The Collatz Conjecture

In this installment I would like to talk about the Collatz conjecture. As usual I would first like to talk about the contributors, and history of the problem. Then I will talk about the problems itself. Finally, I will talk about what the problem means to math, and why its not that big of a deal. Unlike the problems I have previously mentioned, this problem is incredibly simply, and anyone can understand it in its entirety. In fact, if you can do addition and multiplication, then you can understand the problem.

It seems only right to start with Collatz himself. Collatz has a German mathematician, who was born in 1910. Although Collatz made contributions to math outside his conjecture, such as the Collatz-Wielandt formula, or his contributions to the Perron-Frobenius theorem, his conjecture is the most famous. He died in 1990 at the age of 80. It is not until the 1970’s and 1980’s with the emergence of the personal computer that his conjecture gained popularity. The advancement in computations allowed for checking for counter claims to the conjecture to be much easier. Unlike many of the other problems, the Collatz conjecture is not a millennial problem, so there is not a million-dollar prize for proving or disproving the conjecture. However, Paul Erdos offered 500 dollars for solving it, but he also said it would be pointless to even try to solve it. To quote him he said that, “Mathematics may not be ready for such a problem.”  Now let us look at the problem at which math is not ready for.

That is it, that is the entire problem, only two lines. So to explain the two lines. Start by picking a number. Any whole number like seven. So if it is an even number divide by two, if it is an odd number multiply by 3 and add 1. To carry on with seven, multiply by three to get 21, then add one to get 22. 22 is an even number so I will divied by two. to get 11, then 34,17,52,26,13, and 40. At this point it looks like it just keeps getting bigger and bigger over time. However, 40 is the turning point because the next number is 20, then 10,5,16,8,4,2,1,4,2,14,2,1. Notice once you get to one, then it repeats itself in a loop forever. This is not just for the number 7, in fact go ahead and try any number between 1 and infinity. However it is recommend to choose a small number to save time, but it is possible to try any number. The Collatz conjecture simply says that for any number that these conditions are performed to, it will always end up at one. So far it is known that all whole numbers to 2 raised to the 60^th power have been confirmed to follow this conjecture, but there is no proof that all number follow it. It is one of the hardest conjectures in math to prove, but could be easily explained to a fourth grader.

Now as mentioned that there is no million-dollar prize for proving or disproving the conjecture. This is because millennial problems have deep connections to many parts of math, science, and physics. The Collatz conjecture really does not have those kind of connections. So by solving this you will not cure cancer, or fix the flaws of cold fusion, but it will progress math.  Some of you might ask why multiple by three and add one? Or 3n+1, and the answer to that is if you cannot solve a problem you try to generalize the problem, or solve a similar problem. Mathematicians tried to solve the general form of the equation seen as an+b. The mathematicians found that was an even harder problem to solve. So it remains as 3n+1, if they can solve 3n+1 then they might gain how to solve an+b.

So the Collatz conjecture is a conjecture that anyone can understand, but no one can solve. The reason why there is not a large prize for the its solving is it does not need to be solved. Problems like the Riemann hypothesis, and the Navier-Stokes equation have connections to prime numbers and fluid mechanics, unfortunately the Collatz conjecture has no connection. Nevertheless, I encourage you, if you are interested, to go for it. For the million-dollar prize problem that has been solved, the prize was refused. This is because mathematicians do math for the math, not the money.

Sources https://www.youtube.com/watch?v=5mFpVDpKX70
              https://www.youtube.com/watch?v=O2_h3z1YgEU
              http://mathworld.wolfram.com/CollatzProblem.html

The Navier Stokes Equations

This week I would like to talk about the Navier-Stokes existence and smoothness problem. I would first like to talk about the history of the problem, then I would like to talk about the problem itself, then I would like to talk about the physics involved in the problem, and finally I would like to talk about changing the problem to allow for its use.

The problem is named after its contributors Claude-Louis Navier and Georg Gabriel Stokes. Navier was french physicist that lived in the late 1700's and early 1800's. He grew up and took interest in engineering and physics. He made a few contributions to math and science, but his main contribution was the Navier-Stokes equations. Stokes, the other contributor, was born in Ireland. Stokes lived in the 1800's roughly the same time period as Navier. Stokes became the head of mathematics at Cambridge, before dying in 1903. His contributions were similar to that of Navier. They both specifically worked on fluid dynamics. They came up with several equations that to me have too many letters from too many alphabets, but some how someone understands them fully.

