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