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.
Read more about this topic: Four Exponentials Conjecture
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