The problems itself refers to solving the equations created by Navier, and Stokes. As mentioned before the equations deal with fluid dynamics, but there are a few more conditions dealing with the problems. First the fluids are assumed to be non-compressible. Fluids if you are not aware refer to gases and liquids. These fluids are assumed to be incompressible even though fluids are compressible to a small degree, the change in density, income cases, is so small it is negligible. The second condition is the fluid must be viscous. These equations under those conditions then describe fluid mechanics using newton second law of motion, that being force is equal to mass multiplied by acceleration. As you can imagine this is a big deal. Understanding fluids is what gets your airplanes in the air or your water to your house. These equations sometimes have solutions in certain situation, but mathematicians can not seem to prove that these conditions always exist. There is not a certainty that the smoothness condition can always exist in the three dimensions. On top of that if we assume that the smoothness condition always applies then the question remains, if these smoothness solutions exist do the have bounded energy for their mass? Like many of the problems will be talking about, this problem is a millennial problem. That means if you can solve the problem, then you will receive a million dollars. You will also probably be awarded a Field's Medal, and get to be on the front of a newspaper. Society as a whole will still probably care more about what the Kardashians are wearing instead of the solution to an incredibly important explanation of fluid dynamics, but oh well.

At this point it would be good to mention what the equations actually pertain to in fluid dynamics, but the content of the equations would go over a lot of heads and lose people.  Rather then showing all the equations, it would be more appropriate to give a brief description of the general form of the equations. 

This is the general form of the equation. All of the equations can be derived from this equation.

The first "p" represents the density of the fluid. The "v" represent velocity component. The second "p" refers to pressure. The "t" refers to time. The upside down debt refers to gradients or vector derivatives. Now I wish I could fully explain the equation, but I hope it makes more sense to you as a reader than it does to me.

Now these equations are not useless. In fact they are very important. As mentioned there is not a guarantee on our answers. However, there are still ways of getting numerically practical answers out of the equations. For instance, sometime by assuming factors lie change in density or pressure, as negligible you can get a really close answer. So sometimes you do not need to be able to solve the equation straight up, sometimes you can just get close enough for it to work. Hopefully you as an enthusiast will come up with a solution to get rid of these approximations.

Sources http://www.claymath.org/millennium-problems/navier–stokes-equation
              https://www.grc.nasa.gov/www/k-12/airplane/nseqs.html
              https://math.berkeley.edu/~chorin/chorin68.pdf

The Poincaré Conjecture

            For this installment I would like to talk about the Poincaré conjecture. Again I will probably get a comment saying that the Poincaré conjecture has already been solved. Even though the Poincaré conjecture has been solved, it is still a great problem that was not solved until recently as 2003. I would like to first talk about the history of the problem, then I want to continue with the contributors, the problem itself, and finally I would like to talk about the prize associated with solving the Poincaré conjecture.

            The Poincaré conjecture was formulated by Henri Poincaré, a French mathematician. He lived during the 19^th century and had many works in physics and math. The Poincaré conjecture has to do with a field of math know as topology. Topology is the study of geometric properties not concerning a continuous change in shape or size. It means you are a mathematician who plays with play-dough.  Topology is basically about manipulating shapes in all dimensions. For a long time no one could prove the conjecture, so in the 1960’s and 70’s when topology became popular many mathematicians took a swing at it. The problem concerned proving a property of a sphere in all special dimensions. The conjecture was proven in all dimensions except the fourth spacial dimension. It was not until the development of something called Ricci Flow that the fourth dimensional property was finally proven in 2003.

            The man who had solved it was Grigori Perelman. There is not a whole lot of information out there about him. He essentially came, solved, and left. After using Ricci Flow to prove the conjecture, he went on a tour giving lectures to universities about his work, then he stopped after a year, and went back into isolation. When he had posted his proof no one had ever seen him before, and he did not want the publicity with solving it.

