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Six Exponentials Theorem

Let (x_1,x_2) and (y_1,y_2,y_3) be two sets of complex numbers linearly independent over the rationals. Then at least one of

e^(x_1y_1),e^(x_1y_2),e^(x_1y_3),e^(x_2y_1),e^(x_2y_2),e^(x_2y_3)

is transcendental (Waldschmidt 1979, p. 3.5). This theorem is due to Siegel, Schneider, Lang, and Ramachandra. The corresponding statement obtained by replacing y_1,y_2,y_3 with y_1,y_2 is called the four exponentials conjecture and remains unproven.

See also

Four Exponentials Conjecture, Hermite-Lindemann Theorem, Transcendental Number

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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.

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Six Exponentials Theorem

Cite this as:

Weisstein, Eric W. "Six Exponentials Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SixExponentialsTheorem.html

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