534.lean`

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L4DEL: Lean4 Domain Explanation Language

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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, 1 contain sorry, 0 declared as axioms

Erdős Problem 534

References:

erdosproblems.com/534
[Er73] Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.
[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
[AhKh96] Ahlswede, Rudolf and Khachatrian, Levon H., Sets of integers with pairwise common divisor and a factor from a specified set of primes. Acta Arith. (1996), 259--276.

Imports (1)

FormalConjecturesUtil

def Erdos534.IsAdmissible A set A ⊆ {1, …, N} is admissible if it contains N and any two distinct elements of A have a common factor line 40 source
Source documentation
A set A ⊆ {1, …, N} is admissible if it contains N and any two distinct elements of A have a common factor.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset
Proof uses: And, LE.le, Finset.Icc, OfNat.ofNat, Membership.mem, Set.Pairwise, LT.lt, Nat.gcd
10 instances, coercions and auxiliary constants left out.
```
def IsAdmissible (N : ℕ) (A : Finset ℕ) : Prop :=
…
```
def Erdos534.maxCard The largest size of an admissible subset of {1, …, N} line 44 source
Source documentation
The largest size of an admissible subset of {1, …, N}.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, Finset, And, Erdos534.IsAdmissible, Eq, Finset.card
1 instances, coercions and auxiliary constants left out.
```
noncomputable def maxCard (N : ℕ) : ℕ :=
…
```
def Erdos534.primePrefix The prime factors q₁ < ⋯ < qⱼ of N which are at most q line 48 source
Source documentation
The prime factors q₁ < ⋯ < qⱼ of N which are at most q.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset
Proof uses: Finset.filter, LE.le, Nat.primeFactors
2 instances, coercions and auxiliary constants left out.
```
def primePrefix (N q : ℕ) : Finset ℕ := N.primeFactors.filter (· ≤ q)
```
def Erdos534.ahlswedeKhachatrian Documented declaration: ahlswedeKhachatrian line 53 source
Source documentation
The Ahlswede–Khachatrian set associated to a prime factor q of N: those integers in [1, N] which are a multiple of at least one of 2 q₁, …, 2 qⱼ, q₁ ⋯ qⱼ, where q₁ < ⋯ < qⱼ = q are the prime factors of N which are at most q.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset
Proof uses: Finset.filter, Or, Dvd.dvd, Finset.prod, Erdos534.primePrefix, Exists, And, Membership.mem, HMul.hMul, OfNat.ofNat, Finset.Icc
12 instances, coercions and auxiliary constants left out.
```
def ahlswedeKhachatrian (N q : ℕ) : Finset ℕ :=
…
```
theorem Erdos534.erdos_534 sorry line 69 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, And, Membership.mem, Finset, Nat.primeFactors, Erdos534.IsAdmissible, Erdos534.ahlswedeKhachatrian, Eq, Erdos534.maxCard, Finset.card
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.