tag:blogger.com,1999:blog-66066427842279944282018-03-06T05:14:55.085-08:00Mathematical ImmersionExploring the wild, wonderful world of prime numbersBrowng34http://www.blogger.com/profile/13687131439250620317noreply@blogger.comBlogger12125tag:blogger.com,1999:blog-6606642784227994428.post-4346519120905892302014-05-05T19:50:00.002-07:002014-05-05T19:50:55.647-07:00The Congruent Number ProblemAfter a year and a half of independent study, I am happy to share some of what I've learned about the congruent number problem. This will be the first of my independent math papers for math enthusiasts.<br /><br />Read it here: <a href="http://www.graemebrownart.com/congruent.pdf" target="_blank">the congruent number problem, by Graeme Brown</a>.Browng34http://www.blogger.com/profile/13687131439250620317noreply@blogger.com0tag:blogger.com,1999:blog-6606642784227994428.post-16876194706761459052014-01-18T16:09:00.001-08:002014-01-18T16:09:08.349-08:00Nth Root Continued FractionsAs my writing career takes root and pulls more of my energy, my love for math is designated more and more a hobby. Yet, like any hobby one truly enjoys, this does not mean one cannot find enrichment in it.<div><br /></div><div>I am now a part-time student and enjoy tutoring and helping other students. However, as I study more topics on number theory and computer programming, I continue to work on my own projects. Elliptic curves and congruent numbers have given me a taste of number theory, but recently, one day while waiting for late lunch guests, I scribbled some work on a napkin, and that work led to a general continued fraction expansion for the nth root of any number. I showed it to my adviser and he encouraged me that this is an open topic and something just as worthwhile to explore.</div><div><br /></div><div>And there's no rush - just like there's no rush when you have a hobby of building models. You pick away at it, continue to learn and build, and the surprises are the best part.</div>Browng34http://www.blogger.com/profile/13687131439250620317noreply@blogger.com0tag:blogger.com,1999:blog-6606642784227994428.post-86631507398727177282013-06-03T19:37:00.000-07:002013-08-13T22:29:43.910-07:00Trying to Rationalize with TrianglesLast week, my computer broke. No writing, no programming, no email or book promotion. It was a perfect opportunity for me to rediscover how much I love math.<br /><br />I did a lot of it, and good timing. This summer I have been granted a directed research opportunity at the university. The problem of interest: one that has fascinated me since my adviser introduced it to me in 2011, the congruent number problem. Prior to this, I had been working on generating a lot of data for the sake of understanding some of the patterns behind the scenes. Doing algebra and work by hand gave me a fresh angle which has now given me direction for exploration.<br /><br />No, it's not the Reimann Hypothesis, but if anything my studies have taught me that exploring such a topic requires lots of careful development and time; I am simply not ready for it yet. Fortunately, a lot of the work I did - premature and amateur though it was at the time - has useful carry-over to this problem, itself a special case of the Birch-Swinnerton-Dyer Conjecture (which, it turns out, is deeply intertwined with the Riemann Hypothesis).<br /><br />My interest in number theory and, in particular, the patterns and properties of the primes, is leading me into a fascinating little universe, and I am delighted - and excited - to have this opportunity to work full time as an apprentice researcher. The congruent number problem is a great framework for learning about elliptic curves and the properties of group operations they exhibit when analyzed over finite fields. And this is a great starting point for asking the question, "When will an elliptic curve have rational solutions?"<br /><br />So what is the congruent number problem, exactly?<br /><br />You could google it, but most of what you will find might be quite technical. For those of you who are not mathematicians, but are interested, it's not complicated:<br /><br />You have learned about a right angle triangle as a child. Many people know the Pythagorean Theorem: the sum of the squares of the perpendicular sides of a triangle is equal to the square of the hypotenuse. You also might remember the formula for the area of a triangle from high school: area equals one half base times height.<br /><br />What if I said to you: find me a right angle triangle that has area 1, 2, 3, ... and so on? In other words, what if I gave you a whole number and said to find me a right angle triangle whose area equals this number - call it "n".<br /><br />It would be easy to do if you used your calculator. You can just pick any number for one of the perpendicular sides, say 1. Then the other side will be 2n, when you cross multiply, and the hypoteneuse will be the square root of 4n squared plus one.