the reflection of that point. That is, the same x-coordinate, but flip the plus or minus sign for the y-coordinate. (WARNING! Do not call it “the reflection” at your thesis defense! Call it the “conjugate” point. It means “reflection”.) Your elliptic curve will be symmetric around the x-axis. If, say, the point with x-coordinate 4 and y-coordinate 3 is on the curve, so is the point with x-coordinate 4 and y-coordinate -3. So that reflected point is … something special.

    Kind of a curved-out less-than-sign shape.

    y^2 = x^3 - 1 . The water drop bulges out from the surface.

    This lets us do something wonderful. We can think of this reflected point as the sum of your ‘P’ and ‘Q’. You can ‘add’ any two points on the curve and get a third point. This means we can do something that looks like addition for points on the elliptic curve. And this means the points on this curve are a group, and we can bring all our group-theory knowledge to studying them. It’s a commutative group, too; ‘P’ added to ‘Q’ leads to the same point as ‘Q’ added to ‘P’.

    Let me head off some clever thoughts that make fair objections. What if ‘P’ and ‘Q’ are already reflections, so the line between them is vertical? That never touches the original elliptic curve again, right? Yeah, fair complaint. We patch this by saying that there’s one more point, ‘O’, that’s off “at infinity”. Where is infinity? It’s wherever your vertical lines end. Shut up, this can too be made rigorous. In any case it’s a common hack for this sort of problem. When we add that, everything’s nice. The ‘O’ serves the role in this group that zero serves in arithmetic: the sum of point ‘O’ and any point ‘P’ is going to be ‘P’ again.

    Second clever thought to head off: what if ‘P’ and ‘Q’ are the same point? There’s infinitely many lines that go through a single point so how do we pick one to find an intersection with the elliptic curve? Huh? If you did that, then we pick the tangent line to the elliptic curve that touches ‘P’, and carry on as before.

    The curved-out less-than-sign shape has a noticeable c-shaped bulge on the end.

    y^2 = x^3 + 1 . The water drop is close to breaking off, but surface tension has not yet pinched off the falling form.

    There’s more. What kind of number is ‘x’? Or ‘y’? I’ll bet that you figured they were real numbers. You know, ordinary stuff. I didn’t say what they were, so left it to our instinct, and that usually runs toward real numbers. Those are what I meant, yes. But we didn’t have to. ‘x’ and ‘y’ could be in other sets of numbers too. They could be complex-valued numbers. They could be just the rational numbers. They could even be part of a finite collection of possible numbers. As the equation y^2 = x^3 + Ax + B is something meaningful (and some technical points are met) we can carry on. The elliptical curves, and the points we “add” on them, might not look like the curves we started with anymore. They might not look like anything recognizable anymore. But the logic continues to hold. We still create these groups out of the points on these lines intersecting a curve.

    By now you probably admit this is neat stuff. You may also think: so what? We can take this thing you never thought about, draw points and lines on it, and make it look very loosely kind of like just adding numbers together. Why is this interesting? No appreciation just for the beauty of the structure involved? Well, we live in a fallen world.

    It comes back to number theory. The modern study of Diophantine equations grows out of studying elliptic curves on the rational numbers. It turns out the group of points you get for that looks like a finite collection of points with some collection of integers hanging on. How long that collection of numbers is is called the ‘rank’, and there are deep mysteries at work. We know there are elliptic equations that have a rank as big as 28. Nobody knows if the rank can be arbitrary high, though. And I believe we don’t even know if there are any curves with rank of, like, 27, or 25.

    Yeah, I’m still sensing skepticism out there. Fine. We’ll go back to the only part of number theory everybody agrees is useful. Encryption. We have roughly the same goals for every encryption scheme. We want it to be easy to encode a message. We want it to be easy to decode the message if you have the key. We want it to be hard to decode the message if you don’t have the key.

    The curved-out sign has a bulge with convex loops to it, so that it resembles the cut of a jigsaw puzzle piece.

    y^2 = 3x^2 - 3x + 3 . The water drop is almost large enough that its weight overcomes the surface tension holding it to the main body of water.

    Take something inside one of these elliptic curve groups. Especially one that’s got a finite field. Let me call your thing ‘g’. It’s really easy for you, knowing what ‘g’ is and what your field is, to raise it to a power. You can pretty well impress me by sharing the value of ‘g’ raised to some whole number ‘m’. Call that ‘h’.

    Why am I impressed? Because if all I know is ‘h’, I have a heck of a time figuring out what ‘g’ is. Especially on these finite field groups there’s no obvious connection between how big ‘h’ is and how big ‘g’ is and how big ‘m’ is. Start with a big enough finite field and you can encode messages in ways that are crazy hard to crack.

