Four Exponentials Conjecture - Three Exponentials Conjecture

Three Exponentials Conjecture

The four exponentials conjecture rules out a special case of non-trivial, homogeneous, quadratic relations between logarithms of algebraic numbers. But a conjectural extension of Baker's theorem implies that there should be no non-trivial algebraic relations between logarithms of algebraic numbers at all, homogeneous or not. One case of non-homogeneous quadratic relations is covered by the still open three exponentials conjecture. In its logarithmic form it is the following conjecture. Let λ₁, λ₂, and λ₃ be any three logarithms of algebraic numbers and γ be a non-zero algebraic number, and suppose that λ₁λ₂ = γλ₃. Then λ₁λ₂ = γλ₃ = 0.

The exponential form of this conjecture is the following. Let x₁, x₂, and y be non-zero complex numbers and let γ be a non-zero algebraic number. Then at least one of the following three numbers is transcendental:

There is also a sharp three exponentials conjecture which claims that if x₁, x₂, and y are non-zero complex numbers and α, β₁, β₂, and γ are algebraic numbers such that the following three numbers are algebraic

then either x₂y = β₂ or γx₁ = α x₂.

The strong three exponentials conjecture meanwhile states that if x₁, x₂, and y are non-zero complex numbers with x₁/x₂ and y/x₂ both being transcendental, then at least one of the three numbers x₁y, x₂y, x₂/x₁ is not in L∗.

As with the other results in this family, the strong three exponentials conjecture implies the sharp three exponentials conjecture which implies the three exponentials conjecture. However, the strong and sharp three exponentials conjectures are implied by their four exponentials counterparts, bucking the usual trend. And the three exponentials conjecture is neither implied by nor implies the four exponentials conjecture.

The three exponentials conjecture, like the sharp five exponentials conjecture, would imply the transcendence of eπ² by letting (in the logarithmic version) λ₁ = iπ, λ₂ = −iπ, and γ = 1.