53000.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

7 declarations, 1 contain sorry, 0 declared as axioms

$a(n) = (\text{smallest prime} > n^2) - n^2$

The difference between the smallest prime strictly greater than $n^2$ and $n^2$.

References:

A053000

Imports (1)

FormalConjecturesUtil

def OeisA53000.a $a(n) = (\text{smallest prime} > n^2) - n^2$ line 33 source
Source documentation
$a(n) = (\text{smallest prime} > n^2) - n^2$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: HSub.hSub, InfSet.sInf, Set.ofPred, And, Nat.Prime, LT.lt, HPow.hPow, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA53000.a_0 line 37 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA53000.a, OfNat.ofNat
Proof uses: tsub_eq_of_eq_add, InfSet.sInf, Set.ofPred, And, Nat.Prime, LT.lt, HPow.hPow, HAdd.hAdd, IsLeast.csInf_eq, of_decide_eq_true, IsLeast, id, Bool, Decidable.decide, Bool.true, Eq.refl
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA53000.a_1 line 40 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA53000.a, OfNat.ofNat
Proof uses: tsub_eq_of_eq_add, InfSet.sInf, Set.ofPred, And, Nat.Prime, LT.lt, HPow.hPow, HAdd.hAdd, IsLeast.csInf_eq, of_decide_eq_true, IsLeast, id, Bool, Decidable.decide, Bool.true, Eq.refl
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA53000.a_2 line 43 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA53000.a, OfNat.ofNat
Proof uses: tsub_eq_of_eq_add, InfSet.sInf, Set.ofPred, And, Nat.Prime, LT.lt, HPow.hPow, HAdd.hAdd, IsLeast.csInf_eq, of_decide_eq_true, IsLeast, id, Bool, Decidable.decide, Bool.true, Eq.refl
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA53000.a_3 line 46 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA53000.a, OfNat.ofNat
Proof uses: tsub_eq_of_eq_add, InfSet.sInf, Set.ofPred, And, Nat.Prime, LT.lt, HPow.hPow, HAdd.hAdd, IsLeast.csInf_eq, of_decide_eq_true, IsLeast, id, Bool, Decidable.decide, Bool.true, Eq.refl
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA53000.a_4 line 49 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA53000.a, OfNat.ofNat
Proof uses: tsub_eq_of_eq_add, InfSet.sInf, Set.ofPred, And, Nat.Prime, LT.lt, HPow.hPow, HAdd.hAdd, IsLeast.csInf_eq, of_decide_eq_true, IsLeast, id, Bool, Decidable.decide, Bool.true, Eq.refl
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA53000.conjecture Conjecture: $a(n) \le 1 + \phi(n)$ for $n > 0$ sorry line 57 source
Source documentation
Conjecture: $a(n) \le 1 + \phi(n)$ for $n > 0$. This improves on Oppermann's conjecture, which says $a(n) < n$.
- Thomas Ordowski, Dec 17 2014
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, LE.le, OeisA53000.a, HAdd.hAdd, Nat.totient
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture (n : ℕ) (hn : 0 < n) : a n ≤ 1 + n.totient := by
…
```
Mentions: OeisA53000.a