15. Given that $m, n$ are integers, $m, n \in\{1,2, \cdots, 1981\}$, and $\left(n^{2}-m n-\right.$ $\left.m^{2}\right)^{2}=1$, determine the maximum value of $m^{2}+n^{2}$.
15. If $(m, n)$ is a solution to the equation $\left(n^{2}-m n-m^{2}\right)^{2}=1$, then $n \geqslant m$, with equality holding if and only if $m=n=1$. This means $(1,1)$ is a solution to the equation.
When $(m, n) \neq(1,1)$, assume $n-m>0$, then
$$
\left[m^{2}-m(n-m)-(n-m)^{2}\right]^{2}=\left(n^{2}-m n-n^{2}\right)^{2}=1 \text {. }
$$

This implies that if $(m, n)$ is a solution satisfying the above conditions and $(m, n) \neq(1,1)$, then $(n-m, m)$ is also a solution satisfying the above conditions. Similarly, if $(n-m, m) \neq(1,1)$, then $(m-(n-m), n-m)$ is also a solution. Since the number of solutions satisfying the conditions is finite, after a finite number of steps, the solution must be $(1,1)$. Conversely, from a solution $(m, n)$, we can obtain the solution $(m, n+m)$. Thus, starting from the solution $(1,1)$, we can obtain all solutions:
$$
\begin{array}{r}
(1,1) ;(1,2) ;(2,3) ;(3,5) ;(5,8) ;(8,13) ;(13,21) ;(21,34) ;(34,55) ; \\
(55,89) ;(89,144) ;(144,233) ;(233,377) ;(377,610) ;(610,987) ;(987,
\end{array}
$$
1597 ), so the maximum value of $m^{2}+n^{2}$ is
$$
987^{2}+1597^{2}=3524578
$$