What's special about number 3?

$\Large \color{#3D99F6}{m}^{\color{#D61F06}{n}^{\color{#456461}{p}}} \times \color{#D61F06}{n}^{\color{#456461}{p}^{\color{#3D99F6}{m}}} \times \color{#456461}{p}^{\color{#3D99F6}{m}^{\color{#D61F06}{n}}} = \color{#EC7300}3\color{#3D99F6}{m}\color{#D61F06}{n}\color{#456461}{p}$

Given that $\color{#3D99F6}{m}, \color{#D61F06}{n}, \color{#456461}{p}$ are positive integers that satisfy the equation above, what is the value of $\color{#3D99F6}{m} + \color{#D61F06}{n} + \color{#456461}{p}$?