Theorems · Theorem · number theory
irrational_nrt_of_n_not_dvd_multiplicity
∀ {x : ℝ} (n : ℕ) {m : ℤ},
m ≠ 0 → ∀ (p : ℕ) [hp : Fact (Nat.Prime p)], x ^ n = ↑m → multiplicity (↑p) m % n ≠ 0 → Irrational xIf x^n = m is an integer and n does not divide the multiplicity p m, then x
is irrational.
- Defined in
- Mathlib.NumberTheory.Real.Irrational
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 111 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Fact
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites20
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- Factstatement and proof · cited by 2,726
- Nat.Primestatement and proof · cited by 2,059
- LT.lt.ne'proof · cited by 1,417
- pow_zeroproof · cited by 1,094
- Int.cast_oneproof · cited by 371
- zero_powproof · cited by 361
- Fact.outproof · cited by 328
- Irrationalstatement · cited by 139
- multiplicitystatement and proof · cited by 117
- Nat.Prime.ne_oneproof · cited by 61
- Int.cast_powproof · cited by 59
Cited by1
Results whose statement or proof uses this declaration.
- irrational_sqrt_of_multiplicity_oddproof · cited by 0