Six Exponentials Theorem

The six exponentials theorem, in transcendental number theory, guarantees that at least one of six related exponential quantities must be transcendental, given three complex numbers linearly independent over the rationals and two other complex numbers also linearly independent over the rationals: among the six numbers e to the power x_i times y_j formed from these two sets, at least one cannot be algebraic. Leonidas Alaoglu and Paul Erdos noted a special case in 1944 while studying abundant numbers, and Carl Ludwig Siegel reportedly already knew a proof; the theorem in its general form was first explicitly stated and proved independently by Serge Lang and Kanakanahalli Ramachandra in the 1960s. It anchors a family of related results, including the weaker five exponentials theorem and the still-unproven four exponentials conjecture. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/