 Another related result that implies both the six exponentials theorem and the five exponentials theorem is the sharp six exponentials theorem.This theorem is as follows.Let "x"1, "x"2, and "x"3 be complex numbers that are linearly independent over the rational numbers, and let "y"1 and "y"2 be a pair of complex numbers that are linearly independent over the rational numbers, and suppose that β"ij" are six algebraic numbers for 1 ≤ "i" ≤ 3 and 1 ≤ "j" ≤ 2 such that the following six numbers are algebraic