454.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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 454

References:

erdosproblems.com/454
[Po79] Pomerance, Carl, The prime number graph. Math. Comp. (1979), 399-408.

Imports (1)

FormalConjecturesUtil

def Erdos454.f Define f n to be the minimum of (n + i).nth Prime + (n - i).nth Prime over 0 < i < n line 35 source
Source documentation
Define f n to be the minimum of (n + i).nth Prime + (n - i).nth Prime over 0 < i < n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, LE.le, OfNat.ofNat, iInf, Subtype, Fin, LT.lt, HAdd.hAdd, Nat.nth, Prime, HSub.hSub
12 instances, coercions and auxiliary constants left out.
```
noncomputable def f (n : ℕ) : ℕ :=
…
```
theorem Erdos454.erdos_454 Is it true that limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop = ⊤? sorry line 40 source
Source documentation
Is it true that limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop = ⊤?

depends on sorry what this means
Statement uses: Iff, True, Eq, ENat, Filter.limsup, Nat, HSub.hSub, Erdos454.f, HMul.hMul, OfNat.ofNat, Nat.nth, Prime, Filter.atTop, Top.top
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_454 : answer(sorry) ↔ limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop = ⊤ := by
…
```
Mentions: Erdos454.f
theorem Erdos454.erdos_454.variants.two_le_limsup limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop ≥ 2, and this is proved in [Po79] sorry line 45 source
Source documentation
limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop ≥ 2, and this is proved in [Po79].

depends on sorry what this means
Statement uses: LE.le, ENat, OfNat.ofNat, Nat, Filter.limsup, HSub.hSub, Erdos454.f, HMul.hMul, Nat.nth, Prime, Filter.atTop
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_454.variants.two_le_limsup : 2 ≤ limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop := by
…
```
Mentions: Erdos454.f