## Task A-3.3.

How many ordered pairs of natural numbers $(m, n)$ satisfy the equation $m^{2}-n^{2}=$ $2^{2013} ?$
## Solution.

The given equation can be written in the form $(m+n)(m-n)=2^{2013}$.

Consider the system

$$
\left\{\begin{array}{l}
m+n=a \\
m-n=b
\end{array}\right.
$$

where $a$ and $b$ are natural numbers.

For $m$ and $n$ to be natural, $a$ and $b$ must have the same parity, and obviously $a > b$. Under these conditions, the system has a unique solution.

The original equation is equivalent to a series of systems:

$$
\left\{\begin{array}{llllll}
m+n=2^{2012} & 2^{2011} & 2^{2010} & \ldots & 2^{1008} & 2^{1007} \\
m-n=2 & 2^{2} & 2^{3} & \ldots & 2^{1005} & 2^{1006}
\end{array}\right.
$$

Each of these systems has one solution, so the original equation is satisfied by a total of 1006 ordered pairs of natural numbers.