After 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.
Read it here: the congruent number problem, by Graeme Brown.
Monday, May 5, 2014
Saturday, January 18, 2014
Nth Root Continued Fractions
As 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.
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.
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.
Monday, June 3, 2013
Trying to Rationalize with Triangles
Last 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.
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.
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).
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?"
So what is the congruent number problem, exactly?
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:
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.
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".
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.
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.
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?
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).
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.
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 ).
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.
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.
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.
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.
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.
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.
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.
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).
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?"
So what is the congruent number problem, exactly?
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:
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.
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".
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.
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.
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?
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).
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.
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 ).
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.
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.
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.
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.
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.
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.
Wednesday, January 23, 2013
Spiralling into control - A Program to Unwrap Ulam's Spiral
After 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.
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.
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.
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.
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.
At the very least, I hope it is better than a game of Angry Birds.
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.
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.
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.
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.
At the very least, I hope it is better than a game of Angry Birds.
Thursday, December 6, 2012
Building the Foundations
Here 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)
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.
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.
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.
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.
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.
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.
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.
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.
Saturday, October 20, 2012
Formula for Success
Einstein once said, "If A equals success, then the formula is A = X + Y +
Z. X is work. Y is play. Z is keep your mouth shut."
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.
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!
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.
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.
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.
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!
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.
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.
Wednesday, September 26, 2012
September 26, 2012: The Riemann Adventure
Ever 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.
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.
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.
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.
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.
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.
And, I must add, it's keeping my trigonometry and algebra skills sharp!
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.
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.
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.
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.
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.
And, I must add, it's keeping my trigonometry and algebra skills sharp!
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