All three vertices of an equilateral triangle are on the parabola $ y \equal{} x^2$, and one of its sides has a slope of 2. The x-coordinates of the three vertices have a sum of $ m/n$, where $ m$ and $ n$ are relatively prime positive integers. What is the value of $ m \plus{} n$?

$ \textbf{(A)}\ 14\qquad
\textbf{(B)}\ 15\qquad
\textbf{(C)}\ 16\qquad
\textbf{(D)}\ 17\qquad
\textbf{(E)}\ 18$
1. **Identify the vertices on the parabola**: Let the vertices of the equilateral triangle be \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \). Since all vertices lie on the parabola \( y = x^2 \), we have:
   \[
   y_1 = x_1^2, \quad y_2 = x_2^2, \quad y_3 = x_3^2
   \]

2. **Slope of one side**: Given that one side of the triangle has a slope of 2, assume this side is \( AB \). The slope of \( AB \) is:
   \[
   \frac{y_2 - y_1}{x_2 - x_1} = 2 \implies \frac{x_2^2 - x_1^2}{x_2 - x_1} = 2
   \]
   Simplifying the above equation:
   \[
   \frac{(x_2 - x_1)(x_2 + x_1)}{x_2 - x_1} = 2 \implies x_2 + x_1 = 2
   \]

3. **Equilateral triangle properties**: In an equilateral triangle, the slopes of the sides are related by the angle \( 60^\circ \). The slopes of the other two sides can be found using the tangent of \( 60^\circ \):
   \[
   \tan 60^\circ = \sqrt{3}
   \]

4. **Finding the other slopes**: Let the slopes of the other two sides be \( m_1 \) and \( m_2 \). Using the angle subtraction formula for tangent:
   \[
   \sqrt{3} = \frac{2 - m_2}{1 + 2m_2} \implies m_2 = \frac{5\sqrt{3} - 8}{11}
   \]
   Similarly, for the third side:
   \[
   \sqrt{3} = \frac{m_1 - 2}{1 + 2m_1} \implies m_1 = \frac{-(5\sqrt{3} + 8)}{11}
   \]

5. **Sum of the x-coordinates**: The sum of the x-coordinates of the vertices is given by:
   \[
   x_1 + x_2 + x_3
   \]
   Since \( x_1 + x_2 = 2 \), we need to find \( x_3 \). The sum of the slopes of the sides divided by 2 gives the sum of the x-coordinates:
   \[
   \frac{22 + (5\sqrt{3} - 8) - (5\sqrt{3} + 8)}{22} = \frac{22 - 8 - 8}{22} = \frac{6}{22} = \frac{3}{11}
   \]
   Therefore, the sum of the x-coordinates is \( \frac{3}{11} \).

6. **Simplify the fraction**: The sum of the x-coordinates is \( \frac{3}{11} \), where \( m = 3 \) and \( n = 11 \). Thus, \( m + n = 3 + 11 = 14 \).

The final answer is \( \boxed{14} \)