77408.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

10 declarations, 1 contain sorry, 0 declared as axioms

Trajectory of 103 under the Reverse and Add! operation in base 3

The sequence $a(n)$ is the trajectory of $103$ under the Reverse and Add! operation carried out in base $3$, written in base $10$. $a(0) = 103$, and $a(n+1) = a(n) + \text{rev}_3(a(n))$.

References:

A077408

Imports (1)

FormalConjecturesUtil

def OeisA77408.revBase The number whose base-$b$ digits are the reversal of $n$'s base-$b$ digits line 36 source
Source documentation
The number whose base-$b$ digits are the reversal of $n$'s base-$b$ digits.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.ofDigits, List.reverse, Nat.digits
1 instances, coercions and auxiliary constants left out.
```
def revBase (b n : ℕ) : ℕ :=
…
```
def OeisA77408.IsBasePalindrome A natural number $n$ is a base-$b$ palindrome if its base-$b$ digits read the same forwards and backwards line 41 source
Source documentation
A natural number $n$ is a base-$b$ palindrome if its base-$b$ digits read the same forwards and backwards.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Eq, OeisA77408.revBase
```
def IsBasePalindrome (b n : ℕ) : Prop :=
…
```
def OeisA77408.a Trajectory of 103 under the Reverse and Add! line 45 source
Source documentation
Trajectory of 103 under the Reverse and Add! operation in base 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
2 instances, coercions and auxiliary constants left out.
```
def a : ℕ → ℕ
  | 0 => 103
  | n + 1 => a n + revBase 3 (a n)

/-- Value of the sequence `a` at 0. -/
@[category test, AMS 11]
```
Mentions: OeisA77408.revBase, test, AMS
theorem OeisA77408.a_0 Value of the sequence a at 0 line 51 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA77408.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 a_0 : a 0 = 103 := by
…
```
Mentions: OeisA77408.a
theorem OeisA77408.a_1 Value of the sequence a at 1 line 56 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA77408.a, OfNat.ofNat
2 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 = 230 := by
…
```
Mentions: OeisA77408.a
theorem OeisA77408.a_2 Value of the sequence a at 2 line 61 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA77408.a, OfNat.ofNat
2 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 = 436 := by
…
```
Mentions: OeisA77408.a
theorem OeisA77408.a_3 Value of the sequence a at 3 line 66 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA77408.a, OfNat.ofNat
2 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 = 776 := by
…
```
Mentions: OeisA77408.a
theorem OeisA77408.a_4 Value of the sequence a at 4 line 71 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA77408.a, OfNat.ofNat
2 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 = 2424 := by
…
```
Mentions: OeisA77408.a
theorem OeisA77408.a_5 Value of the sequence a at 5 line 76 source
Source documentation
Value of the sequence a at 5.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA77408.a, OfNat.ofNat
2 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 = 3856 := by
…
```
Mentions: OeisA77408.a
theorem OeisA77408.conjecture $103$ is conjectured to be the smallest number such that the Reverse and Add! sorry line 83 source
Source documentation
$103$ is conjectured to be the smallest number such that the Reverse and Add! algorithm in base $3$ does not lead to a palindrome. Its trajectory is conjectured to never reach a palindrome.

depends on sorry what this means
Statement uses: Nat, Not, OeisA77408.IsBasePalindrome, OfNat.ofNat, OeisA77408.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 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) : ¬ IsBasePalindrome 3 (a n) := by
…
```
Mentions: OeisA77408.IsBasePalindrome, OeisA77408.a