Largest possible value of n - m

We have $144^{145} = (12^2)^{145} = (2^4 \times 3^2)^{145} = 2^{580} \times 3^{290}$.

Given $a^m b^n = 2^{580} \times 3^{290}$, where $a, b, m, n \in \mathbb{N}$ and $a, b > 1$.

To maximize $n - m$, we should make $n$ as large as possible and $m$ as small as possible.

Since $b > 1$, the smallest base $b$ can be is $2$. Hence, the maximum possible value for $n$ is $580$ by setting $b = 2$.

Then $a^m = 3^{290}$. The minimum integer value for $m$ is $1$ (which gives $a = 3^{290} > 1$).

Therefore, the largest possible value of $n - m = 580 - 1 = 579$.