407.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

2 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 407

References:

erdosproblems.com/407
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[EGST88] Evertse, J.-H. and Győry, K. and Stewart, C. L. and Tijdeman, R., $S$-unit equations and their applications. New advances in transcendence theory (Durham, 1986) (1988), 110-174.
[TiWa88] Tijdeman, R. and Wang, Lian Xiang, Sums of products of powers of given prime numbers. Pacific J. Math. (1988), 177-193.
[BaBe24] Bajpai, Prajeet and Bennett, Michael A., Effective $S$-unit equations beyond three terms: Newman's conjecture. Acta Arith. (2024), 421-458.

Imports (1)

FormalConjecturesUtil

def Erdos407.w w n counts the ordered quadruples $(a,b,c,d)$ of nonnegative integers with $n=2^a+3^b+2^c3^d$ line 41 source
Source documentation
w n counts the ordered quadruples $(a,b,c,d)$ of nonnegative integers with $n=2^a+3^b+2^c3^d$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Set.ncard, Prod, Set.ofPred, Eq, HAdd.hAdd, HPow.hPow, OfNat.ofNat, Prod.fst, Prod.snd, HMul.hMul
9 instances, coercions and auxiliary constants left out.
```
noncomputable def w (n : ℕ) : ℕ :=
…
```
theorem Erdos407.erdos_407 sorry line 62 source
depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, LE.le, Erdos407.w
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.