<br /><br />That's easy; for a mathematician, easy is not interesting. See, depending on what you choose for n, your calculator might put out something like 1.71415444... for the hypotenuse - in other words, a number that doesn't round nicely. But it's still an answer, and if you were an architect you'd have the measurement you need to make sure the pieces of your triangular frame fit together.<br /><br />So we look for rational numbers - a number that is a fraction. Like 1/3 or 4/7 or 34567 / 356733562984498 - you name it. Now let's ask the question again: what triangles of whole number area exist so that we have all rational sides?<br /><br />This is the congruent number problem. It turns out these numbers are 5,6,7, 13, 14, 15, 20, 21, 22, ... this list goes on and on. 6 is the simplest example - that's the area of a 3,4,5 triangle (3 is a fraction: 3/1, as it 4 and 5, and you can quickly check that the area of this triangle is indeed 6).<br /><br />You can try to find congruent numbers with some algebra, but it's messy. When I started with this problem, I used a spreadsheet and captured some triangles by brute force, but there were others, such as the triangle with area 4, that had no answer. In mathematics, no answer is not good enough - one must prove there is none.<br /><br />Fast-forwarding a little, we can use some algebra tricks to construct a special curve called an elliptic curve (it has nothing to do with an ellipse), where the area is built into a curve that varies with x and y - x is related to the sides of the triangle (specifically, x = (c/2)^2 ).<br /><br />Why go and make it messy like that? Mathematics is a bit like a complicated game of minesweeper. Sometimes you must go around to other corners and work out things you know, before that will open up what you need for the part where you are stuck. And so it is with elliptic curves. After all, it was these wonderful little creatures that led to Andrew Wiles' proof of Fermat's Last Theorem.<br /><br />And because of the mad rush to solve Fermat's Last Theorem, we know a lot about elliptic curves. Particularly, we know that if you analyze them over finite fields, you will find certain properties that give a clue to whether they have rational solutions or not. In other words, the elliptic curves we construct with that integer area of interest, should they have rational points (since x is related to our sides) will thus give us a triangle with rational sides and integer area.<br /><br />Now, the biggest question: why do we care? Who ever needs to understand such things about triangles? After all, the architects are not going to be concerned with 0.00000001 of a centimeter of accuracy.<br /><br />Well, the answer is that it's not really about triangles at all. The congruent number problem is about numbers, particularly prime numbers. A triangle is a nice picture for us to relate a deep number theory problem about special ways that collections of products of primes can be formed; it's just one of many unique questions that represent the universe of numbers and their properties.<br /><br />The devil's in the details, and I have my hands full of devils right now. Though angels would be more appropriate. This project is a marriage of computer programming and the skills I am developing, and it is great fun.<br /><br />I write this tonight, at the end of my day, after getting in my hour-long fix of writing, and I'm already looking forward to getting up early and doing it all over again tomorrow.Browng34http://www.blogger.com/profile/13687131439250620317noreply@blogger.com0tag:blogger.com,1999:blog-6606642784227994428.post-29794400102154071152013-01-23T00:01:00.001-08:002013-01-23T00:10:18.694-08:00Spiralling into control - A Program to Unwrap Ulam's SpiralAfter a difficult study term, I had the privilege during the holidays to see plans from September come to fruition, thanks to the ingenuinty of one of my colleagues. Fascination with Ulam's Spiral, which launched my interest in exploring the properties of prime-generating polynomials and expansions of the Riemann zeta function, has taken me into the realm of computing. <br /><br />Vision has become reality, as I now have all the class files correctly compiled in my computer and can, at any time, run the program. What I get is a GUI display of tiles that fill with numbers, as per the winding of Ulam's Spiral, with the added feature of black squares, representing tiles that the number sequence skips.<br /><br />Playing with the spiral program is a bit like playing Angry Birds, if I can borrow my advisor's analogy. It lots of fun, and at least there's a chance that, by playing with it, you might find something useful. So now I'm playing a game I've dubbed "Prime Chasing", something that is a good way of giving my brain a break from editing. <br /><br />That is just a game, of course. The real work is much more involved. Presently, my colleague is preoccupied with other work, but we have devised two stages of development of our program to systematically explore ranges of black block permutations and to relate these to the lengths of diagonal lines of primes. The algorithm is simple, but the coding takes a bit of time, a thing which neither of us has a the present.