403.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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 403

References:

erdosproblems.com/403
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique (1980).
[Li76] Lin, S., On two problems of Erdős concerning sums of distinct factorials. Bell Laboratories internal memorandum (1960).

Imports (1)

FormalConjecturesUtil

theorem Erdos403.erdos_403 Does the equation $$2^m=a_1!+\cdots+a_k!$$ with $a_1 0$ sorry line 58 source
Source documentation
Does the equation $$2^m=a_1!+\cdots+a_k!$$ with $a_1<a_2<\cdots <a_k$ have only finitely many solutions?
Asked by Burr and Erdős. Frankl and Lin [Li76] independently showed that the answer is yes, and the largest solution is $$2^7=2!+3!+5!.$$ In fact Lin showed that the largest power of $2$ which can divide a sum of distinct factorials containing $2$ is $2^{254}$, and that there are only 5 solutions to $3^m=a_1!+\cdots+a_k!$ (when $m=0,1,2,3,6$).
See also [404].
A solution is encoded below as a pair $(m, s)$ where $s$ is the finite set $\{a_1 < a_2 < \cdots < a_k\}$ of positive integers, so the distinctness of the $a_i$ is given by set membership. The empty set contributes no solutions since $2^m \geq 1 > 0$.
The linked proof gives more than finiteness: it classifies the solutions outright, as $(0,\{1\})$, $(1,\{2\})$, $(3,\{2,3\})$, $(5,\{2,3,4\})$ and $(7,\{2,3,5\})$, so the set below has exactly five elements.

L4DEL explanation
Does the equation $$2^m=a_1!+\cdots+a_k!$$ with $a_1 0$. The linked proof gives more than finiteness: it classifies the solutions outright, as $(0,\{1\})$, $(1,\{2\})$, $(3,\{2,3\})$, $(5,\{2,3,4\})$ and $(7,\{2,3,5\})$, so the set below has exactly five elements.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Iff, True, Set.Finite, Prod, Nat, Finset, Set.ofPred, And, Membership.mem, Prod.snd, LT.lt, OfNat.ofNat, Eq, HPow.hPow, Prod.fst, Finset.sum, Nat.factorial
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_403 : answer(True) ↔
    {p : ℕ × Finset ℕ | (∀ a ∈ p.2, 0 < a) ∧
      2 ^ p.1 = ∑ a ∈ p.2, a.factorial}.Finite := by
…
```