Theorems · Theorem · number theory
Nat.exists_infinite_primes
- #11 of the 100 theorems: The Infinitude of Primes
- 1000+ list: Euclid's theorem
∀ (n : ℕ), ∃ p, n ≤ p ∧ Nat.Prime p
Euclid's theorem on the infinitude of primes.
Here given in the form: for every n, there exists a prime number p ≥ n.
- Defined in
- Mathlib.Data.Nat.Prime.Infinite
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Nat.Primestatement and proof · cited by 2,059
- ne_of_gtproof · cited by 637
- Nat.factorialproof · cited by 616
- Nat.factorial_posproof · cited by 99
- Nat.minFacproof · cited by 72
- le_of_not_geproof · cited by 72
- Nat.minFac_primeproof · cited by 31
- Nat.minFac_dvdproof · cited by 24
- Nat.Prime.not_dvd_oneproof · cited by 13
- Nat.minFac_posproof · cited by 10
- Nat.dvd_factorialproof · cited by 5
Cited by7
Results whose statement or proof uses this declaration.
- Nat.not_bddAbove_setOfPred_primeproof · cited by 2
- Nat.exists_infinite_pseudoprimesproof · cited by 1
- Nat.exists_prime_gt_modEq_oneproof · cited by 1
- Nat.infinite_deficientproof · cited by 0
- Nat.infinite_odd_deficientproof · cited by 0
- Nat.mem_factoredNumbers'proof · cited by 0
- AddCircle.addWellApproximable_ae_empty_or_univproof · cited by 0