Mathoverflow 17560

Reference: mathoverflow/17560 asked by user Alon-Amit

theorem Mathoverflow17560.mathoverflow_17560 {x : ℝ} (hx : ∃ (m : ℕ), 2 ^ x = ↑m) (hx' : ∃ (m : ℕ), 3 ^ x = ↑m) : ∃ (m : ℕ), x = ↑m

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

theorem Mathoverflow17560.mathoverflow_17560.variants.all_nats {x : ℝ} (hx : ∀ (n : ℕ), ∃ (m : ℕ), ↑n ^ x = ↑m) : ∃ (m : ℕ), x = ↑m

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

theorem Mathoverflow17560.mathoverflow_17560.variants.with_5 {x : ℝ} (hx : ∃ (m : ℕ), 2 ^ x = ↑m) (hx' : ∃ (m : ℕ), 3 ^ x = ↑m) (hx'' : ∃ (m : ℕ), 5 ^ x = ↑m) : ∃ (m : ℕ), x = ↑m

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