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import FormalConjecturesUtilExponentials conjectures and theorems
open Complex
namespace Exponentials
Four exponentials conjecture Let $x_0, x_1$ and $y_0, y_1$ be $\mathbb Q$-linearly independent pairs of complex numbers, then some $e^{x_i y_j}$ is transcendental.
@[category research open, AMS 33]
theorem four_exponentials_conjecture (x : Fin 2 → ℂ) (y : Fin 2 → ℂ)
(h1 : LinearIndependent ℚ x) (h2 : LinearIndependent ℚ y) :
∃ i j : Fin 2, Transcendental ℚ (exp (x i * y j)) := x:Fin 2 → ℂy:Fin 2 → ℂh1:LinearIndependent ℚ xh2:LinearIndependent ℚ y⊢ ∃ i j, Transcendental ℚ (cexp (x i * y j))
All goals completed! 🐙
The four exponential conjecture would imply that for any irrational number $t$, at least one of the numbers $2^t$ and $3^t$ is transcendental.
@[category research open, AMS 11]
theorem two_pow_three_pow_transcendental (t : ℝ) (h : Irrational t) :
Transcendental ℚ (2 ^ t : ℝ) ∨ Transcendental ℚ (3 ^ t : ℝ) := t:ℝh:Irrational t⊢ Transcendental ℚ (2 ^ t) ∨ Transcendental ℚ (3 ^ t)
All goals completed! 🐙
end Exponentials