Journal of Integer Sequences, Vol. 17 (2014), Article 14.6.6

On Integral Points on Biquadratic Curves and Near-Multiples of Squares in Lucas Sequences

Max A. Alekseyev
Department of Mathematics
The George Washington University
2115 G St., NW
Washington, DC 20052

Szabolcs Tengely
Mathematical Institute
University of Debrecen
4010 Debrecen, PO Box 12


We describe an algorithmic reduction of the search for integral points on a curve y2 = ax4 + bx2 + c with ac(b2 - 4ac) ≠ 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.

We illustrate this approach with a particular application of finding near-multiples of squares in Lucas sequences. As an example, we establish that among Fibonacci numbers only 2 and 34 are of the form 2m2+2; only 1, 13, and 1597 are of the form m2-3; and so on.

As an auxiliary result, we also give an algorithm for solving a Diophantine equation k^2 = f(m,n)/g(m,n) in integers m, n, k, where f and g are homogeneous quadratic polynomials.

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(Concerned with sequences A000032 A000045 A000129 A002203.)

Received February 22 2014; revised version received May 17 2014. Published in Journal of Integer Sequences, May 18 2014.

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