Summary: We describe an algorithmic reduction of the search for integral points on a curve $y^{2} = ax^{4} + bx^{2} + c$ with $ac(b^{2} - 4ac) \ne 0$ to solving a finite number of Thue equations. While the existence of such a reduction is anticipated from arguments of algebraic number theory, our algorithm is elementary and is, to the best of our knowledge, the first published algorithm of this kind. In combination with other methods and powered by existing Thue equation solvers, it allows one to efficiently compute integral points on biquadratic curves.

11Y50, 11D25, 11B39, 14G05

integral point, biquadratic curve, elliptic curve, thue equation, Fibonacci number, Lucas sequence