FormalConjectures.ErdosProblems.«279»

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import FormalConjecturesUtil

Erdős Problem 279

Reference: erdosproblems.com/279


namespace Erdos279

Let $k\geq 3$. Is there a choice of congruence classes $a_p\pmod{p}$ for every prime $p$ such that all sufficiently large integers can be written as $a_p+tp$ for some prime $p$ and integer $t\geq k$?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_279 :  answer(sorry) ↔ ∀ k : Nat, k ≥ 3 →
    ∃ a : Nat → Nat, ∃ N : Nat, (∀ p : Nat, p.Prime → a p < p) ∧
    ∀ n ≥ N, ∃ p : Nat, ∃ t ≥ k, p.Prime ∧ n = a p + t * p := by⊢ True ↔ ∀ k ≥ 3, ∃ a N, (∀ (p : ℕ), Nat.Prime p → a p < p) ∧ ∀ n ≥ N, ∃ p, ∃ t ≥ k, Nat.Prime p ∧ n = a p + t * p
  sorryAll goals completed! 🐙


end Erdos279