937.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, 1 contain sorry, 0 declared as axioms

Erdős Problem 937

Reference: erdosproblems.com/937

References:

[BBC24] Bajpai, P., Bennett, M. A. and Chan, T. H., Arithmetic progressions in squarefull / powerful numbers, Int. J. Number Theory 20 (2024), 19-45.

Imports (1)

FormalConjecturesUtil

def Erdos937.IsCoprimePowerfulAP4 The four numbers $a, a+d, a+2d, a+3d$ form a four-term arithmetic progression ($d > 0$) of pairwise coprime powerful numbers line 38 source
Source documentation
The four numbers $a, a+d, a+2d, a+3d$ form a four-term arithmetic progression ($d > 0$) of pairwise coprime powerful numbers.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: And, LT.lt, OfNat.ofNat, Nat.Powerful, HAdd.hAdd, HMul.hMul, Nat.Coprime
6 instances, coercions and auxiliary constants left out.
```
def IsCoprimePowerfulAP4 (a d : ℕ) : Prop :=
…
```
theorem Erdos937.erdos_937 sorry line 55 source
depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Prod, Nat, Set.ofPred, Erdos937.IsCoprimePowerfulAP4, Prod.fst, Prod.snd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos937.not_isCoprimePowerfulAP4_zero_one Sanity check for IsCoprimePowerfulAP4: the progression $0, 1, 2, 3$ is not a valid example, since $2$ is not powerful line 62 source
Source documentation
Sanity check for IsCoprimePowerfulAP4: the progression $0, 1, 2, 3$ is not a valid example, since $2$ is not powerful.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, Erdos937.IsCoprimePowerfulAP4, OfNat.ofNat, Nat
Proof uses: LT.lt, And, Nat.Powerful, HAdd.hAdd, HMul.hMul, Nat.Coprime, False, Nat.not_full_of_prime_mod_prime_sq, Nat.prime_two, Mathlib.Meta.NormNum.isNat_eq_true, HMod.hMod, HPow.hPow, Mathlib.Meta.NormNum.isNat_natMod, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.IsNatPowT.run, Mathlib.Meta.NormNum.IsNatPowT.bit0
19 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem not_isCoprimePowerfulAP4_zero_one : ¬ IsCoprimePowerfulAP4 0 1 := by
…
```
Mentions: Erdos937.IsCoprimePowerfulAP4