<br /><br />This project is exciting for me as a student who has a deep interest in prime numbers and their properties. Taking these next two steps will allow me to explore some ideas I have tried to wade through algebraically, to no avail. It is my hypothesis that Ulam's Spiral, as a basis for a larger set of spirals formed by the permutations of skipping blocks, holds some deep answers about the pattern of the primes, and, possibly, to the Riemann Hypothesis. Though the mathematical scope of such analysis is far beyond me, it is my hope that generating such results and refining our project based on the initial set will provide data that will be useful to the community of mathematicians who endeavour to expand our understanding of what numbers are.<br /><br />At the very least, I hope it is better than a game of Angry Birds.Browng34http://www.blogger.com/profile/13687131439250620317noreply@blogger.com0tag:blogger.com,1999:blog-6606642784227994428.post-16590817856363408002012-12-06T12:14:00.000-08:002014-03-29T11:15:45.008-07:00Building the FoundationsHere I am at the end of an intense study term, and I'm diving under for a big feat: the hardest exam I've ever written, for the hardest course I've ever taken. Writing this post - long overdue - was one of my decided break activities in my planned 12-hour work day. (Day one of 5 :S)<br /><br />Studying mathematics rigorously has really taught me a lot about myself, mostly about how I think (or fail to). If not for a deep love and fascination for the topic, I would have jumped ship long ago. I'm forever humbled by (and envious of) the prodigies whose natural aptitude lies in mathematical reasoning. Here comes me the writer trying to piece together the story that the beautiful logic makes, and it's quite a mess.<br /><br />I've realized amidst this, though, that I am a writer at heart. As November passed and I revived my story-telling habits, I realized that, had I to chose one thing to give my life to, it would be writing.<br /><br />Meanwhile, though, there's so much to learn in mathematics, and by that I don't mean theorems and proofs and calculation techniques. Learning how to organize my thoughts has been an important sort of "mental yoga", an often unpleasant experience of wrestling down faulty logic so that what is true and sound comes out on top. The best part of this: it's transfered to my writing, my daily life, and through it I feel I am growing as a person.<br /><br />Alright. Break's over. The nice thing is when this is done - December 10th - I will be launching full-swing on the novel I've started. Boy, I can't wait. I aim to finish the manuscript (or at least get to 50,000 words) by January. But I must confess, after I reach my 5000 word goal for each day, I'll be doing some math problems. After all, I might be a writer at heart, but my love for math will never be far away. In fact, on this road I'm coming to know as my life, they make good companions.Browng34http://www.blogger.com/profile/13687131439250620317noreply@blogger.com0tag:blogger.com,1999:blog-6606642784227994428.post-22231384129050291232012-10-20T12:49:00.003-07:002012-10-25T19:10:55.436-07:00Formula for SuccessEinstein once said, "<span lang="EN-US">If A equals success, then the formula is A = X + Y + Z. X is work. Y is play. Z is keep your mouth shut."</span><br /><br />This more or less reflects the advice I was given when I met to consult on the work I've been doing on mathematical topics that are far over my head. It is good to be enthusiastic about some of the difficult problems, but so many people have been working on them, many who are very gifted and resourceful, and they still remain unsolved. While it is true that I have come up with a unique approach, the formality of checking to see if it leads anywhere is far beyond my present ability.<br /><br />What is within my reach, though, is a wonderful opportunity I was given last week. The professor I work with supported me to come and research full-time next summer on the congruent number problem and related topics. I presented my results to him and he suggested that we look into getting a grant. It was a straightforward process and within the day we were putting the application in. This grant is not as competitive as NSERC, and as I qualified for NSERC, we expect there is a very good chance I will get it. I have my fingers crossed!<br /><br />Meanwhile, all my energy is on my studies. This is a formal second year, with all of the honours math topics, and it is teaching me that my talent is mediocre. Real analysis is a lot of work, particularly because it forces one to part with lazy thinking habits. There is no hand-waving. Mathematics is a language of exactness and precision. Sometimes this leads to a headache, but I suppose its no different than the cramps I remember feelings when I trained to run in the marathon.