281.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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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

4 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 281

References:

erdosproblems.com/281
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[DaEr36] Davenport, H. and Erdős, P., On sequences of positive integers. Acta Arithmetica (1936), 147-151.
[HaRo66] Halberstam, H. and Roth, K. F., Sequences. Vol. I. (1966), xx+291.

Imports (1)

FormalConjecturesUtil

def Erdos281.ResidueChoice Choices of congruence classes $a_i \pmod{n_i}$ line 39 source
Source documentation
Choices of congruence classes $a_i \pmod{n_i}$.

no axioms what this means
Statement uses: Nat
Proof uses: ZMod
```
def ResidueChoice (n : ℕ → ℕ) := ∀ i : ℕ, ZMod (n i)
```
def Erdos281.avoidAll The integers avoiding all congruences $a_i \pmod{n_i}$ line 42 source
Source documentation
The integers avoiding all congruences $a_i \pmod{n_i}$.

standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Erdos281.ResidueChoice, Set, Int
Proof uses: Set.ofPred, Ne, ZMod
5 instances, coercions and auxiliary constants left out.
```
def avoidAll (n : ℕ → ℕ) (a : ResidueChoice n) : Set ℤ :=
…
```
Mentions: Erdos281.ResidueChoice
def Erdos281.avoidPrefix The integers avoiding the first $k$ congruences $a_i \pmod{n_i}$ line 46 source
Source documentation
The integers avoiding the first $k$ congruences $a_i \pmod{n_i}$.

standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Erdos281.ResidueChoice, Set, Int
Proof uses: Set.ofPred, LT.lt, Ne, ZMod
6 instances, coercions and auxiliary constants left out.
```
def avoidPrefix (n : ℕ → ℕ) (a : ResidueChoice n) (k : ℕ) : Set ℤ :=
…
```
Mentions: Erdos281.ResidueChoice
theorem Erdos281.erdos_281 Documented declaration: erdos_281 sorry line 59 source
Source documentation
Let $n_1<n_2<\cdots$ be an infinite sequence such that, for any choice of congruence classes $a_i\pmod{n_i}$, the set of integers not satisfying any of the congruences $a_i\pmod{n_i}$ has density $0$. Is it true that for every $\epsilon>0$ there exists some $k$ such that, for every choice of congruence classes $a_i$, the density of integers not satisfying any of the congruences $a_i\pmod{n_i}$ for $1\leq i\leq k$ is less than $\epsilon$?
The answer is yes; the linked Lean proof formalizes Somani's argument.

L4DEL explanation
Let $n_10$ there exists some $k$ such that, for every choice of congruence classes $a_i$, the density of integers not satisfying any of the congruences $a_i\pmod{n_i}$ for $1\leq i\leq k$ is less than $\epsilon$? The answer is yes; the linked Lean proof formalizes Somani's argument.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Iff, True, Nat, StrictMono, LT.lt, OfNat.ofNat, Erdos281.ResidueChoice, Set.HasIntDensity, Erdos281.avoidAll, Real, Exists, And, Erdos281.avoidPrefix
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_281 : answer(True) ↔
    ∀ (n : ℕ → ℕ), StrictMono n → (∀ i, 0 < n i) →
      (∀ a : ResidueChoice n, Set.HasIntDensity (avoidAll n a) 0) →
        ∀ ε : ℝ, 0 < ε →
          ∃ k : ℕ, ∀ a : ResidueChoice n,
            ∃ d : ℝ, Set.HasIntDensity (avoidPrefix n a k) d ∧ d < ε := by
…
```
Mentions: Erdos281.ResidueChoice, Set.HasIntDensity, Erdos281.avoidAll, Erdos281.avoidPrefix