Let (x_1, x_2) and (y_1, y_2, y_3) be two sets of complex numbers linearly independent over the rationals. Then at least one of e^(x_1 y_1), e^(x_1 y_2), e^(x_1 y_3), e^(x_2 y_1), e^(x_2 y_2), e^(x_2 y_3) is transcendental (Waldschmidt 1979, p. 3.5). This theorem is due to Siegel, Schneider, Lang, and Ramachandra. The corresponding statement obtained by replacing y_1, y_2, y_3 with y_1, y_2 is called the four exponentials conjecture and remains unproven.

four exponentials conjecture | Hermite-Lindemann theorem | transcendental number