Algebra and Topology Seminar

4pm Tuesday 12 March 2002

Keith Matthews
University of Queensland

Solving the diophantine equation x²-Dy²=N for nonsquare D>1

We describe an algorithm of Lagrange (1770) which should be better known and which is based on simple continued fractions. Lagrange's proof of his necessary condition for solubility was long and not easy to follow; instead, we present a simple proof. Solutions divide into finitely many equivalence classes by virtue of Pell's equation x²-Dy²=1. The fundamental solutions (those with least y > 0 in each class) are constructed in the case of solubility. The algorithm has been implemented in the speaker's number theory calculator program CALC (available at http://www.numbertheory.org/calc).




Return to list of seminars