Math Recreation

Team. We have a recreational mathematician on our squad who is interested in honoring an old friend and teacher. The friend was blind, but amazingly gifted. He could play the piano and had a paper route. The teacher was always encouraging and had a gift for teaching college level subjects from Seth Werner's book on Modern Algebra and Oystein Ore's book on The History of Number Theory in simple ways. The procedure is called the Miller-Kovarik Secondary Method. It is useful in resolving Diophantine Equations. It partially answers Hilbert's 10th Problem. Given a Diophantine equation such as C = 2xy + x + y, one can find its solution by first setting it so it equals zero. Then, one must embed it in a parabolic function through composition, g(x) = ( 2xy + x + y - C )^2. Then, one determines the outer bounds of the solution in the first quadrant. Next, one defines a mesh describing the surface which has a single minimum. This mesh is n x n. This means that it has much fewer points than the total surface. It is an approximation of the surface. Afra Zomorodian wrote a wonderful dissertation on describing the salient features of a shape through decimation. It is a wonderful and interesting work in topology. Basically, the next step is taking some statistical measures of the mesh points. After finding the appropriate deviation of the mesh nearest zero, one has made a step at finding the minimum. This smaller mesh is your secondary. The original mesh was the primary. The process is repeated on the surface comprising the smaller mesh with the resolution of the original mesh. Find the third deviation and repeat until the lone root exist at zero or one find a space in which one can search all the points quickly.. Miller first presented the surface embedding idea in secondary school, hence the double entendre. Being blind, he could not fully visualize the mesh. But, during his presentation, he did say that all the points are not need; he had the surface described as a Bull's Eye with the target at zero. A peer, McQuiddy played the devil by confusing him which is sad and unfortunate, otherwise we might he might have reached this result. Kovarik said that some cleverness was needed for resolving this problem. After reading Afra's thesis around 2005, a moment of cleverness occurred. I know this is hand waving, but I imagine most undergraduate computing or math students might derive this quickly. This approach works well for monotonic Diophantine equations.


Hunt. Peck. Think. Happy Coding.