Let
and
be two sets of complex numbers linearly independent over the rationals. Then the
four exponential conjecture posits that at least one of

is transcendental (Waldschmidt 1979, p. 3.5). The corresponding statement obtained by replacing with has been proven and is known as the six
exponentials theorem.

Finch, S. R. "Powers of 3/2 Modulo One." §2.30.1 in Mathematical
Constants. Cambridge, England: Cambridge University Press, pp. 194-199,
2003.Waldschmidt, M. Transcendence
Methods. Queen's Papers in Pure and Applied Mathematics, No. 52. Kingston,
Ontario, Canada: Queen's University, 1979.Waldschmidt, M. "On the
Transcendence Method of Gelfond and Schneider in Several Variables." In New
Advances in Transcendence Theory (Ed. A. Baker). Cambridge, England:
Cambridge University Press, 1988.