889.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

10 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 889

Reference: erdosproblems.com/889

Imports (1)

FormalConjecturesUtil

def Erdos889.v $v(n,k)$ counts the prime factors of $n+k$ which do not divide $n+i$ for all $0 \le i < k$ line 36 source
Source documentation
$v(n,k)$ counts the prime factors of $n+k$ which do not divide $n+i$ for all $0 \le i < k$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Finset.filter, Membership.mem, Finset, Finset.range, Not, Dvd.dvd, HAdd.hAdd, Nat.primeFactors
8 instances, coercions and auxiliary constants left out.
```
def v (n k : ℕ) : ℕ :=
…
```
def Erdos889.v₀ $v_0(n)$ is the supremum of $v(n,k)$ for all $k \ge 0$ line 43 source
Source documentation
$v_0(n)$ is the supremum of $v(n,k)$ for all $k \ge 0$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, ENat
Proof uses: iSup, Erdos889.v
7 instances, coercions and auxiliary constants left out.
```
noncomputable def v₀ (n : ℕ) : ℕ∞ :=
…
```
theorem Erdos889.erdos_889 Let $v(n,k)$ count the prime factors of $n+k$ which do not divide $n+i$ for $0\leq i < k$ sorry line 52 source
Source documentation
Let $v(n,k)$ count the prime factors of $n+k$ which do not divide $n+i$ for $0\leq i < k$. Is it true that $v_0(n)=\max_{k\geq 0}v(n,k)\to \infty$ as $n\to \infty$?

depends on sorry what this means
Statement uses: Filter.Tendsto, Nat, ENat, Erdos889.v₀, Filter.atTop, nhds, Top.top
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_889 : Tendsto v₀ atTop (𝓝 ⊤) := by
…
```
Mentions: Erdos889.v₀
theorem Erdos889.erdos_889.variants.v0_gt_1 $v_0(n) > 1$ for all $n$ except $n$ = 0, 1, 2, 3, 4, 7, 8, 16 [ErSe67] Erdős, P sorry line 61 source
Source documentation
$v_0(n) > 1$ for all $n$ except $n$ = 0, 1, 2, 3, 4, 7, 8, 16
[ErSe67] Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430.

depends on sorry what this means
Statement uses: Nat, Not, Membership.mem, Finset, Insert.insert, OfNat.ofNat, Singleton.singleton, LT.lt, ENat, Erdos889.v₀
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_889.variants.v0_gt_1 :
    ∀ n : ℕ, n ∉ ({0, 1, 2, 3, 4, 7, 8, 16} : Finset ℕ) → 1 < v₀ n := by
…
```
Mentions: Erdos889.v₀
def Erdos889.v_l $v_l(n)$ is the supremum of $v(n,k)$ for all $k \ge l$ line 68 source
Source documentation
$v_l(n)$ is the supremum of $v(n,k)$ for all $k \ge l$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, ENat
Proof uses: iSup, GE.ge, Erdos889.v
8 instances, coercions and auxiliary constants left out.
```
noncomputable def v_l (l n : ℕ) : ℕ∞ :=
…
```
theorem Erdos889.erdos_889.variants.general Let $v_l(n) = \max_{k\geq l} v(n,k)$ sorry line 78 source
Source documentation
Let $v_l(n) = \max_{k\geq l} v(n,k)$. For every fixed $l$, $v_l(n) \to \infty$ as $n \to \infty$
[ErSe67] Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430.

depends on sorry what this means
Statement uses: Nat, Filter.Tendsto, ENat, Erdos889.v_l, Filter.atTop, nhds, Top.top
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_889.variants.general :
    ∀ l, Tendsto (v_l l) atTop (𝓝 ⊤) := by
…
```
Mentions: Erdos889.v_l
theorem Erdos889.erdos_889.variants.v1_eq_1_finite Does $v_1(n) = 1$ have finite solutions? sorry line 88 source
Source documentation
Does $v_1(n) = 1$ have finite solutions?
[ErSe67] Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430.

depends on sorry what this means
Statement uses: Iff, True, Set.Finite, Nat, Set.ofPred, Eq, ENat, Erdos889.v_l, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_889.variants.v1_eq_1_finite :
    answer(sorry) ↔ {n | v_l 1 n = 1}.Finite := by
…
```
Mentions: Erdos889.v_l
def Erdos889.V Documented declaration: V line 98 source
Source documentation
$V(n,k)$ is the number of primes $p$ such that $p^\alpha$ exactly divides $n+k$ and for all $0 \le i < k$, $p^\alpha$ does not divide $n+i$, where $\alpha$ is the multiplicity of $p$ in the factorization of $n+k$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Finset.filter, Membership.mem, Finset, Finset.range, Not, Dvd.dvd, HPow.hPow, Finsupp, Nat.factorization, HAdd.hAdd, Nat.primeFactors
16 instances, coercions and auxiliary constants left out.
```
def V (n k : ℕ) : ℕ :=
…
```
def Erdos889.V_l $V_l(n)$ is the supremum of $V(n,k)$ for all $k \ge l$ line 105 source
Source documentation
$V_l(n)$ is the supremum of $V(n,k)$ for all $k \ge l$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, ENat
Proof uses: iSup, GE.ge, Erdos889.V
8 instances, coercions and auxiliary constants left out.
```
noncomputable def V_l (l n : ℕ) : ℕ∞ :=
…
```
theorem Erdos889.erdos_889.variants.V1_eq_1_finite sorry line 123 source
depends on sorry what this means
Statement uses: Iff, True, Set.Finite, Nat, Set.ofPred, Eq, ENat, Erdos889.V_l, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

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