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\vskip 1cm{\LARGE\bf On Integral Points on Biquadratic Curves \\
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and Near Multiples of Squares in \\
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Lucas Sequences
}
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\large
Max A. Alekseyev\\
Department of Mathematics \\
The George Washington University \\
2115 G St., NW \\
Washington, DC 20052 \\
USA \\
\href{mailto:maxal@gwu.edu}{\tt maxal@gwu.edu} \\
\ \\
Szabolcs Tengely \\
Mathematical Institute \\ 
University of Debrecen \\
4010 Debrecen, PO Box 12 \\
Hungary \\
\href{mailto:tengely@science.unideb.hu}{\tt tengely@science.unideb.hu} \\
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\begin{abstract}
We describe an algorithmic reduction of the search for integral points on a curve $y^2 = ax^4 + bx^2 + c$ 
with $ac(b^2-4ac)\ne 0$
to solving a finite number of Thue equations.
While existence of such reduction is anticipated 
from arguments of algebraic number theory, 
our algorithm is elementary and to best of our knowledge 
is the first published algorithm of this kind.
In combination with other methods and powered by existing software Thue equations 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 $2m^2+2$; 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 $k^2 = \tfrac{f(m,n)}{g(m,n)}$ in integers $m, n, k$, 
where $f$ and $g$ are homogeneous quadratic polynomials.
\end{abstract}