<br /><br />Needless to say, I'm working very hard, and making sure to balance my week out with a bit of exercise, some fun on the weekend, and of course, writing. Please be sure to check out my Twitter feed, as I tweet on a mathematician every day.Browng34http://www.blogger.com/profile/13687131439250620317noreply@blogger.com0tag:blogger.com,1999:blog-6606642784227994428.post-422308031861572062012-09-26T16:22:00.000-07:002012-09-26T16:23:19.113-07:00September 26, 2012: The Riemann AdventureEver since I was introduced to Ulam's Spiral in May, the Riemann Hypothesis, a problem that fascinated me (but which I had decided was too hard to bother with), has had me in its grip. Perhaps it was the confidence I gained after working on the congruent number problem with another professor that made me realize if I really wanted to, undergratuate or not, I can learn what I need to and have some fun. So that's what I've been doing.<br /><br />My work on the Riemann Hypothesis has come in many growth spurts. Each time, it's felt like I am getting closer and closer. The original exploration came from a unique choice I made in how to represent the sum, that later an attempt to understand it geometrically (in images that could be explained to a classroom) led to another interesting connection. At the beginning of this month, I had a conjecture, based on some of the data I had gathered, but it wasn't solid enough.<br /><br />The term has been very busy, so both the RH and my other explorations have only come out now and then - usually when I have an idea that I simply cannot ignore. Last night - or should I say this morning - was one of those times.<br /><br />Prior to this, I had been exploring some of the properties or composite expansions that naturally cropped up in the unique way I had written the zeta function. I noticed that there were certain types of series that could be written to span all the numbers - much like the proof of Euler's infinite product series.<br /><br />It was 1:30am, and the idea came to me like a dream. For a while I did math in my sleep, then I realized I was awake, and the moment was now. Long story short, by 6am I had an outline for how to turn that idea into a proof, and have been exploring it on and off today, in between study sessions for my courses. I had a chance to run my reasoning by a professor who was familiar with series analysis, and he encouraged me to follow up on it.<br /><br />This is very exciting! However, I can't be sure that the correct series can be found easily (if at all). I have some rigorous (and careful) expansions to do, some trial and modification, and perhaps some more surprises from intuition, but regardless of where this goes, there's no doubt it's going to be fun. <br /><br />And, I must add, it's keeping my trigonometry and algebra skills sharp!Browng34http://www.blogger.com/profile/13687131439250620317noreply@blogger.com0tag:blogger.com,1999:blog-6606642784227994428.post-90193869426341467762012-09-18T19:50:00.000-07:002012-09-18T19:50:06.341-07:00September 18, 2012: In the Number LabResearch into the congruent number problem has taken on a fun spin after being given incentive to present my findings in a seminar this spring.<br /><br />I'm starting to develop a sense of how to organize and conduct my research, and this has led to the construction of a "number lab" of sorts. As a student, I have found that organization is my saving grace. By nature, I work chaotically, so I found a way to label my pages, tabbing them for quick access; that allows me to move between problems and go with new ideas without losing myself. <br /><br />I've applied this to my research. Work on the Riemann Hypothesis has its own place, and work on the congruent numbers has another, and as I am conducting an investigation over fields, I've tried to organize everything into tables. So far, I've found and proven three lemmas en route to looking for some sort of theory to explain the pattern of solutions that might be extrapolated to the whole field of integers. It's very much like being a scientist in a lab. When I got to the field of size 17, I took the time to write a computer program to generate larger fields (and verify calculations in others). This was very helpful, because I was able to quickly work on the next field, over 19, which showed there was an exception to a fourth lemma that I had struggled to prove (and now I know why, and am working on proving it prove another angle).<br /><br />This is wonderful fun! I'm studying four math topics, and as I learn, not only am I enjoying the new challenges and techniques I assimilate, but the opportunity to tackle my hobby research from new angles. The best part is, I can bring my lab with me anywhere, and I don't have to worry about anything blowing up...Browng34http://www.blogger.com/profile/13687131439250620317noreply@blogger.com0tag:blogger.com,1999:blog-6606642784227994428.post-90204429656424810742012-09-09T10:11:00.000-07:002012-09-09T10:11:36.317-07:00September 9, 2012: The Second Year RegimenThe study term has started and I simply cannot contain the excitement. This second year is chock-full of some important building blocks that I hope will help me with my independent work on both the Riemann hypothesis and the congruent number problem. I have four courses, but it is the third year course, introduction to topology, which interests me the most. I am hopeful that some of the concepts of metric spaces will assist in the geometric proof I have been trying to put together since May.