So there's this conjecture in math that just fascinates me.
Conjectures:
To clarify, a conjecture in math is something math folks think is true but haven't been able to prove yet. There are a lot of conjectures. You can make up a conjecture pretty easily by just stringing together a bunch of math symbols and numbers and then making some unfounded statement about the monster you just gave birth to.
There are a couple of problems with this kind of conjecture though, first of all, the symbols need to be combined in a way that makes sense: 5++++/3=* doesn't make any sense, whereas 5*5=30 makes sense, but is wrong... Which brings us to the second problem: even if it makes sense, and it's complicated enough that no one has thought to prove it yet, it might be able to be proved very quickly. Consider this conjecture (broken up into pieces to better fit on the screen, but imagine it as one long line):
1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+
1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+
1+1+1+1+1+1+1+1+1+
1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+
1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+
1+1+1+1+1+1+1+1+1+1+1+1+1+11+1+1+
1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+
1+1+1+1=
13+13+13+13+13+26+13+13+13
I'm pretty sure that's true because of the way I made it, but it's fairly long so it's probable that no one has ever bothered to prove EXACTLY that conjecture before. Proving it just involves adding up both sides and seeing if they are equal, but I'm not going to do that, in order to give my conjecture a little more life.
Once somebody (maybe you) does that, it won't be a conjecture any more. If I didn't make any typos and it came out the way I meant it to, then once you prove it, it'll become a theorem. If you add up both sides and find out that they're different, then you will have refuted my conjecture and it will disappear back into the mist where wrong but intelligible math statements come from. As a side note, you could then change my conjecture slightly to reflect what you found: you could change the equals sign to a "not equals" sign, and then the new statement would be a theorem stating that the two sums are not equal to each other.
There is a deeper problem, of course, with the conjecture I made: it's totally uninteresting. Once you prove or disprove it, you don't get anything else out of it. It doesn't shed any light on any other areas of mathematics or the world. It doesn't give you any insights about how to solve other problems. It doesn't even pose the challenge of making you think, "how could I go about proving this?" So the conjecture I just made up will likely not be remembered for very long, even if, by chance and negligence, it goes unproven for many years.
Some conjectures are different though. Some conjectures make sense and are hard to prove and in proving them you have to tie together all kinds of different branches of mathematics that weren't connected before and then, in solving the conjecture, you discover all sorts of new and potentially useful things along the way. Some conjectures are so interesting and hard, that even if you don't prove them or disprove them, just working on them leads you to discover all kinds of things about the math world that you never knew before, and moreover, that nobody ever knew before.
Famous Conjectures:
There are a bunch of famous conjectures. In fact, some of the most famous conjectures have recently been proven, so now they aren't conjectures anymore; now they're theorems. In the 90s, there was Fermat's Last Theorem (which should've been called Fermat's Last Conjecture) which was proved by Andrew Wiles, and recently the Poincare Conjecture which was proved by this crazy Russian guy, Grigori Perelman (who was offered the Fields Medal for solving it, which is like the Nobel prize for math, and became the first person in history to turn it down).
Perelman also stands to win all or part of a million dollars for solving that problem, since the Poincare Conjecture is one of the Millennium Problems posted by the Clay Institute, all of which have a $1,000,000.00 bounty on them. His proof is still pretty recent, so they haven't gotten around to offering the prize to him (there are some other "politics of the math world" reasons and technical reasons due to the prize rules too that contribute to them not offering it to him yet, but it's a huge deal in the math world and they probably will sooner or later), but it's not clear that he'll accept the money even if they offer it to him, since he already turned down the Fields Medal for his work on it.
So anyway, there are a lot of conjectures, some famous, some proven, many still unproven (some with huge prizes for solving them). My favorite is called the Goldbach Conjecture.
The Goldbach Conjecture:
The Goldbach Conjecture is my favorite because it is very easy to state and make sense of (you don't need any advanced math to understand it), but no one has been able to solve it since the time it was posed in 1742, almost 267 years ago!
One of the first people to work on it was Leonard Euler (pronounced "oiler"), one of the greatest mathematicians in history. He couldn't prove it but was certain that it was true.
There are a bunch of ways to state Goldbach's Conjecture. The way he put it was modified by Euler into the way it is most commonly known now:
"Every even number bigger than 2 can be written as the sum of exactly two prime numbers."
Some examples of this being true are 2+2=4, 3+3=6, 3+7=10, 19+23=42, et c. In fact, it's been checked by computers up to several trillion, but still no one is sure why it needs to be the case!
Again, to fill in the math for folks who forget from school, an even number can be divided into two equal whole number parts (parts with nothing left over and nothing after the decimal point). So 10 is even because 5+5=10. 11 is not even (it's odd) because 11/2=5.5 and we aren't allowed to get into the decimals. The best we could do with 11 is 5+6=11 but 5 and 6 aren't the same so we're out of luck.
