An explicit version of Thue's method with applications.
by
Gary Walsh
University of Ottawa
Coauthors: Michael Bennett, Alain Togbe, Paul Voutier

Ljunggren proved that the Diophantine equation X²-DY⁴=1 has at most two solutions in positive integers, and gave precise information on the location of the solutions when two solutions exist. Using an explicit version of Thue's hypergeometric method, we partially solve a parametric family of binary quartic form equations, and apply this result to sharpen Ljunggren's theorem. We also completely solve a subfamily of these Thue equations and use this to generalize Richard Bumby's result on the equation 3X⁴-2Y²=1.

Date received: January 17, 2005