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.
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.