    We trust. At least, if there are any ways to break the code quickly, nobody’s shared them. And there’s one of those enormous-money-prize awards waiting for someone who does know how to break such a code quickly. (I don’t know which. I’m going by what I expect from people.)

    And then there’s fame. These were used to prove Fermat’s Last Theorem. Suppose there are some non-boring numbers ‘a’, ‘b’, and ‘c’, so that for some prime number ‘p’ that’s five or larger, it’s true that a^p + b^p = c^p . (We can separately prove Fermat’s Last Theorem for a power that isn’t a prime number, or a power that’s 3 or 4.) Then this implies properties about the elliptic curve:

    y^2 = x(x - a^p)(x + b^p)

    This is a convenient way of writing things since it showcases the ap and bp. It’s equal to:

    y^2 = x^3 + \left(b^p - a^p\right)x^2 + a^p b^p x

    (I was so tempted to leave an arithmetic error in there so I could make sure someone commented.)

    A little ball off to the side of a curved-out less-than-sign shape.

    y^2 = 3x^3 - 4x . The water drop has broken off, and the remaining surface rebounds to its normal meniscus.

    If there’s a solution to Fermat’s Last Theorem, then this elliptic equation can’t be modular. I don’t have enough words to explain what ‘modular’ means here. Andrew Wiles and Richard Taylor showed that the equation was modular. So there is no solution to Fermat’s Last Theorem except the boring ones. (Like, where ‘b’ is zero and ‘a’ and ‘c’ equal each other.) And it all comes from looking close at these neat curves, none of which looks like an ellipse.

    They’re named elliptic curves because we first noticed them when Carl Jacobi — yes, that Carl Jacobi — while studying the length of arcs of an ellipse. That’s interesting enough on its own. But it is hard. Maybe I could have fit in that anecdote about giving my class an impossible problem after all.

     
  • Joseph Nebus 6:00 pm on Monday, 7 August, 2017 Permalink | Reply
    Tags: , , Fermat's Last Theorem, , , , , ,   

    The Summer 2017 Mathematics A To Z: Diophantine Equations 


    I have another request from Gaurish, of the For The Love Of Mathematics blog, today. It’s another change of pace.

    Diophantine Equations

    A Diophantine equation is a polynomial. Well, of course it is. It’s an equation, or a set of equations, setting one polynomial equal to another. Possibly equal to a constant. What makes this different from “any old equation” is the coefficients. These are the constant numbers that you multiply the variables, your x and y and x2 and z8 and so on, by. To make a Diophantine equation all these coefficients have to be integers. You know one well, because it’s that x^n + y^n = z^n thing that Fermat’s Last Theorem is all about. And you’ve probably seen ax + by = 1 . It turns up a lot because that’s a line, and we do a lot of stuff with lines.

    Diophantine equations are interesting. There are a couple of cases that are easy to solve. I mean, at least that we can find solutions for. ax + by = 1 , for example, that’s easy to solve. x^n + y^n = z^n it turns out we can’t solve. Well, we can if n is equal to 1 or 2. Or if x or y or z are zero. These are obvious, that is, they’re quite boring. That one took about four hundred years to solve, and the solution was “there aren’t any solutions”. This may convince you of how interesting these problems are. What, from looking at it, tells you that ax + by = 1 is simple while x^n + y^n = z^n is (most of the time) impossible?

    I don’t know. Nobody really does. There are many kinds of Diophantine equation, all different-looking polynomials. Some of them are special one-off cases, like x^n + y^n = z^n . For example, there’s x^4 + y^4 + z^4 = w^4 for some integers x, y, z, and w. Leonhard Euler conjectured this equation had only boring solutions. You’ll remember Euler. He wrote the foundational work for every field of mathematics. It turns out he was wrong. It has infinitely many interesting solutions. But the smallest one is 2,682,440^4 + 15,365,639^4 + 18,796,760^4 = 20,615,673^4 and that one took a computer search to find. We can forgive Euler not noticing it.

    Some are groups of equations that have similar shapes. There’s the Fermat’s Last Theorem formula, for example, which is a different equation for every different integer n. Then there’s what we call Pell’s Equation. This one is x^2 - D y^2 = 1 (or equals -1), for some counting number D. It’s named for the English mathematician John Pell, who did not discover the equation (even in the Western European tradition; Indian mathematicians were familiar with it for a millennium), did not solve the equation, and did not do anything particularly noteworthy in advancing human understanding of the solution. Pell owes his fame in this regard to Leonhard Euler, who misunderstood Pell’s revising a translation of a book discussing a solution for Pell’s authoring a solution. I confess Euler isn’t looking very good on Diophantine equations.

