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import FormalConjecturesUtil
namespace Erdos689
Let n be sufficiently large. Is there some choice of congruence class a_p for all primes
2 ≤ p ≤ n such that every integer in [1,n] satisfies at least two of the congruences
≡ a_p (mod p)?
@[category research open, AMS 11]
theorem erdos_689 :
answer(sorry) ↔ ∀ᶠ n in .atTop, ∃ a : ℕ → ℕ, ∀ m ∈ Finset.Icc 1 n,
2 ≤ (Finset.Icc 1 n |>.filter fun p => p.Prime ∧ a p ≡ m [MOD p]).card := ⊢ True ↔
∀ᶠ (n : ℕ) in Filter.atTop, ∃ a, ∀ m ∈ Finset.Icc 1 n, 2 ≤ {p ∈ Finset.Icc 1 n | Nat.Prime p ∧ a p ≡ m [MOD p]}.card
All goals completed! 🐙
end Erdos689