Geodesic
On integral points on biquadratic curves and near-multiples of squares in Lucas sequences
Journal of integer sequences, Tome 17 (2014) no. 6
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.
Classification : 11Y50, 11D25, 11B39, 14G05
Keywords: integral point, biquadratic curve, elliptic curve, thue equation, Fibonacci number, Lucas sequence 
@article{JIS_2014__17_6_a5,
     author = {Alekseyev,  Max A. and Tengely,  Szabolcs},
     title = {On integral points on biquadratic curves and near-multiples of squares in {Lucas} sequences},
     journal = {Journal of integer sequences},
     year = {2014},
     volume = {17},
     number = {6},
     zbl = {1358.11141},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2014__17_6_a5/}
}
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Alekseyev,  Max A.; Tengely,  Szabolcs. On integral points on biquadratic curves and near-multiples of squares in Lucas sequences. Journal of integer sequences, Tome 17 (2014) no. 6. http://geodesic.mathdoc.fr/item/JIS_2014__17_6_a5/