    But nobody looks very good on Diophantine equations. Make up a Diophantine equation of your own. Use whatever whole numbers, positive or negative, that you like for your equation. Use whatever powers of however many variables you like for your equation. So you get something that looks maybe like this:

    7x^2 - 20y + 18y^2 - 38z = 9

    Does it have any solutions? I don’t know. Nobody does. There isn’t a general all-around solution. You know how with a quadratic equation we have this formula where you recite some incantation about “b squared minus four a c” and get any roots that exist? Nothing like that exists for Diophantine equations in general. Specific ones, yes. But they’re all specialties, crafted to fit the equation that has just that shape.

    So for each equation we have to ask: is there a solution? Is there any solution that isn’t obvious? Are there finitely many solutions? Are there infinitely many? Either way, can we find all the solutions? And we have to answer them anew. What answers these have? Whether answers are known to exist? Whether answers can exist? We have to discover anew for each kind of equation. Knowing answers for one kind doesn’t help us for any others, except as inspiration. If some trick worked before, maybe it will work this time.

    There are a couple usually reliable tricks. Can the equation be rewritten in some way that it becomes the equation for a line? If it can we probably have a good handle on any solutions. Can we apply modulo arithmetic to the equation? If it is, we might be able to reduce the number of possible solutions that the equation has. In particular we might be able to reduce the number of possible solutions until we can just check every case. Can we use induction? That is, can we show there’s some parameter for the equations, and that knowing the solutions for one value of that parameter implies knowing solutions for larger values? And then find some small enough value we can test it out by hand? Or can we show that if there is a solution, then there must be a smaller solution, and smaller yet, until we can either find an answer or show there aren’t any? Sometimes. Not always. The field blends seamlessly into number theory. And number theory is all sorts of problems easy to pose and hard or impossible to solve.

    We name these equation after Diophantus of Alexandria, a 3rd century Greek mathematician. His writings, what we have of them, discuss how to solve equations. Not general solutions, the way we might want to solve ax^2 + bx + c = 0 , but specific ones, like 1x^2 - 5x + 6 = 0 . His books are among those whose rediscovery shaped the rebirth of mathematics. Pierre de Fermat’s scribbled his famous note in the too-small margins of Diophantus’s Arithmetica. (Well, a popular translation.)

    But the field predates Diophantus, at least if we look at specific problems. Of course it does. In mathematics, as in life, any search for a source ends in a vast, marshy ambiguity. The field stays vital. If we loosen ourselves to looking at inequalities — x - Dy^2 < A , let's say — then we start seeing optimization problems. What values of x and y will make this equation most nearly true? What values will come closest to satisfying this bunch of equations? The questions are about how to find the best possible fit to whatever our complicated sets of needs are. We can't always answer. We keep searching.

     
  • Joseph Nebus 5:04 pm on Sunday, 3 April, 2016 Permalink | Reply
    Tags: , , Fermat's Last Theorem, , , information   

    Reading the Comics, April 2, 2016: Keeping Me Busy Edition 


    After I made a little busy work for myself posting a Reading the Comics entry the other day, Comic Strip Master Command sent a rush of mathematics themes into the comics. So it goes.

    Chris Browne’s Hagar the Horrible for the 31st of March happens to be funny-because-it’s-true. It’s supposed to be transgressive to see a gambler as the best mathematician available. But quite a few of the great pioneering minds of mathematics were also gamblers looking for an edge. It may shock you to learn that mathematicians in past centuries didn’t have enough money, and would look for ways to get more. And, as ever, knowing something secret about the way cards or dice or any unpredictable event might happen gives one an edge. The question of whether a 9 or a 10 is more likely to be thrown on three dice was debated for centuries, by people as familiar to us as Galileo. And by people as familiar to mathematicians as Gerolamo Cardano.

    Hagar: 'I brought a math tutor for Hamlet!' Helga: 'Great! Does he know his stuff?' Hagar: 'Are you kidding? He's the BEST card counter in the kingdom!'

    It’s funny because this it’s anachronistic for Blaise Pascal to be in this setting.

    Gambling blends imperceptibly into everything people want to do. The question of how to fairly divide the pot of an interrupted game may seem sordid. But recast it as the problem of how to divide the assets of a partnership which had to halt — say, because one of the partners had to stop participating — and we have something that looks respectable. And gambling blends imperceptibly into security. The result of any one project may be unpredictable. The result of many similar ones, on average, often is. Card games or joint-stock insurance companies; the mathematics is the same. A good card-counter might be the best mathematician available.

    Tony Cochran’s Agnes for the 31st name-drops Diophantine equations. It’s in the service of a student resisting class joke. Diophantine equations are equations for which we only allow integer, whole-number, answers. The name refers to Diophantus of Alexandria, who lived in the third century AD. His Arithmetica describes many methods for solving equations, a prototype to algebra as we know it in high school today. Generally, a Diophantine equation is a hard problem. It’s impossible, for example, to say whether an arbitrary Diophantine equation even has a solution. Finding what it might be is another bit of work. Fermat’s Last Theorem is a Diophantine equation, and that took centuries to work out that there isn’t generally an answer.

