Four exponentials conjecture

The four exponentials conjecture says that if each of two pairs of complex numbers is linearly independent over the rationals, at least one of the four exponentials of their pairwise products is transcendental.

Linear independence over the rationals: A set of numbers is linearly independent over the rationals if no nontrivial rational linear combination of them equals zero. The conjecture requires both input pairs to satisfy this precise condition.
Exponential matrix: An exponential matrix has entries obtained by exponentiating products of elements from two lists. The conjecture concerns the four entries formed from two pairs.
Theodor Schneider: Theodor Schneider was a German mathematician whose work established major results in transcendental number theory. His theorem provides an early precedent for transcendence forced by exponential relations.
Six exponentials conjecture: The six exponentials conjecture predicts at least two transcendental entries in a certain six-value exponential matrix. It strengthens a proven six-value result and remains unresolved alongside the four-value problem.
Transcendental number: A complex number is transcendental if it is not a root of any nonzero polynomial with integer coefficients. The conjecture predicts transcendence among the four exponentials.
Lindemann–Weierstrass theorem: The Lindemann–Weierstrass theorem states that exponentials of distinct algebraic numbers are linearly independent over the algebraic numbers. It proves transcendence in important cases where the products are algebraic.
Carl Ludwig Siegel: Carl Ludwig Siegel was a German mathematician whose work spanned number theory, analysis, and Diophantine approximation. Siegel formulated the four exponentials conjecture within transcendental number theory.
Schanuel's conjecture: Schanuel's conjecture predicts a lower bound on the transcendence degree generated by complex numbers and their exponentials. It implies the four exponentials conjecture through a much broader transcendence prediction.
Algebraic number: A complex number is algebraic if it is a root of a nonzero polynomial with integer coefficients. The conclusion rules out all four exponentials being algebraic.
Gelfond–Schneider theorem: The Gelfond–Schneider theorem states that every value of an algebraic number raised to an algebraic irrational power is transcendental. It establishes transcendence for some exponential values covered by special input choices.

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