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 six exponentials.

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

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.

[1] Using the special case they manage to prove that the ratio of consecutive colossally abundant numbers is always either a prime or a semiprime.

The theorem was first explicitly stated and proved in its complete form independently by Serge Lang[2] and Kanakanahalli Ramachandra[3] in the 1960s.

A stronger, related result is the five exponentials theorem,[4] which is as follows.

Then at least one of the following five numbers is transcendental: This theorem implies the six exponentials theorem and in turn is implied by the as yet unproven four exponentials conjecture, which says that in fact one of the first four numbers on this list must be transcendental.

A consequence of this conjecture that isn't currently known would be the transcendence of eπ², by setting x1 = y1 = β11 = 1, x2 = y2 = iπ, and all the other values in the statement to be zero.

A further strengthening of the theorems and conjectures in this area are the strong versions.

The strong six exponentials theorem is a result proved by Damien Roy that implies the sharp six exponentials theorem.

[7] This result concerns the vector space over the algebraic numbers generated by 1 and all logarithms of algebraic numbers, denoted here as L∗.

So L∗ is the set of all complex numbers of the form for some n ≥ 0, where all the βi and αi are algebraic and every branch of the logarithm is considered.

This is stronger than the standard six exponentials theorem which says that one of these six numbers is not simply the logarithm of an algebraic number.

There is also a strong five exponentials conjecture formulated by Michel Waldschmidt.

This conjecture claims that if x1, x2 and y1, y2 are two pairs of complex numbers, with each pair being linearly independent over the algebraic numbers, then at least one of the following five numbers is not in L∗: All the above conjectures and theorems are consequences of the unproven extension of Baker's theorem, that logarithms of algebraic numbers that are linearly independent over the rational numbers are automatically algebraically independent too.

The diagram on the right shows the logical implications between all these results.

The exponential function ez uniformizes the exponential map of the multiplicative group Gm.

Therefore, we can reformulate the six exponential theorem more abstractly as follows: (In order to derive the classical statement, set u(z) = (ey1z; ey2z) and note that Qx1 + Qx2 + Qx3 is a subset of L).

In this way, the statement of the six exponentials theorem can be generalized to an arbitrary commutative group variety G over the field of algebraic numbers.

This generalized six exponential conjecture, however, seems out of scope at the current state of transcendental number theory.

For the special but interesting cases G = Gm × E and G = E × E′, where E, E′ are elliptic curves over the field of algebraic numbers, results towards the generalized six exponential conjecture were proven by Aleksander Momot.

Let G = Gm × E and suppose E is not isogenous to a curve over a real field and that u(C) is not an algebraic subgroup of G(C).

Then L is generated over Q either by two elements x1, x2, or three elements x1, x2, x3 which are not all contained in a real line Rc, where c is a non-zero complex number.