    Mind, we can say for specific cases whether a Diophantine equation has a solution. And those specific cases can be pretty general. If we know integers a and b, then we can find integers x and y that make “ax + by = 1” true, for example.

    Graham Harrop’s Ten Cats for the 31st hurts mathematicians’ feelings on the way to trying to help a shy cat. I’m amused anyway.

    And Jonathan Lemon’s Rabbits Against Magic for the 1st of April mentions Fermat’s Last Theorem. The structure of the joke is fine. If we must ask an irrelevant question of the Information Desk mathematics has got plenty of good questions. The choice makes me suspect Lemon’s showing his age, though. The imagination-capturing power of Fermat’s Last Theorem as a great unknown has to have been diminished since the first proof was found over two decades ago. It’d be someone who grew up knowing there was this mystery about xn plus yn equalling zn who’d jump to this reference.

    Tom Toles’s Randolph Itch, 2 am for the 2nd of April mentions “zero-sum games”. The term comes from the mathematical theory of games. The field might sound frivolous, but that’s because you don’t know how much stuff the field considers to be “games”. Mathematicians who study them consider “games” to be sets of decisions. One or more people make choices, and gain or lose as a result of those choices. That is a pretty vague description. It covers playing solitaire and multiplayer Civilization V. It also covers career planning and imperial brinksmanship. And, for that matter, business dealings.

    “Zero-sum” games refer to how we score the game’s objectives. If it’s zero-sum, then anything gained by one player must be balanced by equal losses by the other player or players. For example, in a sports league’s season standings, one team’s win must balance another team’s loss. The total number of won games, across all the teams, has to equal the total number of lost games. But a game doesn’t have to be zero-sum. It’s possible to create games in which all participants gain something, or all lose something. Or where the total gained doesn’t equal the total lost. These are, imaginatively, called non-zero-sum games. They turn up often in real-world applications. Political or military strategy often is about problems in which both parties can lose. Business opportunities are often intended to see the directly involved parties benefit. This is surely why Randolph is shown reading the business pages.

     
    • Marta Frant 4:06 am on Monday, 4 April, 2016 Permalink | Reply

      I wish comics were as popular in my country as they are in yours.

      Like

      • Joseph Nebus 6:33 pm on Monday, 4 April, 2016 Permalink | Reply

        I’m surprised to find there might be countries without popular comics. But the United States did have some factors that encouraged a lot of comic creation in the past. Mostly, many small population centers with multiple daily newspapers and using comics to entice readers, and that could transmit the comics to national syndicated feeds. The rate of new syndicated comic strips has dropped to almost nothing — I think there were two last year, and several syndicated comics ended — but Gocomics.com particularly looks for web comics and brings them to my view.

        I have been interested in reading more non-United-States-based comic strips. But with a few exceptions — Britain’s Fred Basset and Andy Capp, Australia’s Ginger Meggs, India’s Green Humour, and a bunch of Canadians — they don’t seem to cross over to United States attention. I understand comic strips that are too intensely local not communicating, but it seems to me like there’s an opportunity missing.

        Liked by 1 person

        • Marta Frant 3:36 am on Tuesday, 5 April, 2016 Permalink | Reply

          Yes, I like your point of view. Btw, that’s why Marvels don’t make such a splash at the box office in Russia. We don’t have that strong base for this kind of art.

          Like

          • Joseph Nebus 1:56 am on Saturday, 9 April, 2016 Permalink | Reply

            This does surprise me. I tend to think of comic art as being nearly universal. It has a feeling of inevitability that people would draw sequences of pictures to tell stories, either comic or dramatic. But then I am coming from a cultural heritage that’s got a strong tradition of doing this.

            On the other hand, for example, United States newspapers used to regularly include poems on a daily basis. Some were satiric, some were commentary, some were just about whatever caught the writer’s fancy that day. That’s a tradition that’s so dead in the United States that it would surely look weird if someone tried to revive it. I have the impression that it’s still going in other countries, though. And that’s no worse a medium for writing about the matters of the day than comic strips are.

            Like

            • Marta Frant 12:52 pm on Saturday, 9 April, 2016 Permalink | Reply

              That’s not how Russian art used to work. Censorship of images in the USSR was pervasive and many other art forms were prohibited in the Soviet Union. And then it was strangled by the ‘internets’ and different gadgets.

              Like

              • Joseph Nebus 2:34 am on Friday, 15 April, 2016 Permalink | Reply

                I’m not surprised that images should be heavily censored. I am impressed that, particularly, given the way cartoons can have both an obvious unobjectionable nature and a subtler satiric one, that this couldn’t carry through. The thing comedians in the west tell themselves is that censors are slower to get implications than anyone else. But that’s a self-flattering thing to say and, of course, how would they know?

