1063.lean`

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Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

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

8 declarations, 6 contain sorry, 0 declared as axioms

Erdős Problem 1063

References:

erdosproblems.com/1063
[ErSe83] Erdos, P. and Selfridge, J. L., Problem 6447. Amer. Math. Monthly (1983), 710.
[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), Problem B31.
[Mo85] Monier, Jean-Marie, Problems and Solutions: Solutions of Advanced Problems: 6447. Amer. Math. Monthly 92 (1985), 435-436.
[Ku+26] Kung, P.-N., Song, L., Hwang, D., Yoon, J., Li, C.-L., Severini, S., Olšák, M., Lockhart, E., Le, Q. V., Gokturk, B., Luong, T., Pfister, T., & Peng, N. (2026). LEAP: Supercharging LLMs for Formal Mathematics with Agentic Frameworks. arXiv:2606.03303.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

def Erdos1063.n Let $n_k$ be the least $n \ge 2k$ such that all but one of the integers $n - i$ with $0 \le i < k$ divide $\binom{n}{k}$ line 46 source
Source documentation
Let $n_k$ be the least $n \ge 2k$ such that all but one of the integers $n - i$ with $0 \le i < k$ divide $\binom{n}{k}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, And, LE.le, HMul.hMul, OfNat.ofNat, Exists, LT.lt, Not, Dvd.dvd, HSub.hSub, Nat.choose, Ne
9 instances, coercions and auxiliary constants left out.
```
noncomputable def n (k : ℕ) : ℕ :=
…
```
theorem Erdos1063.erdos_1063.better_upper Estimate $n_k$ by finding a better upper bound than Cambie's $n_k \leq k \cdot \operatorname{lcm}(1, \dotsc, k-1)$ sorry line 59 source
Source documentation
Estimate $n_k$ by finding a better upper bound than Cambie's $n_k \leq k \cdot \operatorname{lcm}(1, \dotsc, k-1)$.
The comparator takes its least common multiple in ℕ and casts the result. Writing the ascription as ((… ).lcm (fun n : ℕ => n) : ℝ) instead puts it on the Finset.lcm application, so the coercion lands on n and the lcm is taken in ℝ, where lcm of non-zero elements is 1 and the whole comparator collapses to k.

depends on sorry what this means
Statement uses: Nat, Real, sorryAx, Bool.false, And, Asymptotics.IsBigO, Filter.atTop, Erdos1063.n, Asymptotics.IsLittleO, HMul.hMul, Finset.lcm, Finset.Icc, OfNat.ofNat, HSub.hSub, id
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1063.better_upper :
    let upper_bound : ℕ → ℝ := answer(sorry)
…
```
theorem Erdos1063.erdos_1063.variants.subexponential_upper_bound sorry line 76 source
depends on sorry what this means
Statement uses: Nat, Real, Real.exp, HMul.hMul, HDiv.hDiv, Real.log, HAdd.hAdd, OfNat.ofNat, And, Asymptotics.IsBigO, Filter.atTop, Erdos1063.n, Asymptotics.IsLittleO, Finset.lcm, Finset.Icc, HSub.hSub, id
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos1063.erdos_1063.variants.exists_exception sorry line 92 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, HMul.hMul, Exists, And, LT.lt, Not, Dvd.dvd, HSub.hSub, Nat.choose
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos1063.erdos_1063.variants.small_values The initial values satisfy $n_2 = 4$, $n_3 = 6$, $n_4 = 9$, and $n_5 = 12$ ([Gu04], Problem B31) line 98 source
Source documentation
The initial values satisfy $n_2 = 4$, $n_3 = 6$, $n_4 = 9$, and $n_5 = 12$ ([Gu04], Problem B31).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: And, Eq, Nat, Erdos1063.n, OfNat.ofNat
Proof uses: And.intro, le_antisymm, Nat.sInf_le, Set.ofPred, LE.le, HMul.hMul, Exists, LT.lt, Not, Dvd.dvd, HSub.hSub, Nat.choose, Ne, of_decide_eq_true, Membership.mem, Set, id, Bool, Decidable.decide, Bool.true and 16 more
34 instances, coercions and auxiliary constants left out.

Tactic trace: 60 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1063.variants.small_values :
    n 2 = 4 ∧ n 3 = 6 ∧ n 4 = 9 ∧ n 5 = 12 := by
…
```
Mentions: Erdos1063.n
theorem Erdos1063.erdos_1063.variants.monier_upper_bound sorry line 142 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Erdos1063.n, Nat.factorial
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos1063.erdos_1063.variants.cambie_upper_bound sorry line 157 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Erdos1063.n, HMul.hMul, Finset.lcm, Finset.Icc, HSub.hSub, id
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos1063.erdos_1063.variants.exp_upper_bound sorry line 165 source
depends on sorry what this means
Statement uses: Exists, Nat, Real, And, Filter.Tendsto, Filter.atTop, nhds, OfNat.ofNat, LE.le, Erdos1063.n, Real.exp, HMul.hMul, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
31 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.