A prime number is a munber with nothing after the decimal point, that, when you try to divide it by any whole number smaller than itself (but bigger than 1) you always get a number with stuff after the decimal point (or in other words, you never get another whole number). For example, 6 is not a prime number because we can divide 6 by 2 and get 3 which is another whole number. 5 is a prime number because dividing 5 by any of the numbers between 1 and 5 gives you some decimal parts: 5/4 is 1.25, 5/3 is 1.66666666..., 5/2 is 2.5. So since none of those work, 5 is prime.
Prime numbers are really important in math, it ends up turning out. They come up all over the place even in places where you wouldn't expect them (kinda like Pi, that number you get from working with circles). So Goldbach's conjecture is interesting to math people because it says something about primes that people don't understand. Since primes are important, math people want to know everything they can about them, and, eventhough there are lots of things people have figured out about primes over the course of human history, there are lots of things people still don't know about them so the mathematicians feel like if they figure out Goldbach's Conjecture, maybe that will help them figure out more stuff they want to know. ::deep breath::
Back to me:
So I think about Goldbach's Conjecture a lot.
I think about it because it seems so deceptively simple: primes, even numbers, adding. What could be simpler, right? Fermat's Last Theorem was at least a little intimidating. He said, approximately, "x^n+y^n=z^n has no whole number solutions when n is bigger than 2." This has variables, it has exponents, you have to prove that something is never true, which is usually harder than proving something is true. Goldbach's Conjecture gives you the impression that if only you thought about it the right way, it would be very easy and the proof would, in a sense, jump out at you.
So instead of trying to compete with math giants like Euler and such who know much much more math than I will likely ever know and who are much better at using that knowledge to prove stuff, I amuse myself with looking for "the unlocked backdoor" to the problem, the way of visualizing the idea in just the right way, or picture, or metaphor, so that the answer falls right out on it's own.
In this blog, I'll probably talk a lot about different ways I've come up with for thinking about and visualizing this conjecture. I think they are really interesting. Maybe some of you will think so, too.
Conjectures:
To clarify, a conjecture in math is something math folks think is true but haven't been able to prove yet. There are a lot of conjectures. You can make up a conjecture pretty easily by just stringing together a bunch of math symbols and numbers and then making some unfounded statement about the monster you just gave birth to.
There are a couple of problems with this kind of conjecture though, first of all, the symbols need to be combined in a way that makes sense: 5++++/3=* doesn't make any sense, whereas 5*5=30 makes sense, but is wrong... Which brings us to the second problem: even if it makes sense, and it's complicated enough that no one has thought to prove it yet, it might be able to be proved very quickly. Consider this conjecture (broken up into pieces to better fit on the screen, but imagine it as one long line):
1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+
1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+
1+1+1+1+1+1+1+1+1+
1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+
1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+
1+1+1+1+1+1+1+1+1+1+1+1+1+11+1+1+
1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+
1+1+1+1=
13+13+13+13+13+26+13+13+13
I'm pretty sure that's true because of the way I made it, but it's fairly long so it's probable that no one has ever bothered to prove EXACTLY that conjecture before. Proving it just involves adding up both sides and seeing if they are equal, but I'm not going to do that, in order to give my conjecture a little more life.
Once somebody (maybe you) does that, it won't be a conjecture any more. If I didn't make any typos and it came out the way I meant it to, then once you prove it, it'll become a theorem. If you add up both sides and find out that they're different, then you will have refuted my conjecture and it will disappear back into the mist where wrong but intelligible math statements come from. As a side note, you could then change my conjecture slightly to reflect what you found: you could change the equals sign to a "not equals" sign, and then the new statement would be a theorem stating that the two sums are not equal to each other.
There is a deeper problem, of course, with the conjecture I made: it's totally uninteresting. Once you prove or disprove it, you don't get anything else out of it. It doesn't shed any light on any other areas of mathematics or the world. It doesn't give you any insights about how to solve other problems. It doesn't even pose the challenge of making you think, "how could I go about proving this?" So the conjecture I just made up will likely not be remembered for very long, even if, by chance and negligence, it goes unproven for many years.
Some conjectures are different though. Some conjectures make sense and are hard to prove and in proving them you have to tie together all kinds of different branches of mathematics that weren't connected before and then, in solving the conjecture, you discover all sorts of new and potentially useful things along the way. Some conjectures are so interesting and hard, that even if you don't prove them or disprove them, just working on them leads you to discover all kinds of things about the math world that you never knew before, and moreover, that nobody ever knew before.