                Liked by 1 person

  • Joseph Nebus 3:00 pm on Monday, 7 March, 2016 Permalink | Reply
    Tags: , , Dedekind rings, , Fermat's Last Theorem, , , ,   

    A Leap Day 2016 Mathematics A To Z: Dedekind Domain 


    When I tossed this season’s A To Z open to requests I figured I’d get some surprising ones. So I did. This one’s particularly challenging. It comes fro Gaurish Korpal, author of the Gaurish4Math blog.

    Dedekind Domain

    A major field of mathematics is Algebra. By this mathematicians don’t mean algebra. They mean studying collections of things on which you can do stuff that looks like arithmetic. There’s good reasons why this field has that confusing name. Nobody knows what they are.

    We’ve seen before the creation of things that look a bit like arithmetic. Rings are a collection of things for which we can do something that works like addition and something that works like multiplication. There are a lot of different kinds of rings. When a mathematics popularizer tries to talk about rings, she’ll talk a lot about the whole numbers. We can usually count on the audience to know what they are. If that won’t do for the particular topic, she’ll try the whole numbers modulo something. If she needs another example then she talks about the ways you can rotate or reflect a triangle, or square, or hexagon and get the original shape back. Maybe she calls on the sets of polynomials you can describe. Then she has to give up on words and make do with pictures of beautifully complicated things. And after that she has to give up because the structures get too abstract to describe without losing the audience.

    Dedekind Domains are a kind of ring that meets a bunch of extra criteria. There’s no point my listing them all. It would take several hundred words and you would lose motivation to continue before I was done. If you need them anyway Eric W Weisstein’s MathWorld dictionary gives the exact criteria. It also has explanations for all the words in those criteria.

    Dedekind Domains, also called Dedekind Rings, are aptly named for Richard Dedekind. He was a 19th century mathematician, the last doctoral student of Gauss, and one of the people who defined what we think of as algebra. He also gave us a rigorous foundation for what irrational numbers are.

    Among the problems that fascinated Dedekind was Fermat’s Last Theorem. This can’t surprise you. Every person who would be a mathematician is fascinated by it. We take our innings fiddling with cases and ways to show an + bn can’t equal cn for interesting whole numbers a, b, c, and n. We usually go about this by saying, “Suppose we have the smallest a, b, and c for which this is true and for which n is bigger than 2”. Then we do a lot of scribbling that shows this implies something contradictory, like an even number equals an odd, or that there’s some set of smaller numbers making this true. This proves the original supposition was false. Mathematicians first learn that trick as a way to show the square root of two can’t be a rational number. We stick with it because it’s nice and familiar and looks relevant. Most of us get maybe as far as proving there aren’t any solutions for n = 3 or maybe n = 4 and go on to other work. Dedekind didn’t prove the theorem. But he did find new ways to look at numbers.

    One problem with proving Fermat’s Last Theorem is that it’s all about integers. Integers are hard to prove things about. Real numbers are easier. Complex-valued numbers are easier still. This is weird but it’s so. So we have this promising approach: if we could prove something like Fermat’s Last Theorem for complex-valued numbers, we’d get it up for integers. Or at least we’d be a lot of the way there. The one flaw is that Fermat’s Last Theorem isn’t true for complex-valued numbers. It would be ridiculous if it were true.

    But we can patch things up. We can construct something called Gaussian Integers. These are complex-valued numbers which we can match up to integers in a compelling way. We could use the tools that work on complex-valued numbers to squeeze out a result about integers.

    You know that this didn’t work. If it had, we wouldn’t have had to wait for the 1990s for the proof of Fermat’s Last Theorem. And that proof would have anything to do with this stuff. It hasn’t. One of the problems keeping this kind of proof from working is factoring. Whole numbers are either prime numbers or the product of prime numbers. Or they’re 1, ruled out of the universe of prime numbers for reasons I get to after the next paragraph. Prime numbers are those like 2, 5, 13, 37 and many others. They haven’t got any factors besides themselves and 1. The other whole numbers are the products of prime numbers. 12 is equal to 2 times 2 times 3. 35 is equal to 5 times 7. 165 is equal to 3 times 5 times 11.

    If we stick to whole numbers, then, these all have unique prime factorizations. 24 is equal to 2 times 2 times 2 times 3. And there are no other combinations of prime numbers that multiply together to give us 24. We could rearrange the numbers — 2 times 3 times 2 times 2 works. But it will always be a combination of three 2’s and a single 3 that we multiply together to get 24.

