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