Six exponentials theorem
In mathematics, specifically transcendental number theory, the six exponentials theorem is a result that, given the right conditions on the exponents, guarantees the transcendence of at least one of a set of exponentials.
If x1, x2,..., xd are d complex numbers that are linearly independent over the rational numbers, and y1, y2,...,yl are l complex numbers that are also linearly independent over the rational numbers, and if dl > d + l, then at least one of the following dl numbers is transcendental:
The most interesting case is when d = 3 and l = 2, in which case there are six exponentials, hence the name of the result. The theorem is weaker than the related but thus far unproved four exponentials conjecture, whereby the strict inequality dl > d + l is replaced with dl ≥ d + l, thus allowing d = l = 2.
The theorem can be stated in terms of logarithms by introducing the set L of logarithms of algebraic numbers:
The theorem then says that if λij are elements of L for i = 1, 2 and j = 1, 2, 3, such that λ11, λ12, and λ13 are linearly independent over the rational numbers, and λ11 and λ21 are also linearly independent over the rational numbers, then the matrix
has rank 2.
A special case of the result where x1, x2, and x3 are logarithms of positive integers, y1 = 1, and y2 is real, was first mentioned in a paper by Leonidas Alaoglu and Paul Erdős from 1944 in which they try to prove that the ratio of consecutive colossally abundant numbers is always prime. They claimed that Carl Ludwig Siegel knew of a proof of this special case, but it is not recorded. Using the special case they manage to prove that the ratio of consecutive colossally abundant numbers is always either a prime or a semiprime.
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