    (This is a reason we don’t consider 1 a prime number. If we did consider a prime number, then “three 2’s and a single 3” would be a prime factorization of 24, but so would “three 2’s, a single 3, and two 1’s”. Also “three 2’s, a single 3, and fifteen 1’s”. Also “three 2’s, a single 3, and one 1”. We have a lot of theorems that depend on whole numbers having a unique prime factorization. We could add the phrase “except for the count of 1’s in the factorization” to every occurrence of the phrase “prime factorization”. Or we could say that 1 isn’t a prime number. It’s a lot less work to say 1 isn’t a prime number.)

    The trouble is that if we work with Gaussian integers we don’t have that unique prime factorization anymore. There are still prime numbers. But it’s possible to get some numbers as a product of different sets of prime numbers. And this point breaks a lot of otherwise promising attempts to prove Fermat’s Last Theorem. And there’s no getting around that, not for Fermat’s Last Theorem.

    Dedekind saw a good concept lurking under this, though. The concept is called an ideal. It’s a subset of a ring that itself satisfies the rules for being a ring. And if you take something from the original ring and multiply it by something in the ideal, you get something that’s still in the ideal. You might already have one in mind. Start with the ring of integers. The even numbers are an ideal of that. Add any two even numbers together and you get an even number. Multiply any two even numbers together and you get an even number. Take any integer, even or not, and multiply it by an even number. You get an even number.

    (If you were wondering: I mean the ideal would be a “ring without identity”. It’s not required to have something that acts like 1 for the purpose of multiplication. If we insisted on looking at the even numbers and the number 1, then we couldn’t be sure that adding two things from the ideal would stay in the ideal. After all, 2 is in the ideal, and if 1 also is, then 2 + 1 is a peculiar thing to consider an even number.)

    It’s not just even numbers that do this. The multiples of 3 make an ideal in the integers too. Add two multiples of 3 together and you get a multiple of 3. Multiply two multiples of 3 together and you get another multiple of 3. Multiply any integer by a multiple of 3 and you get a multiple of 3.

    The multiples of 4 also make an ideal, as do the multiples of 5, or the multiples of 82, or of any whole number you like.

    Odd numbers don’t make an ideal, though. Add two odd numbers together and you don’t get an odd number. Multiply an integer by an odd number and you might get an odd number, you might not.

    And not every ring has an ideal lurking within it. For example, take the integers modulo 3. In this case there are only three numbers: 0, 1, and 2. 1 + 1 is 2, uncontroversially. But 1 + 2 is 0. 2 + 2 is 1. 2 times 1 is 2, but 2 times 2 is 1 again. This is self-consistent. But it hasn’t got an ideal within it. There isn’t a smaller set that has addition work.

    The multiples of 4 make an interesting ideal in the integers. They’re not just an ideal of the integers. They’re also an ideal of the even numbers. Well, the even numbers make a ring. They couldn’t be an ideal of the integers if they couldn’t be a ring in their own right. And the multiples of 4 — well, multiply any even number by a multiple of 4. You get a multiple of 4 again. This keeps on going. The multiples of 8 are an ideal for the multiples of 4, the multiples of 2, and the integers. Multiples of 16 and 32 make for even deeper nestings of ideals.

    The multiples of 6, now … that’s an ideal of the integers, for all the reasons the multiples of 2 and 3 and 4 were. But it’s also an ideal of the multiples of 2. And of the multiples of 3. We can see the collection of “things that are multiples of 6” as a product of “things that are multiples of 2” and “things that are multiples of 3”. Dedekind saw this before us.

    You might want to pause a moment while considering the idea of multiplying whole sets of numbers together. It’s a heady concept. Trying to do proofs with the concept feels at first like being tasked with alphabetizing a cloud. But we’re not planning to prove anything so you can move on if you like with an unalphabetized cloud.

    A Dedekind Domain is a ring that has ideals like this. And the ideals come in two categories. Some are “prime ideals”, which act like prime numbers do. The non-prime ideals are the products of prime ideals. And while we might not have unique prime factorizations of numbers, we do have unique prime factorizations of ideals. That is, if an ideal is a product of some set of prime ideals, then it can’t also be the product of some other set of prime ideals. We get back something like unique factors.

    This may sound abstract. But you know a Dedekind Domain. The integers are one. That wasn’t a given. Yes, we start algebra by looking for things that work like regular arithmetic do. But that doesn’t promise that regular old numbers will still satisfy us. We can, for instance, study things where the order matters in multiplication. Then multiplying one thing by a second gives us a different answer to multiplying the second thing by the first. Still, regular old integers are Dedekind domains and it’s hard to think of being more familiar than that.

    Another example is the set of polynomials. You might want to pause for a moment here. Mathematics majors need a pause to start thinking of polynomials as being something kind of like regular old numbers. But you can certainly add one polynomial to another, and you get a polynomial out of it. You can multiply one polynomial by another, and you get a polynomial out of that. Try it. After that the only surprise would be that there are prime polynomials. But if you try to think of two polynomials that multiply together to give you “x + 1” you realize there have to be.

