Six exponentials theorem

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In mathematics, specifically transcendence theory, the six exponentials theorem is a result which, given the right conditions on the exponents, guarantees the transcendence of at least one of six exponentials.

[edit] Statement

If x1,x2,x3 are three complex numbers that are linearly independent over the rational numbers, and y1,y2 are two complex numbers that are also linearly independent over the rational numbers, then at least one of the following numbers is transcendental:
e^{x_1y_1}, e^{x_1y_2}, e^{x_2y_1}, e^{x_2y_2}, e^{x_3y_1}, e^{x_3y_2}.

[edit] History

The theorem was first explicitly stated and proved in its complete form independently by Lang[1] and Ramachandra[2] in the 1960s. The theorem is weaker than the related but thus far unproved four exponentials conjecture.

[edit] References

  1. ^ S. Lang, Introduction to transcendental numbers, Chapter 2 ยง1, Addison-Wesley Publishing Co., Reading, Mass., 1966.
  2. ^ K. Ramachandra, Contributions to the theory of transcendental numbers. I, II., Acta Arith. 14 (1967/68), pp.65-72.