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Four Exponentials Conjecture

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

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

is transcendental (Waldschmidt 1979, p. 3.5). The corresponding statement obtained by replacing y_1,y_2 with y_1,y_2,y_3 has been proven and is known as the six exponentials theorem.

See also

Hermite-Lindemann Theorem, Six Exponentials 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.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.

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Four Exponentials Conjecture

Cite this as:

Weisstein, Eric W. "Four Exponentials Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FourExponentialsConjecture.html

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