            Now to get to the actual problem. The conjecture is closely related to understanding the shape of our universe. So the universe could be a sphere, or it could not be a sphere. Instead it could be a more tube kind of shape with twists and turns and holes. Poincaré said if someone traveled around the edge of the universe and left a rope behind them as they went, once they have completed their journey then there is a close loop of some kind. At this point, pull on either end and synched up the rope.  Continue to pull and pull and if the rope can not be pulled anymore then the universe is not a sphere. If they can pull all the way, then it is. That is what the conjecture entails. Another perspective on this is if a have a single piece of clay with no holes in it I can make it a sphere. Simply by rolling the clay between my hands it can be done. If I have a doughnut shape or a shape with more holes I cannot turn it into a sphere without pinches or tears in the surface. That is simple to see in 3 dimensions, but Poincaré suggested that, that is the case in all dimensions. The conjecture could be proven in every dimension except the fourth special dimension, until Perelman did in 2003.

            Perelman’s discovery was a big deal in the math community. This problem is known as a millennial problem. In the year 2000 the Clay Institute of Mathematics realeased seven unsolved problems of math. These were considered the greatest unsolved problems at the time. The Poincaré conjecture is the only one to be solved since then. Each problem has a million dollar set aside for the winner. In a turn of events Perelman refused the money. Perelman was also awarded the fields medal. The fields medal is the equivalent of a Nobel prize in math. However, he refused both the medal and the money. It goes to show you that Perelman was not interested in the glory or the fame, but rather in the math itself.  The man is considered a hero in topology, if he wanted to he could get a job at any university he wanted. All he had to do is put on his resume that he solved this problem, and he would instantly be hired. Like mentioned before, Perlman was not about the fame, so he went back to his isolation. Hopefully this can show you that mathematicians do math for the sake of math not anything else, or it can also show you what math means to them.

Fermat's Last Theorem

            Today’s blog post will about Fermat’s Last Theorem. I know right off the bat I am going to get a comment saying that Fermat’s last theorem has been solved, and you would be correct. However, it is still a great mathematical problem that was unsolved for a couple hundred years. So no this is not an unsolved problem, but it is one of the simpler problems to explain, that was once an unsolved problem. First I would like to talk about some background information, such as the origin of the problem, the problems contributors, etc. Then I want to talk about the problem itself, and how it works. Then I want to talk about how common it comes up in society. I will not be talking about the proof in this post because, it is too long for a single post, in the future I might post another, or maybe two more posts to explain it in depth but that will not be happening in this post.

            It seems only right to start with Fermat himself. Pierre de Fermat was a French man who lived during the 17^th century in France. He himself was not a mathematician, in fact he was a judge. He was a judge that had an interest in math during his spare time. He would read these published mathematical texts and write notes in the margin. On one occasion he was reading a book called Arimetica by Diophantus. He stumbled on an equation that was very similar to that of the Pythagoras, that being X^2+Y^2=Z^2. He was curious to an alteration of the problem he wanted to know if the equation had any other solutions that would satisfy a higher order of this equation. He thought about it, and then in the margin wrote something to the extent of, “I can prove that there is no solution to a higher order, but there is not room for me to write it in the margin.” His book was littered with these little marginal notes. He would write,” I have a proof to this, but I am too busy to write it out.” Fermat then without proving anything died. His son found his annotated copy of the book and had it republished with his father’s notes added to it. Because of this, mathematicians started working on these proofs that Fermat claimed to have been able to prove. Now proofs are flying in left and right, one after another, the marginal notes are being ratified. That was all good until, they got to this one involving Pythagoras’ equation. The theorem could not be solved.  This problem remained un solved for centuries afterward. The problem itself is not that hard to explain, but by the 20^th century it is understood that this problem does not have a simple solution. Many mathematicians have tried a crack at the problem, but to no avail, until Andrew Wiles comes along. Andrew Wiles grows up learning and loving the problem. So after going through college getting a PHD and going on with life. Someone proves that if you can solve The Taniyama Shimura conjecture, you would also be proving Fermat’s last Theorem as well in doing so. Now it is not important to know about the Conjecture other than by solving it Wile’s would get what he wanted. He spent several years working on it alone, then gave a lecture solving the conjecture. Unfortunately, he had made a mistake. It took him another year, but then he fixed his mistake. After 8 years he had gotten what had been trying to do since the age of ten.

