FormalConjectures.Wikipedia.Irrational

/-
Copyright 2025 The Formal Conjectures Authors.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    https://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/

import FormalConjecturesUtil

Open questions on irrationality of numbers

Reference: Wikipedia

open Reallocal 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 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 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 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 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 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 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 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 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 irrational_catalanConstant :
    answer(sorry) ↔ Irrational catalanConstant := by⊢ True ↔ Irrational catalanConstant
  sorryAll goals completed! 🐙

end Irrational