34694.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

Smallest prime $\equiv 1 \pmod n$

$$a(n) = \min \{p \in \mathbb{P} \mid p \equiv 1 \pmod n\}$$

References:

A034694

Imports (1)

FormalConjecturesUtil

def OeisA34694.a Smallest prime $\equiv 1 \pmod n$ line 33 source
Source documentation
Smallest prime $\equiv 1 \pmod n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, And, Nat.Prime, Dvd.dvd, HSub.hSub, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA34694.a_1 Value of the sequence a at 1 line 38 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA34694.a, OfNat.ofNat
Proof uses: IsLeast.csInf_eq, Set.ofPred, And, Nat.Prime, Dvd.dvd, HSub.hSub, of_decide_eq_true, IsLeast, id, Bool, Decidable.decide, Bool.true, Eq.refl
17 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 = 2 := IsLeast.csInf_eq <| by decide
…
```
Mentions: OeisA34694.a
theorem OeisA34694.a_2 Value of the sequence a at 2 line 42 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA34694.a, OfNat.ofNat
Proof uses: IsLeast.csInf_eq, Set.ofPred, And, Nat.Prime, Dvd.dvd, HSub.hSub, of_decide_eq_true, IsLeast, id, Bool, Decidable.decide, Bool.true, Eq.refl
17 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 = 3 := IsLeast.csInf_eq <| by decide
…
```
Mentions: OeisA34694.a
theorem OeisA34694.a_3 Value of the sequence a at 3 line 46 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA34694.a, OfNat.ofNat
Proof uses: IsLeast.csInf_eq, Set.ofPred, And, Nat.Prime, Dvd.dvd, HSub.hSub, of_decide_eq_true, IsLeast, id, Bool, Decidable.decide, Bool.true, Eq.refl
17 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 = 7 := IsLeast.csInf_eq <| by decide
…
```
Mentions: OeisA34694.a
theorem OeisA34694.a_4 Value of the sequence a at 4 line 50 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA34694.a, OfNat.ofNat
Proof uses: IsLeast.csInf_eq, Set.ofPred, And, Nat.Prime, Dvd.dvd, HSub.hSub, of_decide_eq_true, IsLeast, id, Bool, Decidable.decide, Bool.true, Eq.refl
17 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 = 5 := IsLeast.csInf_eq <| by decide
…
```
Mentions: OeisA34694.a
theorem OeisA34694.a_5 Value of the sequence a at 5 line 54 source
Source documentation
Value of the sequence a at 5.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA34694.a, OfNat.ofNat
Proof uses: IsLeast.csInf_eq, Set.ofPred, And, Nat.Prime, Dvd.dvd, HSub.hSub, of_decide_eq_true, IsLeast, id, Bool, Decidable.decide, Bool.true, Eq.refl
17 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 = 11 := IsLeast.csInf_eq <| by decide
…
```
Mentions: OeisA34694.a
theorem OeisA34694.conjecture "Conjecture: $a(n) < n^2$ for $n > 1$ sorry line 59 source
Source documentation
"Conjecture: $a(n) < n^2$ for $n > 1$. - Thomas Ordowski, Dec 19 2016"

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, OeisA34694.a, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 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 : 1 < n) : a n < n ^ 2 := by
…
```
Mentions: OeisA34694.a