            The problem itself is rather straight forward. If you have had junior high math you know Pythagoras’ equation X^2+Y^2=Z^2. Well Fermat thought to himself, "I wonder if there are solutions to equations like X^3+Y^3=Z^3 or to the fourth power." He could not find any whole number solutions. He said that there were no whole number solutions to any power greater than two so in math terms, there is no solution to X^N+Y^N=Z^N where X,Y and Z are not equal, and N is greater than two. That is all the problem is. I said it was a simple problem. If you do not believe me, go ahead try to find a solution.

            Now I mentioned that this problem actually comes up in the media. There are a few famous instances of this. The main one being the Simpsons. In one the episodes when Homer is writing on the chalk board doing calculations it flashes an apparent solution to Fermat’s unsolvable equation. The equation shown on the board is obviously not correct, but rather to pay tribute to the mathematician. Another example in a book called, The Devil and Simon Flagg, where Simon makes a deal with the devil that he gets to keep his soul if the devil can prove Fermat’s Last theorem. Also the problem crops up in star trek. The point is, the problem is more common than you think it is. So maybe next time someone makes reference to the problem you will be able to spot it.

            

Why Math?

I think that I have always been interested in math. I think my interest in math comes from two sources. The first source being my own curiosity. From a young I have always liked to take things apart and look at the guts of things. This is because I like to understand how things work. If you think about it math is that exact same thing. Math can be used to explain anything. My math teacher taught me something that changed the way I look at math. Some people see math as just a bunch of numbers, because of my math teacher I do not see it that way. So my math teacher walked up to the board and drew a big 2 on the board, and then he asked what is that? The obvious response is well that is a two, but he said, no that is the symbol that represents the idea of two. The numbers are just place holders for the actual manipulation of physical things. That is a little crazy to think about, but it is a great way of looking at it. The second source of my interest in math comes from my dad. My dad is a math teacher. He has taught at Park City at the junior high, high school, and even for UVU at the college level. Because of that he has always talked to us about math. Most families do not sit around the table and talk about complex numbers, or have math competitions. Because of that I feel that I have been more exposed to math then most kids.

You may say, oh this kid is a high school student, what does he know? I am only a senior, so I am only taking BC calculus this year, but I have interest in some of these unsolved problems. To give you some perspective on my prior knowledge I plan to talk about the Riemann hypothesis zero’s, the zeta function included in those. I also plan to talk about Fermat’s last Theorem, and how it deals with elliptical curves. Or maybe the P vs. NP problem and how it deals with computer science computation. Now if you know any of these problems, you know that I know what I am talking about, and if you do not know anything i just said, you will after I am done with this blog. Through my research I probably will not be able to explain a full in depth analysis of every problem, the math would kill some of my audience. However, it is not too complicated to explain the basic idea of these problems and what their discoveries are. For example, Fermat’s last theorem deals with high integers values of the Pythagoras’ theorem. Now that is rather brief, but it did not require me to know very much math. Hopefully though, I will gain an advancement in my understanding of math, and I am excited to do so. 

I plan on learning this information from books, videos, and other enthusiasts of math. For example, my calculus teacher has several books on Euler’s and Fermat’s conjectures, that he let me borrow. Another great source for math junkies is YouTube. Channels such as Numberphile, Mathlogger, and 3Blue1Brown are all great sources on these unsolved problems, and math in general. The most common source for information these days is the internet, so I will be looking through many websites as well for the information to compose this blog. Sometimes when mathematicians put math in terms for a general audience in these posts, they over simplify for conceptual reasoning. I hope to be able to overcome that as much as possible. However, I am human so I will probably make a mistake of two but, out of the 9 to 12 people who will ever stumble upon my blog, I am sure someone will correct me in the comments like they always do.

I hope to get a few things out this blog. My first hope is that I personally learn a lot about math, and its workings. Not only the material but also its contributors, and the legacy of these great mathematicians. Second, I hope The people who reads this gain something. Knowing how bad of a writer I am, I can guarantee I will lose some people in my rambling of some complex obscure math equations. Even though that may happen I hope people will gain a new perspective on math. Maybe I have this unrealistic standard but, I see math as a beautiful representation of nature itself. Out of all of this I hope to show some people what I see.