Six Exponentials Theorem - Sharp Six Exponentials Theorem

Another related result that implies both the six exponentials theorem and the five exponentials theorem is the sharp six exponentials theorem. This theorem is as follows. Let x₁, x₂, and x₃ be complex numbers that are linearly independent over the rational numbers, and let y₁ and y₂ be a pair of complex numbers that are linearly independent over the rational numbers, and suppose that β_ij are six algebraic numbers for 1 ≤ i ≤ 3 and 1 ≤ j ≤ 2 such that the following six numbers are algebraic:

Then x_i y_j = β_ij for 1 ≤ i ≤ 3 and 1 ≤ j ≤ 2. The six exponentials theorem then follows by setting β_ij = 0 for every i and j, while the five exponentials theorem follows by setting x₃ = γ/x₁ and using Baker's theorem to ensure that the x_i are linearly independent.

There is a sharp version of the five exponentials theorem as well, although it as yet unproven so is known as the sharp five exponentials conjecture. This conjecture implies both the sharp six exponentials theorem and the five exponentials theorem, and is stated as follows. Let x₁, x₂ and y₁, y₂ be two pairs of complex numbers, with each pair being linearly independent over the rational numbers, and let α, β₁₁, β₁₂, β₂₁, β₂₂, and γ be six algebraic numbers with γ ≠ 0 such that the following five numbers are algebraic:

Then x_i y_j = β_ij for 1 ≤ i, j ≤ 2 and γx₂ = αx₁.

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.