The Problem with 33

   

     In this weeks installment, the topic of the post will be on a problem concerning the number thirty three. This is a simpler problem, with not a large amount of backstory so the focus will primarily be on the problem itself, and not its contributors. This is partially due to the fact that there is limited information on the founders. In order to compensate for this the founders of the math used in the problem will be covered. That will be covered secondly  The first part of the post will deal with the problem itself. 

     The problem of 33 seems simple on the outside, but it has rather complex working behind it. This is because the problem stems from number theory, which deserves many blog posts about by itself, but will be mentioned briefly in this one. The problem starts out with a list, or sequence of numbers. The sequence stars out with one, then two, then three, then skips 4 and 5. There will be several numbers that are skipped that seem a bit random, but this will be addressed later on. The sequence continues without being interrupted, until it skips thirteen and fourteen, then it skips twenty two and twenty three, and finally it skips thirty one and thirty two. The sequence does continue on, but the sequence has reached thirty three, the important part of the problem. All the numbers on this sequence are thought to be sums of three cubic whole numbers. Meaning that they can be expressed as the sum of three whole numbers raised to the third power. In math terms the numbers on the sequence can be written as equaling a^3 + b^3 +c^3.  This is a Diophantus equation, and has been brought up in a previous post about Fermat's last theorem. In this particular case the negative values of the numbers are also allowed in order to allow for subtraction. In order to illustrate this the example of the solution to the number three will be provided. Three can be written as one cubed, plus one cubed, plus one cubed, or 1 + 1 + 1. This may seem simple, and many of the smaller integers do have simple solutions that can be worked out, but some numbers on the sequence have very large number solutions. It was not until 1999 when the solution for the number thirty was confirmed by the use of computers. The solution to the number thirty is, (2,220,422,9320)^3 + (-2,218,888,517)^3 + (-283,059,965)^3. This seems like a rather large sum of numbers, so it shows the scale of some of the answers to this sequence. 

     This is where the number thirty three comes along. Thirty three does not yet have a solution. Not only is the solution not know, whether there is a solution is also unknown. Not being able to determine if the number in the sequence can have a whole number solution to the Diphantus equation is irregular. So far it is know to all numbers up to 10^14 can be proven to have a solution or not have a solution. Now backtracking to the sequence itself there where a few numbers that were skipped. This is because they were proven to not have a solution to the Diophantus equation. It has been proven that if the number on this sequence can be written as nine multiple by a whole number plus four, or plus five, then it can not be written as a sum of three cubes. In mathematical terms if the number can written as 9n + 4, or 9n + 5, with n being a whole number, then it can not be a sum of three cubes. This has been proved, but the proof will not be covered in this post. So the number four was skipped because it can be written as 9(0) + 4. That is the problem concerning the number thirty three.

     As mentioned previously, this has to do with a branch of math called number theory. Number theory is the study of integers. Many of its contributor's work can be seen in this problem, or in work closely related to this problem. Diophantus equations are the biggest ones, but the work of Euler and  Gauss, with their summations can be seen.  Also the work of Ramanujan can be seen in the partitions of this problem, partitions being the ways you can break down number into sums and multiples. This means that even though this problem seems rather insignificant it has many ties to great mathematicians, and great math.

      Sources https://www.youtube.com/watch?v=wymmCdLdPvM
                    http://www.mathpages.com/home/kmath071.htm