26.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

14 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 26

References:

erdosproblems.com/26
Te19 G. Tenenbaum, Some of Erdős' unconventional problems in number theory, thirty-four years later, arXiv:1908.00488 [math.NT] (2019)

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

def Erdos26.IsThick A sequence of naturals $(a_i)$ is thick if their sum of reciprocals diverges: $$ \sum_i \frac{1}{a_i} = \infty $$ line 38 source
Source documentation
A sequence of naturals $(a_i)$ is thick if their sum of reciprocals diverges: $$ \sum_i \frac{1}{a_i} = \infty $$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Not, Summable, Real, HDiv.hDiv, OfNat.ofNat, SummationFilter.unconditional
11 instances, coercions and auxiliary constants left out.
```
def IsThick {ι : Type*} (A : ι → ℕ) : Prop := ¬Summable (fun i ↦ (1 : ℝ) / A i)
…
```
theorem Erdos26.not_isThick_of_finite line 41 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finite, Nat, Not, Erdos26.IsThick
Proof uses: Eq.mpr, Summable, Real, Inv.inv, SummationFilter.unconditional, id, Eq, Eq.trans, congrArg, HDiv.hDiv, OfNat.ofNat, congrFun', SummationFilter, optParam, funext, one_div, Summable.of_finite
19 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos26.not_isThick_of_geom_one_lt line 45 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, Not, Erdos26.IsThick, HPow.hPow
Proof uses: Eq.mpr, Summable, Real, Inv.inv, SummationFilter.unconditional, id, Eq, Eq.trans, congrArg, HDiv.hDiv, congrFun', SummationFilter, optParam, funext, Nat.cast_pow, one_div, Eq.mp, inv_pow, summable_geometric_of_lt_one, of_eq_true and 8 more
80 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem Erdos26.isThick_const line 50 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Infinite, Nat, GT.gt, OfNat.ofNat, Erdos26.IsThick
Proof uses: Eq.mpr, Not, Eq, id, congrArg, Summable, Real, HDiv.hDiv, SummationFilter.unconditional, Eq.trans, Inv.inv, congrFun', SummationFilter, optParam, funext, one_div, Nat.ne_zero_of_lt
56 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
def Erdos26.MultiplesOf The set of multiples of a sequence $(a_i)$ is $\{na_i | n \in \mathbb{N}, i\}$ line 55 source
Source documentation
The set of multiples of a sequence $(a_i)$ is $\{na_i | n \in \mathbb{N}, i\}$.

