Four exponentials conjecture

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In mathematics, specifically transcendence theory, the four exponentials conjecture is a conjecture which, given the right conditions on the exponents, would guarantee the transcendence of at least one of four exponentials.

[edit] Statement

If x1,x2 and y1,y2 are two pairs of complex numbers, with each pair being 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}.

[edit] History

The related six exponentials theorem was first explicitly mentioned in the 1960s by Lang[1], and after proving the theorem he mentions the difficulty in dropping the number of exponents from six to four - the proof used for six exponentials “just misses” when one tries to apply it to four.

[edit] References

  1. ^ S. Lang, Introduction to transcendental numbers, Chapter 2 §1, Addison-Wesley Publishing Co., Reading, Mass., 1966.