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.

REFERENCES:

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.