Differences in Squares (with factorials)

Let us look at the value of $\dfrac{2014!}{2012!!}$ more closely. This simplifies to $2013!! \cdot 2014$ . Notice that $2^1$ is the largest power of $2$ for this value, since $2013!!$ is a product of odd numbers and $2014=2 \cdot 19 \cdot 53$ . Furthermore, $a^2-b^2=(a+b)(a-b)$ , and in order for $a$ and $b$ to be both integers, $a+b$ and $a-b$ must have the same parity (proof: the difference between $a+b$ and $a-b$ is $2b$ , which is even). However, $a+b$ and $a-b$ cannot have the same parity, because only one can be even while the other is odd. Therefore, $a$ and $b$ cannot be positive integers, so there are $\boxed{0}$ possible ways.