Erdős Problem 204

References:

• erdosproblems.com/204
• [Ad25] S. Adenwalla, A Question of Erdős and Graham on Covering Systems. arXiv:2501.15170 (2025).

theorem Erdos204.erdos_204 : False ↔ ∃ (n : ℕ) (a : ℕ → ℤ), have 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']) → d.gcd d' = 1

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 $n$.