Unsolvable and Unsolved Problems
Date: 02/19/98 at 18:16:57 From: D-Jitsu Subject: 11th Grade Research Paper Dear Dr. Math, I need help getting started on a research paper, as well as on doing the project. The topic of my report is "The Impossible and Unsolved in Mathematics (Goldbach's Conjecture, Squaring the Circle, etc.). I was just wondering if I could get some references, tips, or ideas on how to get started. This is for a pre-calculus class.
Date: 02/20/98 at 16:31:35 From: Doctor Martin Subject: Re: 11th Grade Research Paper This is a very interesting topic to explore - it would be good to distinguish between problems which have been proven unsolvable, and those unsolved. For instance, "squaring the circle" and the trisection of general angles with straightedge and compass are examples of problems which have been proven unsolvable. These are problems of construction, and it has been proven that certain things are inconstructable. Goldbach's Conjecture, on the other hand, is not necessarily unsolvable, but just hasn't yet been solved. In other words, no one has proved Goldbach's Conjecture, but no one has proved that it's unprovable either. I would begin with this distinction, and discuss specific mathematical problems, some which are unsolvable, and some unsolved. Unsolvable problems include the squaring of the circle, the trisection of general angles, solving the quintic with radicals, and the continuum hypothesis. Unsolved problems that aren't too hard to understand include Goldbach's conjecture, the 3n+1 conjecture, and the general Poincare conjecture. A good first place to look would be a history of mathematics book, such as Carl Boyer's book, _A History of Mathematics_, or Morris Kline's _Mathematical Thought from Ancient to Modern Times_. Most of the problems mentioned above are in these sources, with the exception of the most recent 3n+1 conjecture. There is, however, some material on this problem (which also goes under the name "The Collatz Problem") on the Web. A good starting point would be: http://ug.cs.dal.ca/~campbell/colllink.html Even though the Collatz problem is not as historically important as the others mentioned, it may be especially interesting for your discussion. First of all, it is fairly easy to understand. Second, it is an example of a problem that is unsolved, and there is a little evidence that it may be unsolvable (due to J. H. Conway). Even though this response is already getting somewhat long, maybe I should describe this problem specifically. It goes like this: Choose a positive integer n. Submit it to the following process: (1) If n is odd, let your new number equal 3n + 1 (2) If n is even, let your new number equal n/2 (3) If your new number isn't equal to 1, put it back into step (1). So for the number 7, this process yields the following sequence: 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. The 3n+1 problem, or the Collatz Conjecture, states that no matter what number you start with, you eventually arrive at 1. Most people think that this is true, but it may be unprovable. This problem, I think, conveys the distinction between unsolved problems and unsolvable problems particularly well. Good luck on your project - it's a great topic to write about! -Doctor Martin, The Math Forum http://mathforum.org/dr.math/
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