Take a square from a square

$N=m^2-n^2$ .Consider $m=n+k$

$N=(n+k)^-n^2=2nk+k^2=k(2n+k)$ .

We have to find smallest $N$ so, it can be expressed as difference between two perfect squares in three different ways.So, there will be $3$ distinct values of $k$ such that there will be an integral $n$ then the given condition will be met.

For minimum $N$ should be comprised of three factors(not necessarily distinct).If $k$ is chosen odd then other part should be odd because $n$ shoul be integer..

If $k$ is chosen even then other part should be even forthe above reason.So,either $N$ will contain only even numbers or odd numbers. $N$ can be $2 \times 4 \times 6$ or $3 \times 3 \times 5$ .Form the first part $k$ will be $2,4,6$ .From second option $k$ will be $1,3,5$ .

So, minimum is $N=45$ because $45=9^2-6^2=7^2-2^2=23^2-22^2$