Let  and  be two sets of complex numbers linearly independent over the rationals. Then at least one of

is transcendental (Waldschmidt 1979, p. 3.5). This theorem is due to Siegel, Schneider, Lang, and Ramachandra. The corresponding statement obtained by replacing  with  is called the four exponentials conjecture and remains unproven.

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.

Ramachandra, K. "Contributions to the Theory of Transcendental Numbers. I, II." Acta Arith. 14, 65-78, 1967-68.

Ramachandra, K. and Srinivasan, S. "A Note to a Paper: 'Contributions to the Theory of Transcendental Numbers. I, II' by Ramachandra on Transcendental Numbers." Hardy-Ramanujan J. 6, 37-44, 1983.

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 Gel'fond and Schneider in Several Variables." In New Advances in Transcendence Theory (Ed. A. Baker). Cambridge, England: Cambridge University Press, pp. 375-398, 1988.