An Infinite Family of Thue Equations With at Most Two Solutions

For an odd positive integer , defineWe prove that, for ≥ 5, the Thue equation ( , ) = 1 has exactly two solutions in integers and , namely (1, 0) and (-1, 0), and that the Thue equation ( , ) = -1 has no solutions.Our result is a consequence of the famous theorem of Bilu, Hanrot and Voutier on Lucas sequences.