164.lean`

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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 164

References:

erdosproblems.com/164
[Er76g] Erdős, P., Problems and results on combinatorial number theory. II. J. Indian Math. Soc. (N.S.) (1976), 285-298.
[Er86] Erdős, P., Problémes et résultats en théorie des nombres. (1986).
[Va99] Various, Some of Paul's favorite problems. Booklet produced for the conference "Paul Erdős and his mathematics", Budapest, July 1999 (1999).
[ABLLPSTT26] B. Alexeev, K. Barreto, Y. Li, J. D. Lichtman, L. Price, J. I. Shah, Q. Tang, and T. Tao, Primitive sets and Von Mangoldt Chains: Erdős problem #1196 and beyond. arXiv:2605.00301 (2026).
[Er35] Erdős, Paul, Note on Sequences of Integers No One of Which is Divisible By Any Other. J. London Math. Soc. (1935), 126-128.
[Li23] Lichtman, J. D., A proof of the Erdős primitive set conjecture. arXiv:2202.02384 (2023).

Imports (2)

theorem Erdos164.erdos_164 A set $A\subset \mathbb{N}$ is primitive if no member of $A$ divides another sorry line 53 source
Source documentation
A set $A\subset \mathbb{N}$ is primitive if no member of $A$ divides another. Is the sum $$\sum_{n\in A}\frac{1}{n\log n}$$ maximised over all primitive sets when $A$ is the set of primes?
Erdős [Er35] proved that this sum always converges for a primitive set. Lichtman [Li23] proved that the answer is yes. An alternative, simpler, proof is given by Alexeev, Barreto, Li, Lichtman, Price, Shah, Tang, and Tao [ABLLPSTT26].

depends on sorry what this means
Statement uses: Iff, True, Set, Nat, Membership.mem, LE.le, OfNat.ofNat, Erdos1196.IsPrimitive, Real, tsum, HDiv.hDiv, HMul.hMul, Real.log, SummationFilter.unconditional, Set.ofPred, Nat.Prime
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_164 : answer(True) ↔
    ∀ A : Set ℕ, (∀ a ∈ A, 2 ≤ a) → Erdos1196.IsPrimitive A →
      (∑' a : A, 1 / ((a : ℕ) * Real.log (a : ℕ))) ≤
        ∑' p : {p : ℕ | p.Prime}, 1 / ((p : ℕ) * Real.log (p : ℕ)) := by
…
```
Mentions: Erdos1196.IsPrimitive