Four exponentials conjecture
- Four exponentials conjecture
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.
tatement
If "x"1, "x"2 and "y"1, "y"2 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::
History
The related six exponentials theorem was first explicitly mentioned in the 1960s by Lang [S. Lang, "Introduction to transcendental numbers", Chapter 2 §1, Addison-Wesley Publishing Co., Reading, Mass., 1966.] , 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.
Corollaries
Using Euler's identity this conjecture implies the transcendence of many numbers involving "e" and π. For example, taking "x"1 = 1, "x"2 = √Overline|2, "y"1 = iπ, and "y"2 = iπ√Overline|2, the conjecture — if true — implies that one of the following four numbers is transcendental::The first of these is just −1, and the fourth is 1, so the conjecture implies that "e"iπ√Overline|2 is transcendental.
Notes
External links
*planetmath reference|id=4349|title=Four exponentials conjecture
*MathWorld|urlname=FourExponentialsConjecture|title=Four Exponentials Conjecture
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