109845.lean`

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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

Conjectures associated with A109845

$a(1) = 2$; $a(2n)$ = lcm of all previous terms + 1; $a(2n+1)$ = lcm of all previous terms - 1.

References:

A109845

Imports (1)

FormalConjecturesUtil

def OeisA109845.a $a(1) = 2$; $a(2n)$ = lcm of all previous terms + 1; $a(2n+1)$ = lcm of all previous terms - 1 line 40 source
Source documentation
$a(1) = 2$; $a(2n)$ = lcm of all previous terms + 1; $a(2n+1)$ = lcm of all previous terms - 1. The sequence is correctly generated by the recurrence relation $a(1)=2$, $a(2)=3$, and for $n \ge 3$: $$a(n) = a(n-1)^2 + (-1)^n a(n-1) + (-1)^n.$$ We use $0$-based indexing here, so $a(n)$ corresponds to the OEIS sequence $a(n+1)$.

no axioms what this means
Statement uses: Nat
2 instances, coercions and auxiliary constants left out.
```
def a : ℕ → ℕ
  | 0 => 2
  | 1 => 3
  | n + 2 =>
    let prev := a (n + 1)
…
```
theorem OeisA109845.a_0 line 51 source
no axioms what this means
Statement uses: Eq, Nat, OeisA109845.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA109845.a_1 line 54 source
no axioms what this means
Statement uses: Eq, Nat, OeisA109845.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA109845.a_2 line 57 source
no axioms what this means
Statement uses: Eq, Nat, OeisA109845.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA109845.a_3 line 60 source
no axioms what this means
Statement uses: Eq, Nat, OeisA109845.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA109845.a_4 line 63 source
no axioms what this means
Statement uses: Eq, Nat, OeisA109845.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA109845.conjecture Conjecture: There are infinitely many primes in this sequence sorry line 69 source
Source documentation
Conjecture: There are infinitely many primes in this sequence.

depends on sorry what this means
Statement uses: Set.Infinite, Nat, Set.ofPred, Nat.Prime, OeisA109845.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture : Set.Infinite {n : ℕ | (a n).Prime} := by
…
```
Mentions: OeisA109845.a