My textbook says that $(X_1)^2 +(X_2)^2+(X_3)^2$ would be written $X_i \cdot X_i$ in Einstein notation. It also says that you can only have 2 of the same index. But what if I wanted to express $(X_1)^3 +(X_2)^3+(X_3)^3$
Automatically, I would want to write $X_i \cdot X_i\cdot X_i$, but this is not allowed.
Why does Einstein notation not allow an index to be repeated three times? Is there some sort of ambiguity that it would cause?
The reason for that not being allowed is that the result is not a tensor. It doesn't transform in a good way when the coordinate system is changed.