FormalConjectures.Mathoverflow.«17560»

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import FormalConjecturesUtil

Mathoverflow 17560

Reference: mathoverflow/17560 asked by user Alon-Amit


namespace Mathoverflow17560

If $2^x$ and $3^x$ are integers, then $x$ must be an integer.

@[category research open, AMS 11 13]
theorem declaration uses 'sorry'mathoverflow_17560 {x : ℝ} (hx : ∃ m : ℕ, (2 : ℝ) ^ x = m) (hx' : ∃ m : ℕ, (3 : ℝ) ^ x = m) :
    ∃ m : ℕ, x = m := byx:ℝhx:∃ m, 2 ^ x = ↑mhx':∃ m, 3 ^ x = ↑m⊢ ∃ m, x = ↑m
  sorryAll goals completed! 🐙

If for each natural number $n$ the number $n^x$ is an integer then $x$ must also be an integer.

@[category textbook, AMS 11 13]
theorem declaration uses 'sorry'mathoverflow_17560.variants.all_nats {x : ℝ} (hx : ∀ n : ℕ, ∃ m : ℕ, (n : ℝ) ^ x = m) :
    ∃ m : ℕ, x = m := byx:ℝhx:∀ (n : ℕ), ∃ m, ↑n ^ x = ↑m⊢ ∃ m, x = ↑m
  sorryAll goals completed! 🐙

If $2^x$, $3^x$ and $5^x$ are integers, then $x$ must be an integer.

@[category textbook, AMS 11 13]
theorem declaration uses 'sorry'mathoverflow_17560.variants.with_5 {x : ℝ} (hx : ∃ m : ℕ, (2 : ℝ) ^ x = m)
    (hx' : ∃ m : ℕ, (3 : ℝ) ^ x = m) (hx'' : ∃ m : ℕ, (5 : ℝ) ^ x = m) : ∃ m : ℕ, x = m := byx:ℝhx:∃ m, 2 ^ x = ↑mhx':∃ m, 3 ^ x = ↑mhx'':∃ m, 5 ^ x = ↑m⊢ ∃ m, x = ↑m
  sorryAll goals completed! 🐙

end Mathoverflow17560

-- TODO(Paul-Lez): add the implications