Erdős Problem 279

Reference: erdosproblems.com/279

theorem Erdos279.erdos_279 : True ↔ ∀ k ≥ 3, ∃ (a : ℕ → ℕ) (N : ℕ), (∀ (p : ℕ), Nat.Prime p → a p < p) ∧ ∀ n ≥ N, ∃ (p : ℕ), ∃ t ≥ k, Nat.Prime p ∧ n = a p + t * p

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 $t\geq k$?