Four Exponentials Conjecture

The four exponentials conjecture asserts that if $x_1,x_2$ are complex numbers linearly independent over the rationals, and $y_1,y_2$ are likewise linearly independent over the rationals, then at least one of the four numbers $e^{x_iy_j}$ with $1\le i,j\le 2$ is transcendental. The problem entered the transcendence literature in the mid-1960s, appearing in Lang's book [Lang1966Transcendental], though no single first-proposal date can be fixed with certainty.

The natural point of comparison is the six exponentials theorem: the same conclusion is a proved theorem when one of the two families has three members instead of two, that is, for a $2\times 3$ array of exponentials. The gap between the proved $2\times 3$ case and the conjectured $2\times 2$ case has resisted all attempts, as the auxiliary-function constructions that yield six exponentials lack the room to work with only four. The conjecture would follow from Schanuel's conjecture, and it occupies a central place in Waldschmidt's accounts of transcendence and open Diophantine problems [Waldschmidt2000Diophantine], [Waldschmidt2006Open]. It remains open; a resolution requires pushing transcendence methods past the six-exponentials barrier.