Four exponentials conjecture

In mathematics, specifically the field of transcendental number 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. The conjecture, along with two related, stronger conjectures, is at the top of a hierarchy of conjectures and theorems concerning the arithmetic nature of a certain number of values of the exponential function.

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, e^π, and e^π ² is transcendental (which in this case is already known since e is transcendental).

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