The Diophantine equation aX4-bY2=1

  • Date: 11/02/2006
Lecturer(s):

Shabnam Akhtari (University of British Columbia)

Location: 

University of British Columbia

Topic: 

In a series of papers over nearly forty years, Ljunggren derived
remarkably sharp bounds for the number of solutions to various quartic
Diophantine equations, particularly those of the shape aX4-bY2=±1,
typically via a sophisticated application of Skolem's p-adic method.
More recent results along these lines are well surveyed in a paper of
Walsh. For general a and b, however, there is no absolute upper bound
for the number of integral solutions to aX4-bY2=1 available in the
literature. Computations and assorted heuristics suggest the following
conjecture of Walsh: For any positive integers a and b, the equation
aX4-bY2=1 has at most two solutions in positive integers X and Y. In
this talk, we will appeal to a classical result of Thue from the theory
of Diophantine approximation to deduce the following result: For any
positive integers a and b, the equation aX4-bY2=1 has at most three
solutions in positive integers X and Y.

Other Information: 

SFU/UBC Number Theory Seminar 2006

Sponsor: 

<a href="http://pims.math.ca" rel="blank"><img src="/files/users/PIMS_Logo_Colour_tiny.gif" alt="pims" height="72" width="46" /></a>