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import FormalConjecturesUtilErdős Problem 204
[Ad25] S. Adenwalla, A Question of Erdős and Graham on Covering Systems. arXiv:2501.15170 (2025).
namespace Erdos204
Are there $n$ such that there is a covering system with moduli the divisors of $n$ which is 'as disjoint as possible'?
That is, for all $d\mid n$ with $d>1$ there is an associated $a_d$ such that every integer is congruent to some $a_d\pmod{d}$, and if there is some integer $x$ with $$x\equiv a_d\pmod{d}\textrm{ and }x\equiv a_{d'}\pmod{d'}$$then $(d,d')=1$.
The density of such $n$ is zero. Erdős and Graham believed that no such $n$ exist.
Adenwalla [Ad25] has proved there are no such $n$.
This was formalized by van Doorn in Lean using Aristotle.
@[category research solved, AMS 5, formal_proof using lean4 at "https://github.com/Woett/Lean-files/blob/main/ErdosProblem204.lean"]
theorem erdos_204 : answer(False) ↔ ∃ (n : ℕ) (a : ℕ → ℤ),
let D := {d : ℕ | d ∣ n ∧ d > 1}
(∀ x : ℤ, ∃ d ∈ D, x ≡ a d [ZMOD d]) ∧
(∀ d ∈ D, ∀ d' ∈ D, d ≠ d' → (∃ x : ℤ, x ≡ a d [ZMOD d] ∧ x ≡ a d' [ZMOD d']) →
Nat.gcd d d' = 1) := ⊢ False ↔
∃ n a,
let D := {d | d ∣ n ∧ d > 1};
(∀ (x : ℤ), ∃ d ∈ D, x ≡ a d [ZMOD ↑d]) ∧
∀ d ∈ D, ∀ d' ∈ D, d ≠ d' → (∃ x, x ≡ a d [ZMOD ↑d] ∧ x ≡ a d' [ZMOD ↑d']) → d.gcd d' = 1
All goals completed! 🐙
end Erdos204