## The Diophantine equation aX4-bY2=1

- Date: 11/02/2006

Shabnam Akhtari (University of British Columbia)

University of British Columbia

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.

SFU/UBC Number Theory Seminar 2006

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