69922.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

Number of primes $p$ such that $n^n \le p \le n^n + n^2$

The sequence $a(n)$ counts the number of prime numbers in the interval $[n^n, n^n + n^2]$: $$a(n) = |\{p \text{ prime} \mid n^n \le p \le n^n + n^2\}|$$

References:

A069922

Imports (1)

FormalConjecturesUtil

def OeisA69922.a Number of primes $p$ such that $n^n \le p \le n^n + n^2$ line 36 source
Source documentation
Number of primes $p$ such that $n^n \le p \le n^n + n^2$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Finset.filter, Nat.Prime, Finset.Icc, HPow.hPow, HAdd.hAdd, OfNat.ofNat
10 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA69922.a_1 Value of the sequence a at 1 line 41 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA69922.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 1 := by
…
```
Mentions: OeisA69922.a
theorem OeisA69922.a_2 Value of the sequence a at 2 line 46 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA69922.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 2 := by
…
```
Mentions: OeisA69922.a
theorem OeisA69922.a_3 Value of the sequence a at 3 line 51 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA69922.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 2 := by
…
```
Mentions: OeisA69922.a
theorem OeisA69922.a_4 Value of the sequence a at 4 line 56 source
Source documentation
Value of the sequence a at 4.

extra axioms: _private.0.OeisA69922.a_4._native.decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA69922.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 4 := by
…
```
Mentions: OeisA69922.a
theorem OeisA69922.a_5 Value of the sequence a at 5 line 61 source
Source documentation
Value of the sequence a at 5.

extra axioms: _private.0.OeisA69922.a_5._native.decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA69922.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_5 : a 5 = 1 := by
…
```
Mentions: OeisA69922.a
theorem OeisA69922.conjecture Question: for any $n > 0$, is there at least one prime $p$ such that $n^n \le p \le n^n + n^2$? sorry line 69 source
Source documentation
Question: for any $n > 0$, is there at least one prime $p$ such that $n^n \le p \le n^n + n^2$? In this case, that would be stronger than the Schinzel conjecture: "for $m > 1$ there's at least one prime $p$ such that $m \le p \le m + \log(m)^2$" since $n^2 < \log(n^n)^2 = n^2 \log(n)^2$.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, LE.le, OeisA69922.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 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) : 1 ≤ a n := by
…
```
Mentions: OeisA69922.a