4290.lean`

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

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

6 declarations, 1 contain sorry, 0 declared as axioms

Least positive multiple of $n$ in base 10 with digits 0 and 1

Least positive multiple of $n$ that when written in base 10 uses only 0's and 1's.

References:

A004290

Imports (1)

FormalConjecturesUtil

def OeisA4290.a Least positive multiple of $n$ using only 0's and 1's in base 10 line 34 source
Source documentation
Least positive multiple of $n$ using only 0's and 1's in base 10.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, And, LT.lt, OfNat.ofNat, Dvd.dvd, Membership.mem, List, Nat.digits, Or, Eq
5 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA4290.a_0 line 38 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA4290.a, OfNat.ofNat
Proof uses: id, Set, Set.ofPred, And, LT.lt, Dvd.dvd, Membership.mem, List, Nat.digits, Or, EmptyCollection.emptyCollection, Set.ext, Eq.mpr, Iff, Not, Eq.trans, False, congrArg, iff_false, implies_congr and 4 more
11 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
theorem OeisA4290.a_1 line 49 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA4290.a, OfNat.ofNat
Proof uses: id, IsLeast, Set.ofPred, And, LT.lt, Dvd.dvd, Membership.mem, List, Nat.digits, Or, And.intro, Set, lowerBounds, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, dvd_rfl, Or.inr and 14 more
20 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
theorem OeisA4290.a_2 line 62 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA4290.a, OfNat.ofNat
Proof uses: id, IsLeast, Set.ofPred, And, LT.lt, Dvd.dvd, Membership.mem, List, Nat.digits, Or, And.intro, Set, lowerBounds, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, HMul.hMul, HPow.hPow and 30 more
45 instances, coercions and auxiliary constants left out.

Tactic trace: 100 steps
Open to load the checked before-and-after goal states.
theorem OeisA4290.a_ten_pow $a(10^k) = 10^k$ for all $k$ line 109 source
Source documentation
$a(10^k) = 10^k$ for all $k$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, OeisA4290.a, HPow.hPow, OfNat.ofNat
Proof uses: id, IsLeast, Set.ofPred, And, LT.lt, Dvd.dvd, Membership.mem, List, Nat.digits, Or, And.intro, Set, lowerBounds, Nat.pow_pos, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, dvd_rfl and 22 more
38 instances, coercions and auxiliary constants left out.

Tactic trace: 19 steps
Open to load the checked before-and-after goal states.
```
theorem a_ten_pow (k : ℕ) : a (10 ^ k) = 10 ^ k := by
…
```
Mentions: OeisA4290.a
theorem OeisA4290.conjecture It is known that $a(10^k - 1) = (10^{9k} - 1) / 9$ for all $k$ sorry line 129 source
Source documentation
It is known that $a(10^k - 1) = (10^{9k} - 1) / 9$ for all $k$. Is $a(n) < a(10^k - 1)$ for all $n < 10^k - 1$?
- David Radcliffe, Aug 01 2025
depends on sorry what this means
Statement uses: Nat, LT.lt, HSub.hSub, HPow.hPow, OfNat.ofNat, OeisA4290.a
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 conjecture (k : ℕ) :
    ∀ n < 10 ^ k - 1, a n < a (10 ^ k - 1) := by
…
```
Mentions: OeisA4290.a