Primes congruent to one

We prove that, for any positive k : ℕ, there are infinitely many primes p such that p ≡ 1 [MOD k].

theorem Nat.exists_prime_gt_modEq_one {k : ℕ} (n : ℕ) (hk0 : k ≠ 0) : ∃ (p : ℕ), Prime p ∧ n < p ∧ p ≡ 1 [MOD k]

For any positive k : ℕ there exists an arbitrarily large prime p such that p ≡ 1 [MOD k].

theorem Nat.frequently_atTop_modEq_one {k : ℕ} (hk0 : k ≠ 0) : ∃ᶠ (p : ℕ) in Filter.atTop, Prime p ∧ p ≡ 1 [MOD k]

theorem Nat.infinite_setOfPred_prime_modEq_one {k : ℕ} (hk0 : k ≠ 0) : {p : ℕ | Prime p ∧ p ≡ 1 [MOD k]}.Infinite

For any positive k : ℕ there are infinitely many primes p such that p ≡ 1 [MOD k].

@[deprecated Nat.infinite_setOfPred_prime_modEq_one (since := "2026-07-09")]

theorem Nat.infinite_setOf_prime_modEq_one {k : ℕ} (hk0 : k ≠ 0) : {p : ℕ | Prime p ∧ p ≡ 1 [MOD k]}.Infinite

Alias of Nat.infinite_setOfPred_prime_modEq_one.

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For any positive k : ℕ there are infinitely many primes p such that p ≡ 1 [MOD k].