    Other examples start getting more exotic. They’re things like the Gaussian integers I mentioned before. Gaussian integers are themselves an example of a structure called algebraic integers. Algebraic integers are — well, think of all the polynomials you can out of integer coefficients, and with a leading coefficient of 1. So, polynomials that look like “x3 – 4 x2 + 15 x + 6” or the like. All of the roots of those, the values of x which make that expression equal to zero, are algebraic integers. Yes, almost none of them are integers. We know. But the algebraic integers are also a Dedekind Domain.

    I’d like to describe some more Dedekind Domains. I am foiled. I can find some more, but explaining them outside the dialect of mathematics is hard. It would take me more words than I am confident readers will give me.

    I hope you are satisfied to know a bit of what a Dedekind Domain is. It is a kind of thing which works much like integers do. But a Dedekind Domain can be just different enough that we can’t count on factoring working like we are used to. We don’t lose factoring altogether, though. We are able to keep an attenuated version. It does take quite a few words to explain exactly how to set this up, however.

     
    • gaurish 3:33 pm on Monday, 7 March, 2016 Permalink | Reply

      Wow! I just couldn’t believe my eyes while reading this post. It’s beautiful. Thanks for satisfying my curiosity. :)

      Like

      • Joseph Nebus 7:54 am on Wednesday, 9 March, 2016 Permalink | Reply

        I’m glad to be of service, and I hope that you were satisfied. I’d tried a couple times to find a way to describe all the properties of a Dedekind domain in conversational English, and gave up with reluctance. I just got too many thousands of words in without being near the ending and had to try a different goal.

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  • Joseph Nebus 7:27 pm on Thursday, 24 July, 2014 Permalink | Reply
    Tags: Andrew Wiles, , Fermat's Last Theorem, , , , , , , rockets   

    Reading the Comics, July 24, 2014: Math Is Just Hard Stuff, Right? Edition 


    Maybe there is no pattern to how Comic Strip Master Command directs the making of mathematics-themed comic strips. It hasn’t quite been a week since I had enough to gather up again. But it’s clearly the summertime anyway; the most common theme this time seems to be just that mathematics is some hard stuff, without digging much into particular subjects. I can work with that.

    Pab Sungenis’s The New Adventures of Queen Victoria (July 19) brings in Erwin Schrödinger and his in-strip cat Barfly for a knock-knock joke about proof, with Andrew Wiles’s name dropped probably because he’s the only person who’s gotten to be famous for a mathematical proof. Wiles certainly deserves fame for proving Fermat’s Last Theorem and opening up what I understand to be a useful new field for mathematical research (Fermat’s Last Theorem by itself is nice but unimportant; the tools developed to prove it, though, that’s worthwhile), but remembering only Wiles does slight Richard Taylor, whose help Wiles needed to close a flaw in his proof.

    Incidentally I don’t know why the cat is named Barfly. It has the feel to me of a name that was a punchline for one strip and then Sungenis felt stuck with it. As Thomas Dye of the web comic Newshounds said, “Joke names’ll kill you”. (I’m inclined to think that funny names can work, as the Marx Brotehrs, Fred Allen, and Vic and Sade did well with them, but they have to be a less demanding kind of funny.)

    John Deering’s Strange Brew (July 19) uses a panel full of mathematical symbols scrawled out as the representation of “this is something really hard being worked out”. I suppose this one could also be filed under “rocket science themed comics”, but it comes from almost the first problem of mathematical physics: if you shoot something straight up, how long will it take to fall back down? The faster the thing starts up, the longer it takes to fall back, until at some speed — the escape velocity — it never comes back. This is because the size of the gravitational attraction between two things decreases as they get farther apart. At or above the escape velocity, the thing has enough speed that all the pulling of gravity, from the planet or moon or whatever you’re escaping from, will not suffice to slow the thing down to a stop and make it fall back down.

    The escape velocity depends on the size of the planet or moon or sun or galaxy or whatever you’re escaping from, of course, and how close to the surface (or center) you start from. It also assumes you’re talking about the speed when the thing starts flying away, that is, that the thing doesn’t fire rockets or get a speed boost by flying past another planet or anything like that. And things don’t have to reach the escape velocity to be useful. Nothing that’s in earth orbit has reached the earth’s escape velocity, for example. I suppose that last case is akin to how you can still get some stuff done without getting out of the recliner.

    Mel Henze’s Gentle Creatures (July 21) uses mathematics as the standard for proving intelligence exists. I’ve got a vested interest in supporting that proposition, but I can’t bring myself to say more than that it shows a particular kind of intelligence exists. I appreciate the equation of the final panel, though, as it can be pretty well generalized.

    To disguise a sports venue it's labelled ``Math Arena'', with ``lectures on the actual odds of beating the casino''.

