FormalConjectures.ErdosProblems.«275»

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import FormalConjecturesUtil

Erdős Problem 275

References:

erdosproblems.com/275

[CrVE70] R.B. Crittenden and C.L. Vanden Eynden, Any n arithmetic progressions covering the first 2^n integers cover all integers, Proc. Amer. Math. Soc. 24 (1970), 475-481.


open Set


namespace Erdos275

If a finite system of $r$ congruences ${ a_i\pmod{n_i} : 1\leq i\leq r}$ (the $n_i$ are not necessarily distinct) covers $2^r$ consecutive integers then it covers all integers.

This is best possible as the system $2^{i-1}\pmod{2^i}$ shows. This was proved independently by Selfridge and Crittenden and Vanden Eynden [CrVE70].

This was formalized in Lean by Alexeev using Aristotle.

@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos275.lean"]
theorem declaration uses 'sorry'erdos_275 (r : ℕ) (a : Fin r → ℤ) (n : Fin r → ℕ)
    (H : ∃ k : ℤ, ∀ x ∈ Ico k (k + 2 ^ r), ∃ i, x ≡ a i [ZMOD n i]) (x : ℤ) :
    ∃ i, x ≡ a i [ZMOD n i] := byr:ℕa:Fin r → ℤn:Fin r → ℕH:∃ k, ∀ x ∈ Ico k (k + 2 ^ r), ∃ i, x ≡ a i [ZMOD ↑(n i)]x:ℤ⊢ ∃ i, x ≡ a i [ZMOD ↑(n i)]
  sorryAll goals completed! 🐙


end Erdos275