Four Exponentials Conjecture - Bertrand's Conjecture

Bertrand's Conjecture

Many of the theorems and results in transcendental number theory concerning the exponential function have analogues involving the modular function j. Writing q = e2πiτ for the nome and j(τ) = J(q), Daniel Bertrand conjectured that if q₁ and q₂ are non-zero algebraic numbers in the complex unit disc that are multiplicatively independent, then J(q₁) and J(q₂) are algebraically independent over the rational numbers. Although not obviously related to the four exponentials conjecture, Bertrand's conjecture in fact implies a special case known as the weak four exponentials conjecture. This conjecture states that if x₁ and x₂ are two positive real algebraic numbers, neither of them equal to 1, then π² and the product (logx₁)(logx₂) are linearly independent over the rational numbers. This corresponds to the special case of the four exponentials conjecture whereby y₁ = iπ, y₂ = −iπ, and x₁ and x₂ are real. Perhaps surprisingly, though, it is also a corollary of Bertrand's conjecture, suggesting there may be an approach to the full four exponentials conjecture via the modular function j.