no axioms what this means
Statement uses: Nat, Set
Proof uses: Set.range, Prod, Erdos26.MultiplesOf.match_1, HMul.hMul
2 instances, coercions and auxiliary constants left out.
```
def MultiplesOf {ι : Type*} (A : ι → ℕ) : Set ℕ := Set.range fun (n, i) ↦ n * A i
…
```
theorem Erdos26.multiplesOf_eq_univ line 58 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Membership.mem, Set, Set.range, OfNat.ofNat, Eq, Erdos26.MultiplesOf, Set.univ
Proof uses: top_unique, Top.top, Exists.intro, Prod, Erdos26.MultiplesOf.match_1, HMul.hMul, Prod.mk, of_eq_true, Eq.trans, True, congrFun', congrArg, mul_one, eq_self
17 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
def Erdos26.IsBehrend A sequence of naturals $(a_i)$ is Behrend if almost all integers are a multiple of some $a_i$ line 65 source
Source documentation
A sequence of naturals $(a_i)$ is Behrend if almost all integers are a multiple of some $a_i$. In other words, if the set of multiples has natural density $1$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Set.HasDensity, Erdos26.MultiplesOf, OfNat.ofNat, Real, Set.univ
6 instances, coercions and auxiliary constants left out.
```
def IsBehrend {ι : Type*} (A : ι → ℕ) : Prop := (MultiplesOf A).HasDensity 1
```
Mentions: Erdos26.MultiplesOf
def Erdos26.IsWeaklyBehrend A sequence of naturals $(a_i)$ is weakly Behrend with respect to $\varepsilon \in \mathbb{R}$ if at least $1 - \varepsilon$ density of all numbers are a multiple of $A$ line 69 source
Source documentation
A sequence of naturals $(a_i)$ is weakly Behrend with respect to $\varepsilon \in \mathbb{R}$ if at least $1 - \varepsilon$ density of all numbers are a multiple of $A$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: LE.le, HSub.hSub, OfNat.ofNat, Set.lowerDensity, Erdos26.MultiplesOf, Set.univ
9 instances, coercions and auxiliary constants left out.
```
def IsWeaklyBehrend {ι : Type*} (A : ι → ℕ) (ε : ℝ) : Prop := 1 - ε ≤ (MultiplesOf A).lowerDensity
…
```
Mentions: Erdos26.MultiplesOf
theorem Erdos26.isBehrend_of_contains_one line 72 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Membership.mem, Set, Set.range, OfNat.ofNat, Erdos26.IsBehrend
Proof uses: Eq.mpr, Set.HasDensity, Erdos26.MultiplesOf, Real, Set.univ, id, Eq, congrArg, Erdos26.IsBehrend.eq_1, Filter.Tendsto, Set.partialDensity, Filter.atTop, nhds, Set.HasDensity.eq_1, tendsto_atTop_of_eventually_const, GE.ge, Not, Eq.trans, Ne, HDiv.hDiv and 9 more
41 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem Erdos26.isWeaklyBehrend_of_ge_one line 80 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, LE.le, OfNat.ofNat, Erdos26.IsWeaklyBehrend
Proof uses: LE.le.trans, HSub.hSub, Set.lowerDensity, Erdos26.MultiplesOf, Set.univ, Iff.mpr, sub_nonpos, Set.lowerDensity_nonneg
25 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos26.not_isWeaklyBehrend_of_neg line 85 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, LT.lt, OfNat.ofNat, Not, Erdos26.IsWeaklyBehrend
Proof uses: Eq.mpr, HAdd.hAdd, Set.lowerDensity, Erdos26.MultiplesOf, Set.univ, id, Eq, Eq.trans, LE.le, congrArg, LT.lt.trans_le, add_lt_of_neg_right, Set.lowerDensity_le_one
39 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem Erdos26.erdos_26 Let $A\subset\mathbb{N}$ be infinite such that $\sum_{a \in A} \frac{1}{a} = \infty$ sorry line 98 source
Source documentation
Let $A\subset\mathbb{N}$ be infinite such that $\sum_{a \in A} \frac{1}{a} = \infty$. Must there exist some $k\geq 1$ such that almost all integers have a divisor of the form $a+k$ for some $a\in A$?
This was formalized in Lean by Alexeev using Aristotle.

depends on sorry what this means
Statement uses: Iff, False, Nat, StrictMono, Erdos26.IsThick, Exists, Erdos26.IsBehrend, HAdd.hAdd
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_26 : answer(False) ↔ ∀ A : ℕ → ℕ, StrictMono A → IsThick A →
    ∃ k, IsBehrend (A · + k) := by
…
```
Mentions: Erdos26.IsThick, Erdos26.IsBehrend
theorem Erdos26.erdos_26.variants.rusza sorry line 107 source
depends on sorry what this means
Statement uses: Exists, Nat, And, StrictMono, Not, Erdos26.IsThick, Erdos26.IsBehrend, HAdd.hAdd
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 Erdos26.erdos_26.variants.tenenbaum sorry line 120 source
depends on sorry what this means
Statement uses: Iff, False, Nat, StrictMono, Erdos26.IsThick, Real, GT.gt, OfNat.ofNat, Exists, Erdos26.IsWeaklyBehrend, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
13 instances, coercions and auxiliary constants left out.

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