<br /><br />Meanwhile, I began an investigation into modularity patterns for the elliptic curves associated with the congruent numbers, and have thus far found and proven three lemmas that have half-formulated a theorem about the relationship between cubic and quadratic residues. I have analysed curves for all n up to the field formed by 13 elements, and hope to find some useful generalization that extends between fields as they increase (thus approximating the integers), with the hypothesis that there will be some pattern emerging for values of n that correspond to congruent numbers.<br /><br />It is all a little bit of undergraduate fun, which I enjoy. Not only do I get to learn lots of neat and useful things, but my innate curiosity for problem-solving is satisfied. There's nothing like taking abstract concepts and finding a practical use for them. After all, that's what mathematicians do.Browng34http://www.blogger.com/profile/13687131439250620317noreply@blogger.com0tag:blogger.com,1999:blog-6606642784227994428.post-38086375803972459432012-08-29T11:13:00.002-07:002012-08-29T11:13:58.638-07:00August 29, 2012: Onward to ModularityAfter a month of independent work, I had a meeting with the professor who has offered to help me with my research. This morning I got on the wrong bus, so ended up taking an hour to get to the university. It meant I arrived right on time. It also meant I managed to get through the rest of Chahal and Osserman's paper about the Riemann Hypothesis for Elliptic curves.<br /><br />Because I am an undergraduate with a deep curiosity for particular topics (RH, BSD conjecture), I am presented with the challenge of grappling with concepts that are beyond what I have (formally) learned. It means I have to teach myself, and that I get to constantly upgrade what I'm doing as I learn ways to do it better. I am very grateful to Dr. Padmanabhan for his time, as it gives me an idea where to look to learn the concepts I need to. I have also started adding to my daily word research routine the task of looking up one mathematician a day. Usually, this leads to a whole bunch of mischief. Yesterday, for instance, reading about David Hilbert led to researching the meaning of transfinite numbers and how proof theory complements model theory, recursion theory and axiomatic set theory to form the four pillars of modern mathematics. I have a notebook devoted to mathematicians' names, and usually get 5 to 10 for every mathematician I look up (let's not mention the two pages I filled when I stumbled on the list of those who have won the Fields Medal and the Wolf Prize). This new part of my routine is very exciting, because it is something I can build upon daily. With Godel on my radar for today, who can say what new things I will know about math at bedtime? As the short-lived Niels Heinrik Abel put it, "Study the masters, not their pupils".<br /><br />For those who follow my writing blog (<a href="http://www.fantasywritingjourney.wordpress.com/" target="_blank">Graeme Brown Winnipeg fantasy writer</a>), you are likely aware of how I have wrestled with to fit writing - a passion that pulls at my heart - in with another passion that I want to devote my life to. The best advice I have ever heard from someone was, "Do both." With the start of my publishing career last month, I can no longer say writing is just a hobby. So, as the month of August has passed, I've been devoting a lot of time to the learning curve involved with establishing an online routine that is sufficient. My number one purpose of blogging is to provide interaction with those who are interested in the work I do, and because I do different kinds of work, I have made different blogs to suit the context. This means a lot of maintenance, and it means a need for efficiency.<br /><br />As the term resumes, I look forward to having a well-balanced diet of writing and study. My work on the Riemann Hypothesis and congruent number problem continues, and, as always, I am looking every day to take my mathematical toolbag to a new edge. I was happy to hear that one of my proposed directions of research on the congruent number problem looks promising, so that will be my task for now. Unfortunately, the problem of understanding the Riemann Hypothesis is one I cannot part with, and this means a neverending kick at the can. My current intuition on how to approach the problem has evolved from an idea that came to me during a particularly bad sermon at church. At the time I was taking an algebraic approach, breaking Euler's product expansion into chunks and developing a notation for a combinatorial function (then analyzing the holomorphic functions comprising the numerator and denominator), and though I found some interesting convergence properties, there was nothing telling. My idea was to look for a modularity pattern in the ring 2pi as the primes evolved, but I could not find a useful connection between the natural logarithm and a prime to a given exponent, even though intuition would say that e and pi's relation should be linked through some form of trigonometric analysis. Later, I