On a certain family of quadratic Thue equations
We consider the parameterized Thue equation X^4 − 4sX^3Y − (2ab + 4(a + b)s)X^2Y^2 − 4absXY^3 + a^2b^2Y^4 = ±1, with a, b ∈ 1/4Z such that ab ∈ Z. By the hypergeometric method and a method of Tzanakis we find all solutions, if s is large with respect to |a| and |b|.
Thue  who proved that Diophantine equation (1.1) has only finitely many solutions (X, Y) ∈ Z^2. The proof of this theorem is based on Thue’s approximation theorem.
Diophantine equations, parameterized Thue equations, norm form equations, simultaneous Pellian equations
Ziegler, V. (2006). On a certain family of quadratic Thue equations with three parameters. Pg 9-30.