PollocksConjecture.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

5 declarations, 3 contain sorry, 0 declared as axioms

Pollock's (tetrahedral numbers) conjecture

Every positive integer is the sum of at most 5 tetrahedral numbers.

References:

Wikipedia
A797
L. E. Dickson, History of the Theory of Numbers, Vol. II: Diophantine Analysis, Dover (2005), pp. 22–23
Frederick Pollock, On the extension of the principle of Fermat's theorem on the polygonal numbers to the higher order of series whose ultimate differences are constant, Abstracts of the Papers Communicated to the Royal Society of London 5 (1850), 922–924
H. E. Salzer and N. Levine, Table of integers not exceeding 100000 that are not expressible as the sum of four tetrahedral numbers, Math. Comp. 12 (1958), 141–144
MathWorld: Pollock's Conjecture

Imports (1)

FormalConjecturesUtil

def PollocksConjecture.tetrahedral The $n$-th tetrahedral number: $T_n = \frac{n(n+1)(n+2)}{6}$ line 43 source
Source documentation
The $n$-th tetrahedral number: $T_n = \frac{n(n+1)(n+2)}{6}$.

no axioms what this means
Statement uses: Nat
Proof uses: HDiv.hDiv, HMul.hMul, HAdd.hAdd, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.
```
def tetrahedral (n : ℕ) : ℕ :=
…
```
def PollocksConjecture.NotSumOfFourTetrahedral The set of natural numbers that are not a sum of $4$ tetrahedral numbers line 49 source
Source documentation
The set of natural numbers that are not a sum of $4$ tetrahedral numbers.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat
Proof uses: Set.ofPred, Fin, OfNat.ofNat, Ne, Finset.sum, Finset.univ, PollocksConjecture.tetrahedral
3 instances, coercions and auxiliary constants left out.
```
def NotSumOfFourTetrahedral : Set ℕ :=
…
```
theorem PollocksConjecture.pollock_tetrahedral Pollock's (tetrahedral numbers) conjecture: every integer is the sum of at most $5$ tetrahedral numbers sorry line 59 source
Source documentation
Pollock's (tetrahedral numbers) conjecture: every integer is the sum of at most $5$ tetrahedral numbers.

depends on sorry what this means
Statement uses: Nat, Exists, Fin, OfNat.ofNat, Eq, Finset.sum, Finset.univ, PollocksConjecture.tetrahedral
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 pollock_tetrahedral (N : ℕ) :
    ∃ f : Fin 5 → ℕ, N = ∑ i, tetrahedral (f i) := by
…
```
Mentions: PollocksConjecture.tetrahedral
theorem PollocksConjecture.pollock_tetrahedral.salzer_levine Salzer–Levine strengthening (as stated on Wikipedia/OEIS): there are exactly $241$ integers that are not a sum of $4$ tetrahedral numbers, and the largest is $343867$ sorry line 68 source
Source documentation
Salzer–Levine strengthening (as stated on Wikipedia/OEIS): there are exactly $241$ integers that are not a sum of $4$ tetrahedral numbers, and the largest is $343867$.

depends on sorry what this means
Statement uses: IsGreatest, Nat, PollocksConjecture.NotSumOfFourTetrahedral, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem pollock_tetrahedral.salzer_levine :
    IsGreatest NotSumOfFourTetrahedral 343867 := by
…
```
Mentions: PollocksConjecture.NotSumOfFourTetrahedral
theorem PollocksConjecture.pollock_tetrahedral.ncard_exceptions As stated on Wikipedia/OEIS (A797), the set of exceptions has cardinality $241$ sorry line 74 source
Source documentation
As stated on Wikipedia/OEIS (A797), the set of exceptions has cardinality $241$.

depends on sorry what this means
Statement uses: Iff, IsGreatest, Nat, PollocksConjecture.NotSumOfFourTetrahedral, OfNat.ofNat, Eq, Set.ncard
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem pollock_tetrahedral.ncard_exceptions :
    type_of% pollock_tetrahedral.salzer_levine ↔
    NotSumOfFourTetrahedral.ncard = 241 := by
…
```
Mentions: PollocksConjecture.pollock_tetrahedral.salzer_levine