Six Exponentials Theorem - Five Exponentials Theorem

A stronger, related result is the five exponentials theorem, which is as follows. Let x₁, x₂ and y₁, y₂ be two pairs of complex numbers, with each pair being linearly independent over the rational numbers, and let γ be a non-zero algebraic number. Then at least one of the following five numbers is transcendental:

This theorem implies the six exponentials theorem and in turn is implied by the as yet unproven four exponentials conjecture, which says that in fact one of the first four numbers on this list must be transcendental.