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

Department of Mathematics

The George Washington University

2115 G St., NW

Washington, DC 20052

USA

Szabolcs Tengely

Mathematical Institute

University of Debrecen

4010 Debrecen, PO Box 12

Hungary

**Abstract:**

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} - 4*ac*) ≠ 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 2*m*^{2}+2; only 1, 13, and 1597 are of
the form *m*^{2}-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.

(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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