Four Exponentials Conjecture - Statement

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 four numbers is transcendental:

An alternative way of stating the conjecture in terms of logarithms is the following. For 1 ≤ i,j ≤ 2 let λ_ij be complex numbers such that exp(λ_ij) are all algebraic. Suppose λ₁₁ and λ₁₂ are linearly independent over the rational numbers, and λ₁₁ and λ₂₁ are also linearly independent over the rational numbers, then

An equivalent formulation in terms of linear algebra is the following. Let M be the 2×2 matrix

where exp(λ_ij) is algebraic for 1 ≤ i,j ≤ 2. Suppose the two rows of M are linearly independent over the rational numbers, and the two columns of M are linearly independent over the rational numbers. Then the rank of M is 2.

While a 2×2 matrix having linearly independent rows and columns usually means it has rank 2, in this case we require linear independence over a smaller field so the rank isn't forced to be 2. For example, the matrix

has rows and columns that are linearly independent over the rational numbers, since π is irrational. But the rank of the matrix is 1. So in this case the conjecture would imply that at least one of e, eπ, and eπ ² is transcendental (which in this case is already known since e is transcendental).