A Tale of Two by Two Matrices

Abstract

We formulate the Euclidean algorithm using directed graphs on integer points in the plane and operations on a particular semigroup of two by two matrices. The properties of the graphs and the semigroup provide surprisingly effective tools for solving certain classical diophantine equations, among other applications.

MSC::

• 11D09
• 15A99

Acknowledgment

The authors thank the referees for many helpful comments and suggestions.

Notes

1 This problem dates at least to 1730, when it was discussed in letters between Goldbach and Euler [5, 6].

Additional information

Notes on contributors

Dan Kalman

DAN KALMAN and ROBERT MENA have been friends, colleagues, and coauthors since the late 1980s, when Mena was a new department chair at Cal State Long Beach and Kalman was a member of the technical staff at the Aerospace Corporation. They share an appreciation of discrete mathematics, number theory, and the history of math, with a special affinity for matrices. With another coauthor, Shahriar Shahriari, they won an Allendoerfer award in 1998. Now retired after teaching for 30 years and 48 years, respectively, they continue to enjoy studying, talking about, and writing about mathematics. Among their non-mathematical pursuits, Kalman solves crossword puzzles and Mena solves acrostics. [email protected], [email protected]

Robert Mena

DAN KALMAN and ROBERT MENA have been friends, colleagues, and coauthors since the late 1980s, when Mena was a new department chair at Cal State Long Beach and Kalman was a member of the technical staff at the Aerospace Corporation. They share an appreciation of discrete mathematics, number theory, and the history of math, with a special affinity for matrices. With another coauthor, Shahriar Shahriari, they won an Allendoerfer award in 1998. Now retired after teaching for 30 years and 48 years, respectively, they continue to enjoy studying, talking about, and writing about mathematics. Among their non-mathematical pursuits, Kalman solves crossword puzzles and Mena solves acrostics. [email protected], [email protected]