FormalConjectures.Wikipedia.Irrational

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import FormalConjecturesUtil

Open questions on irrationality of numbers

Reference: Wikipedia


open Real


local notation "e" => exp 1


-- See also corresponding transcendence conjectures
-- in `FormalConjectures.Wikipedia.SchanuelsConjecture`

namespace Irrational

Are $e$ and $\pi$ algebraically independent?

@[category research open, AMS 33]
theorem declaration uses 'sorry'algebraicIndependent_e_pi :
    answer(sorry) ↔ AlgebraicIndependent ℚ ![e, π] := by⊢ True ↔ AlgebraicIndependent ℚ ![e, π]
  sorryAll goals completed! 🐙

Is $e + \pi$ irrational?

@[category research open, AMS 33]
theorem declaration uses 'sorry'irrational_e_plus_pi :
    answer(sorry) ↔ Irrational (e + π) := by⊢ True ↔ Irrational (e + π)
  sorryAll goals completed! 🐙

Is $e \pi$ irrational?

@[category research open, AMS 33]
theorem declaration uses 'sorry'irrational_e_times_pi :
    answer(sorry) ↔ Irrational (e * π) := by⊢ True ↔ Irrational (e * π)
  sorryAll goals completed! 🐙

Is $e ^ e$ irrational?

@[category research open, AMS 33]
theorem declaration uses 'sorry'irrational_e_to_e :
    answer(sorry) ↔ Irrational (e ^ e) := by⊢ True ↔ Irrational (e ^ e)
  sorryAll goals completed! 🐙

Is $\pi ^ e$ irrational?

@[category research open, AMS 33]
theorem declaration uses 'sorry'irrational_pi_to_e :
    answer(sorry) ↔ Irrational (π ^ e) := by⊢ True ↔ Irrational (π ^ e)
  sorryAll goals completed! 🐙

Is $\pi ^ \pi$ irrational?

@[category research open, AMS 33]
theorem declaration uses 'sorry'irrational_pi_to_pi :
    answer(sorry) ↔ Irrational (π ^ π) := by⊢ True ↔ Irrational (π ^ π)
  sorryAll goals completed! 🐙

Is $\ln(\pi)$ irrational?

@[category research open, AMS 33]
theorem declaration uses 'sorry'irrational_ln_pi :
    answer(sorry) ↔ Irrational (log π) := by⊢ True ↔ Irrational (log π)
  sorryAll goals completed! 🐙

Is the Euler-Mascheroni constant $\gamma$ irrational?

@[category research open, AMS 33]
theorem declaration uses 'sorry'irrational_eulerMascheroniConstant :
    answer(sorry) ↔ Irrational eulerMascheroniConstant := by⊢ True ↔ Irrational eulerMascheroniConstant
  sorryAll goals completed! 🐙

Is the Catalan constant $$G = \sum_{n=0}^∞ (-1)^n / (2n + 1)^2 \approx 0.91596$$ irrational?

@[category research open, AMS 11 33]
theorem declaration uses 'sorry'irrational_catalanConstant :
    answer(sorry) ↔ Irrational catalanConstant := by⊢ True ↔ Irrational catalanConstant
  sorryAll goals completed! 🐙


end Irrational