Written by Michael Rouleau
Willimantic, Conn. - Two members of Eastern Connecticut State University's Mathematics Department have made a discovery in the field of mathematics known as "number theory." Eastern mathematics professor Mizan Khan and Richard "Ricky" Magner, a junior majoring in mathematics, will have their discovery published in Volume 14 of the electronic journal INTEGERS.
The research, titled "Two Combinatorial Geometric Problems Involving Modular Hyperbolas," was a collaborative effort among four scholars, including Khan and Magner, as well as Steven Senger of the University of Delaware and Arne Winterhof of the Austrian Academy of Sciences.
"The research concerned two problems, and Ricky answered one of them," said Khan. "Ricky's discovery is quite pretty; he is very clever."
The questions leading their research were: "given a finite collection of points on a two-dimensional grid, how many distinct lines can you draw connecting two or more points in that collection?"; and "what conditions ensure that a line connecting two points in that collection do not meet a third point?"
The answers to these seemingly simple questions are indecipherable for those without some background in number theory--which deals with the properties and relationships of integers, or whole numbers.
"Questions in number theory are easy to state," said Magner, "but they are difficult to answer, and their implications are often unknown."
During Magner's freshman year, Khan approached him after class with an excerpt from a book dealing with "modular hyperbolas"--an area of number theory that Khan has been working on since 1999. "At first I didn't really know what we were looking for," said Magner, "but in the fall of my sophomore year, after spending much time with modular hyperbolas, things started to come together."
Both Khan and Magner agree that the discovery alone has no practical application. "While the solution is elegant, this is a minor discovery," said Khan. "In this case, it is the process that is important, not the solution. My hope is that Ricky will build on this experience to prove bigger theorems in the future when he is in graduate school."
"Through solving problems you develop skill and build an 'arsenal,' which can lead to new discoveries and expand the field of mathematics," said Magner. "At the time, you may not know if the discovery is useful. It may be years before its use is realized."
Magner is currently taking graduate mathematics courses at UConn, in addition to his full-time workload at Eastern. After obtaining his master's degree, he plans to apply for a PhD program in mathematics, while still investing time in his passion for writing and other intellectual pursuits.