FormalConjectures/ErdosProblems/68.lean

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Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

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

2 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 68

Reference: erdosproblems.com/68

Imports (1)
  1. theorem Erdos68.erdos_68 Is $$\sum_{n=2}^\infty \frac{1}{n!-1}$$ irrational? sorry line 36 source

    Source documentation

    Is $$\sum_{n=2}^\infty \frac{1}{n!-1}$$ irrational?

    depends on sorry what this means

    Statement uses: Iff, True, Irrational, tsum, Real, Nat, HDiv.hDiv, OfNat.ofNat, HSub.hSub, Nat.factorial, HAdd.hAdd, SummationFilter.unconditional

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

    32 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_68 :
    answer(sorry) ↔ Irrational (∑' n : ℕ, 1 / ((n + 2).factorial - 1 : ℝ)) := by
…
  2. theorem Erdos68.sum_factorial_inv_eq_geometric $$\sum_{n=2}^\infty \frac{1}{n!-1} = \sum_{n=2}^\infty \sum_{k=1}^\infty \frac{1}{(n!)^k}$$ line 44 source

    Source documentation

    $$\sum_{n=2}^\infty \frac{1}{n!-1} = \sum_{n=2}^\infty \sum_{k=1}^\infty \frac{1}{(n!)^k}$$

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Nat, Real, HDiv.hDiv, OfNat.ofNat, HPow.hPow, Nat.factorial, HAdd.hAdd, Eq, tsum, HSub.hSub, SummationFilter.unconditional

    Proof uses: tsum_congr, Eq.symm, Inv.inv, rfl, LE.le, le_of_lt, inv_pos_of_pos, Iff.mpr, LT.lt, Nat.cast_pos', Nat.factorial_pos, inv_lt_one_of_one_lt₀, of_eq_true, Eq.trans, True, HasSum, hasSum_geometric_of_lt_one, HMul.hMul, HasSum.mul_left, Eq.mpr and 91 more

    169 instances, coercions and auxiliary constants left out.

    Tactic trace: 17 steps

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

    theorem sum_factorial_inv_eq_geometric :
    let f (n k : ℕ) : ℝ := 1 / ((n + 2).factorial : ℝ) ^ (k + 1)
…