analyzed its general term for various values of the complex coefficient as a composition of three functions, but this did not lead to the node of convergence I expected. As I was working through the paper by Chahal and Osserman, I returned to Ulam's Spiral (which kick-started my interest in this new approach to RH), looking for a connection between limited sums of squares. This, too, led to nothing useful. Then, yesterday morning, while reading and enjoying coffee, I had a new idea to move from squares to circles in the geometric model (which has an interesting link to the Bernoulli numbers) of the proof, and I devised a four-step research strategy. Inevitably, I will explore this one as well, and if it leads to a wall, I'm sure there will come another cup of coffee and a new idea. There's always more coffee when it comes to math. That's why I have a mug that has pi on it.<br /><br />Now, if you are a mathematician and want a cool mathematical mug, I've made something special and it's available at Cafepress: <a href="http://www.cafepress.com/mathmilestones.685946703" target="_blank">Coffee cup for math lovers: relationship between pi, e and i</a>Browng34http://www.blogger.com/profile/13687131439250620317noreply@blogger.com0tag:blogger.com,1999:blog-6606642784227994428.post-9760163413898836312012-08-20T08:56:00.002-07:002012-08-20T08:56:20.289-07:00August 20, 2012: Mixing Math and HobbiesAre you passionate about Math? You might want to check out the latest project I've started. <a href="http://www.cafepress.com/mathmilestones" target="_blank">Math Milestones: from Pi to Euler to the Imaginary</a> is a shop I will be building (about one product a month) which will feature a design based on one of the mathematicians or mathematical topics I've looked up (I do one a day). For the launch, I have featured the logo, depicting the relationship of Euler's constant, pi and the imaginary symbol, three milestones of math that are as might as Gauss, Euler and Reimann (though I say mightier).<br /><br />The next (potential) project will be based on the fractal Sierpinski curve. I am a vector graphics artist, and the idea for this project has been evolving over the last several months as a way to lateralize what I already do, while also tieing into my studies. As my university doesn't have a history of math program, I have taken it upon myself to research mathematicians and the relationships between topics in mathematics to build the picture for myself. I hope that in five years time, doing this daily, I will have somewhat adequate knowledge to tackle some of the deep mathematical problems that often wake me up early in the morning.Browng34http://www.blogger.com/profile/13687131439250620317noreply@blogger.com0tag:blogger.com,1999:blog-6606642784227994428.post-18615952028374837712012-08-16T13:01:00.002-07:002012-08-16T19:14:49.544-07:00August 16, 2012: Groups and Elliptic CurvesAs my first post here, I think it would be fitting to tell you how I got to where I am today, so here goes.<br /><br />My journey with math took at exciting turn in the fall of 2011 when I took a number theory course, after a few years away from school. I did extremely well in the course and maintained a connection with the professor, meeting biweekly to consult on a problem he gave me. Little did I know, this was in fact the Birch-Swinnerton-Dyer conjecture.<br /><br />I have had a deep love for prime numbers for the last several years, but now that I have studied number theory and taken a number of higher-level math courses, I realize this is a true passion that will lead me deeper down the rabbit hole every day. And it's a good rabbit hole.<br /><br />As the summer began, I put the congruent number problem aside, realizing I needed more formal training in groups and abstract algebra. I studied computer programming, and was acquainted with Ulam's spiral. On my early morning walks to work at Starbucks, I contemplated this spiral and some of the patterns it exhibited, realizing there was a deep connection between prime-generating polynomials and, ultimately, the Riemann Hypothesis. I did a great deal of work on this before setting it aside. Of note was a modified Ulam Spiral that evolved along the Riemann line (square root sequence), but I was unable to formalize my intuition that fractional modifications of this addition would lead to an infinite set of convergent areas.<br /><br />After a meeting with my professor, we discussed Hasse's Theorem and I realized, after going full circle, that the kernel to gaining further insight into the Riemann Hypothesis lay in first understanding Birch-Swinnerton-Dyer, which meant tackling groups and modular forms of elliptic curves on my own. I have spent the last week and a half of my summer working on this, and it is going quite well. I feel like a mathematical teenager going through growth spurts, the type that sometimes have me waking up at 1am and working until dawn (though only occasionally, and with the help of coffee).<br /><br />So, here I am. I look forward to sharing, now and then.Browng34http://www.blogger.com/profile/13687131439250620317noreply@blogger.com0