    Bill Holbrook’s _Safe Havens_ for the 22nd of July, 2014.

    Bill Holbrook’s Safe Havens (July 22) plays on mathematics’ reputation of being not very much a crowd-pleasing activity. That’s all right, although I think Holbrook makes a mistake by having the arena claim to offer a “lecture on the actual odds of beating the casino”, since the mathematics of gambling is just the sort of mathematics I think would draw a crowd. Probability enjoys a particular sweet spot for popular treatment: many problems don’t require great amounts of background to understand, and have results that are surprising, but which have reasons that are easy to follow and don’t require sophisticated arguments, and are about problems that are easy to imagine or easy to find interesting: cards being drawn, dice being rolled, coincidences being found, or secrets being revealed. I understand Holbrook’s editorial cartoon-type point behind the lecture notice he put up, but the venue would have better scared off audiences if it offered a lecture on, say, “Chromatic polynomials for rigidly achiral graphs: new work on Yamada’s invariant”. I’m not sure I could even explain that title in 1200 words.

    Missy Meyer’s Holiday Doodles (July 22) revelas to me that apparently the 22nd of July was “Casual Pi Day”. Yeah, I suppose that passes. I didn’t see much about it in my Twitter feed, but maybe I need some more acquaintances who don’t write dates American-fashion.

    Thom Bluemel’s Birdbrains (July 24) again uses mathematics — particularly, Calculus — as not just the marker for intelligence but also as The Thing which will decide whether a kid goes on to success in life. I think the dolphin (I guess it’s a dolphin?) parent is being particularly horrible here, as it’s not as if a “B+” is in any way a grade to be ashamed of, and telling kids it is either drives them to give up on caring about grades, or makes them send whiny e-mails to their instructors about how they need this grade and don’t understand why they can’t just do some make-up work for it. Anyway, it makes the kid miserable, it makes the kid’s teachers or professors miserable, and for crying out loud, it’s a B+.

    (I’m also not sure whether a dolphin would consider a career at Sea World success in life, but that’s a separate and very sad issue.)

     
    • BunnyHugger 7:40 pm on Thursday, 24 July, 2014 Permalink | Reply

      I take Holbrook’s point not as “math is unpopular” but as “math is an anathema to gamblers” — because it demonstrates that their behavior is irrational, while they prefer to hold on to superstition and vain hope.

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      • Joseph Nebus 8:13 pm on Saturday, 26 July, 2014 Permalink | Reply

        I think that you’re right about Holbrook’s point.

        But I suspect the claimed lecture title would still attract gamblers, out of a belief that if they knew a little more about how the industry works then they’d be able to beat the house. The belief may be irrational but it’s hard to fight, and sometimes (as with various Blackjack card-counting schemes) it even pays out.

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    • Donna 1:47 am on Sunday, 27 July, 2014 Permalink | Reply

      My friend Vanessa put casual pi day in her Facebook feed. Does that help? I didn’t know about this before this year. I can get you guys to friend each other if need be. She’s also a math prof.

      Like

      • Joseph Nebus 8:17 pm on Sunday, 27 July, 2014 Permalink | Reply

        Helps some, sure. I don’t have any particular reason to disapprove of Casual Pi day, ahouth I suppose it’s worse-timed for fitting into classes than the fourteenth of March will be.

        And, thank you, although I haven’t gotten on Facebook yet. I’m still getting Twitter to make sense, I’m afraid.

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    • Donna 1:48 am on Sunday, 27 July, 2014 Permalink | Reply

      Oh, and I’m so glad I guessed my wordpress password… hmmm maybe it’s not too secure after all…

      Like

      • Joseph Nebus 8:18 pm on Sunday, 27 July, 2014 Permalink | Reply

        Well, if nobody else guessed it first, that’s as secure as it needs to be, isn’t it?

        Like

    • elkement 9:02 am on Sunday, 3 August, 2014 Permalink | Reply

      “Barfly” looks like “Garfield” – isn’t that a copyright issue? (I did not research any background, such as if both cartoons are perhaps created by the same illustrator so it is maybe a stupid comment…)

      Like

      • Joseph Nebus 6:39 am on Monday, 4 August, 2014 Permalink | Reply

        Yeah, “Barfly” is just Garfield, and there’s no connection I’m aware of between the Queen Victoria cartoonist and the Garfield owners. Possibly Pab Sugenis uses an image that was given out for free use for publicity purposes. Possibly the original use of the character was for satirical purposes (I could easily imagine making a joke about Garfield repeating the same pose for the characters over and over, since the comic strip does do that), which would make Barfly at least start out as a fair use. Possibly Sugenis just figures, if they were going to sue him, they’d have done it by now, so what’s he got to lose? Those big web-cartoonist paychecks?

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