277.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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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 277

References:

erdosproblems.com/277
[Ha79] Haight, J. A., Covering systems of congruences, a negative result. Mathematika (1979), 53--61.

Imports (1)

FormalConjecturesUtil

theorem Erdos277.erdos_277 Is it true that, for every $c$, there exists an $n$ such that $\sigma(n)>cn$ but there is no covering system whose moduli all divide $n$? sorry line 42 source
Source documentation
Is it true that, for every $c$, there exists an $n$ such that $\sigma(n)>cn$ but there is no covering system whose moduli all divide $n$?
This was answered affirmatively by Haight [Ha79].

depends on sorry what this means
Statement uses: Iff, True, Real, Exists, Nat, And, GT.gt, ArithmeticFunction, ArithmeticFunction.sigma, OfNat.ofNat, HMul.hMul, StrictCoveringSystem, Int, CoveringSystem.ι, StrictCoveringSystem.toCoveringSystem, Not, Membership.mem, Ideal, CoveringSystem.moduli
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_277 :
   answer(True) ↔ ∀ c : ℝ, ∃ n : ℕ, (σ 1 n : ℝ) > c * n ∧
      ∀ (m : StrictCoveringSystem ℤ), ∃ i, (n : ℤ) ∉ m.moduli i  := by
…
```
Mentions: c, StrictCoveringSystem