FormalConjectures/ErdosProblems/204.lean

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L4DEL
Lean4 Domain Explanation Language

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 204

References:

Imports (1)
  1. theorem Erdos204.erdos_204 Are there $n$ such that there is a covering system with moduli the divisors of $n$ which is 'as disjoint as possible'? sorry line 47 source

    Source documentation

    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.

    depends on sorry what this means

    Statement uses: Iff, False, Exists, Nat, Int, Set, Set.ofPred, And, Dvd.dvd, GT.gt, OfNat.ofNat, Membership.mem, Int.ModEq, Ne, Eq, Nat.gcd

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous

    12 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem erdos_204 : answer(False) ↔ ∃ (n : ℕ) (a : ℕ → ℤ),
    let D := {d : ℕ | d ∣ n ∧ d > 1}
…