Given a square matrix \(A\) of order \(n\), consider the notation \(r_A(m):=\mathrm{rank}\left(A^m\right)\), where \(m\) is a positive integer. This paper introduces the definition of rank function equations as being a system of equations of the form \[ f\left(r_{A_1}(m)\right)+\cdots+f\left(r_{A_k}(m)\right)=g\left(r_{B}(m)\right),\;\forall m\in S, \] where the unknowns are the square matrices \(A_1,\dots,A_k, B\) of order \(n\), \(f\) and \(g\) are functions over the positive integers and \(S\) is a set of positive integers. The nilpotent solutions of some particular rank function equations are found or characterized.