405.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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 405

References:

erdosproblems.com/405
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980)
[BrEr91] Brindza, B. and Erdős, P., On some {D}iophantine problems involving powers and factorials. J. Austral. Math. Soc. Ser. A (1991), 1--7.
[YuLi96] Yu, Kunrui and Liu, Dehua, A complete resolution of a problem of {E}rdős and {G}raham. Rocky Mountain J. Math. (1996), 1235--1244.

Imports (1)

FormalConjecturesUtil

theorem Erdos405.erdos_405 Let $p$ be an odd prime sorry line 48 source
Source documentation
Let $p$ be an odd prime. Is it true that the equation $(p-1)!+a^{p-1}=p^k$ has only finitely many solutions?
Originally proposed by Erdős and Graham [ErGr80]. Brindza and Erdős [BrEr91] proved that there are finitely many such solutions.

depends on sorry what this means
Statement uses: Set.Finite, Prod, Nat, Set.ofPred, Erdos405.erdos_405.match_1, And, Nat.Prime, Odd, Eq, HAdd.hAdd, Nat.factorial, HSub.hSub, OfNat.ofNat, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_405 :
    Set.Finite { x : ℕ × ℕ × ℕ | let (a, k, p) := x; p.Prime ∧ Odd p ∧
…
```
theorem Erdos405.erdos_405.variants.nonodd_p Erdős and Graham [ErGr80] ask this allowing $p=2$, but this is presumably an oversight, since clearly there are infinitely many solutions when $p=2$ line 60 source
Source documentation
Erdős and Graham [ErGr80] ask this allowing $p=2$, but this is presumably an oversight, since clearly there are infinitely many solutions when $p=2$.
Observe that this creates $1! + a = 2^k$. For all $k$, fix $a = 2^k - 1$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set.Infinite, Prod, Nat, Set.ofPred, Erdos405.erdos_405.match_1, And, Nat.Prime, Eq, HAdd.hAdd, Nat.factorial, HSub.hSub, OfNat.ofNat, HPow.hPow
Proof uses: Set.infinite_of_injective_forall_mem, Prod.mk, And.left, True, And.right, Eq.mp, Eq.trans, Prod.mk.injEq, congrArg, eq_self, id, Membership.mem, Set, And.intro, Nat.prime_two, Eq.mpr, pow_one, LT.lt, pow_pos, Mathlib.Meta.Positivity.pos_of_isNat and 6 more
35 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_405.variants.nonodd_p :
    Set.Infinite { x : ℕ × ℕ × ℕ | let (a, k, p) := x; p.Prime ∧
…
```
theorem Erdos405.erdos_405.variants.yu_liu Yu and Liu [YuLi96] showed that the only solutions to $(p-1)! + a^(p-1) = p^k$ for an odd prime p are: $$ 2! + 1^2 = 3\\ 2! + 5^2 = 3^3\\ 4! + 1^4 = 5^2 $$ sorry line 94 source
Source documentation
Yu and Liu [YuLi96] showed that the only solutions to $(p-1)! + a^(p-1) = p^k$ for an odd prime p are: $$ 2! + 1^2 = 3\\ 2! + 5^2 = 3^3\\ 4! + 1^4 = 5^2 $$

depends on sorry what this means
Statement uses: Eq, Set, Prod, Nat, Set.ofPred, Erdos405.erdos_405.match_1, And, Nat.Prime, Odd, HAdd.hAdd, Nat.factorial, HSub.hSub, OfNat.ofNat, HPow.hPow, Insert.insert, Prod.mk, Singleton.singleton
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_405.variants.yu_liu :
    { x : ℕ × ℕ × ℕ | let (a, k, p) := x; p.Prime ∧ Odd p ∧
…
```