427.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

Erdős Problem 427

Reference: erdosproblems.com/427

Imports (1)

FormalConjecturesUtil

def Erdos427.erdos427 The predicate that for every $n$ and $d$, there exists $k$ such that $$ d \mid p_{n + 1} + \cdots + p_{n + k}, $$ where $p_r$ denotes the $r$th prime? line 37 source
Source documentation
The predicate that for every $n$ and $d$, there exists $k$ such that $$ d \mid p_{n + 1} + \cdots + p_{n + k}, $$ where $p_r$ denotes the $r$th prime?

standard axioms: Classical.choice, Quot.sound, propext what this means
Proof uses: Nat, Ne, OfNat.ofNat, Exists, And, Dvd.dvd, Finset.sum, Finset.Ico, HAdd.hAdd, Nat.nth, Nat.Prime
7 instances, coercions and auxiliary constants left out.
```
def erdos427 : Prop := ∀ (n d : ℕ),
…
```
def Erdos427.ShiuTheorem Documented declaration: ShiuTheorem line 50 source
Source documentation
The statement of Shiu's theorem: for any $k \geq 1$ and $(a, q) = 1$ there exist infinitely many $k$-tuples of consecutive primes $p_m, \dots, p_{m + k - 1}$ all of which are congruent to $a$ modulo $q$.
[Sh00] Shiu, D. K. L., Strings of congruent primes. J. London Math. Soc. (2) (2000), 359-373.

standard axioms: Classical.choice, Quot.sound, propext what this means
Proof uses: Nat, LE.le, OfNat.ofNat, Eq, Nat.gcd, Set.Infinite, Set.ofPred, Membership.mem, Finset, Finset.image, Nat.nth, Nat.Prime, Finset.Ico, HAdd.hAdd, Nat.ModEq
9 instances, coercions and auxiliary constants left out.
```
def ShiuTheorem : Prop := ∀ (k a q : ℕ), 1 ≤ k → 1 ≤ q → a.gcd q = 1 →
…
```
theorem Erdos427.erdos_427.variants.shiu Documented declaration: shiu sorry line 64 source
Source documentation
Shiu's theorem: for any $k \geq 1$ and $(a, q) = 1$ there exist infinitely many $k$-tuples of consecutive primes $p_m, \dots, p_{m + k - 1}$ all of which are congruent to $a$ modulo $q$.
This is stated ahead of erdos_427 because the formal proof linked there assumes it, and the assuming clause must name a declaration that already exists.
[Sh00] Shiu, D. K. L., Strings of congruent primes. J. London Math. Soc. (2) (2000), 359-373.

depends on sorry what this means
Statement uses: Erdos427.ShiuTheorem
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_427.variants.shiu : ShiuTheorem := by
…
```
Mentions: Erdos427.ShiuTheorem
theorem Erdos427.erdos_427 sorry line 78 source
depends on sorry what this means
Statement uses: Iff, True, Erdos427.erdos427
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos427.erdos_427.variants.of_shiu Cedric Pilatte has observed that a positive solution to Erdős Problem 427 follows from Shiu's theorem sorry line 86 source
Source documentation
Cedric Pilatte has observed that a positive solution to Erdős Problem 427 follows from Shiu's theorem.

depends on sorry what this means
Statement uses: Erdos427.ShiuTheorem, Erdos427.erdos427
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_427.variants.of_shiu (H : ShiuTheorem) : erdos427 := by
…
```
Mentions: Erdos427.ShiuTheorem, Erdos427.erdos427