Famous Conjectures:
There are a bunch of famous conjectures. In fact, some of the most famous conjectures have recently been proven, so now they aren't conjectures anymore; now they're theorems. In the 90s, there was Fermat's Last Theorem (which should've been called Fermat's Last Conjecture) which was proved by Andrew Wiles, and recently the Poincare Conjecture which was proved by this crazy Russian guy, Grigori Perelman (who was offered the Fields Medal for solving it, which is like the Nobel prize for math, and became the first person in history to turn it down).
Perelman also stands to win all or part of a million dollars for solving that problem, since the Poincare Conjecture is one of the Millennium Problems posted by the Clay Institute, all of which have a $1,000,000.00 bounty on them. His proof is still pretty recent, so they haven't gotten around to offering the prize to him (there are some other "politics of the math world" reasons and technical reasons due to the prize rules too that contribute to them not offering it to him yet, but it's a huge deal in the math world and they probably will sooner or later), but it's not clear that he'll accept the money even if they offer it to him, since he already turned down the Fields Medal for his work on it.
So anyway, there are a lot of conjectures, some famous, some proven, many still unproven (some with huge prizes for solving them). My favorite is called the Goldbach Conjecture.
The Goldbach Conjecture:
The Goldbach Conjecture is my favorite because it is very easy to state and make sense of (you don't need any advanced math to understand it), but no one has been able to solve it since the time it was posed in 1742, almost 267 years ago!
One of the first people to work on it was Leonard Euler (pronounced "oiler"), one of the greatest mathematicians in history. He couldn't prove it but was certain that it was true.
There are a bunch of ways to state Goldbach's Conjecture. The way he put it was modified by Euler into the way it is most commonly known now:
"Every even number bigger than 2 can be written as the sum of exactly two prime numbers."
Some examples of this being true are 2+2=4, 3+3=6, 3+7=10, 19+23=42, et c. In fact, it's been checked by computers up to several trillion, but still no one is sure why it needs to be the case!
Again, to fill in the math for folks who forget from school, an even number can be divided into two equal whole number parts (parts with nothing left over and nothing after the decimal point). So 10 is even because 5+5=10. 11 is not even (it's odd) because 11/2=5.5 and we aren't allowed to get into the decimals. The best we could do with 11 is 5+6=11 but 5 and 6 aren't the same so we're out of luck.
A prime number is a munber with nothing after the decimal point, that, when you try to divide it by any whole number smaller than itself (but bigger than 1) you always get a number with stuff after the decimal point (or in other words, you never get another whole number). For example, 6 is not a prime number because we can divide 6 by 2 and get 3 which is another whole number. 5 is a prime number because dividing 5 by any of the numbers between 1 and 5 gives you some decimal parts: 5/4 is 1.25, 5/3 is 1.66666666..., 5/2 is 2.5. So since none of those work, 5 is prime.
Prime numbers are really important in math, it ends up turning out. They come up all over the place even in places where you wouldn't expect them (kinda like Pi, that number you get from working with circles). So Goldbach's conjecture is interesting to math people because it says something about primes that people don't understand. Since primes are important, math people want to know everything they can about them, and, eventhough there are lots of things people have figured out about primes over the course of human history, there are lots of things people still don't know about them so the mathematicians feel like if they figure out Goldbach's Conjecture, maybe that will help them figure out more stuff they want to know. ::deep breath::
Back to me:
So I think about Goldbach's Conjecture a lot.
I think about it because it seems so deceptively simple: primes, even numbers, adding. What could be simpler, right? Fermat's Last Theorem was at least a little intimidating. He said, approximately, "x^n+y^n=z^n has no whole number solutions when n is bigger than 2." This has variables, it has exponents, you have to prove that something is never true, which is usually harder than proving something is true. Goldbach's Conjecture gives you the impression that if only you thought about it the right way, it would be very easy and the proof would, in a sense, jump out at you.
So instead of trying to compete with math giants like Euler and such who know much much more math than I will likely ever know and who are much better at using that knowledge to prove stuff, I amuse myself with looking for "the unlocked backdoor" to the problem, the way of visualizing the idea in just the right way, or picture, or metaphor, so that the answer falls right out on it's own.
In this blog, I'll probably talk a lot about different ways I've come up with for thinking about and visualizing this conjecture. I think they are really interesting. Maybe some of you will think so, too.
I saw the word "math" and failed to read your blog. I just wanted to let you know that I will be reading it....except when you talk about math. Because I am math-retarded.
ReplyDeletehaha, fair enough. though I really go to great lengths to make the math understandable by anyone who made it out of lower school, so give it a try sometime you might be surprised how much you understand. :)
ReplyDeleteI will try...but it is very hard for me to enjoy math or to even be slightly interested in it. BUT since you are making such a gallant effort, I'll see what I can do. :)
ReplyDelete