Rainbow numbers of \([m] \times [n]\) for \(x_1 + x_2 = x_3\)

Consider the set \([m]\times [n] = \{(i,j)\, : 1\le i \le m, 1\le j \le n\}\) and the equation \(x_1+x_2 = x_3\), namely \(eq\). The \emph{rainbow number of \([m] \times [n]\) for \(eq\)}, denoted \(\text{rb}([m]\times [n],eq)\), is the smallest number of colors such that for every surjective \(\text{rb}([m]\times[n], eq)\)-coloring of \([m]\times [n]\) there must exist a solution to \(eq\), with component-wise addition, where every element of the solution set is assigned a distinct color. This paper determines that \(\text{rb}([m]\times [n], eq) = m+n+1\) for all values of \(m\) and \(n\) that a greater than or equal to \(2\).