690.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

7 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 690

References:

erdosproblems.com/690
[Er79e] Erdős, Paul, Some unconventional problems in number theory. Astérisque (1979), 73-82.
[Ca25] S. Cambie, Resolution of Erdős' problems about unimodularity. arXiv:2501.10333 (2025).

Imports (1)

FormalConjecturesUtil

def Erdos690.kthPrimeFactorSet The set of positive integers whose k-th smallest prime factor is p: if $p_1<p_2<\cdots$ are the primes dividing $n$ then $p_k=p$ line 37 source
Source documentation
The set of positive integers whose k-th smallest prime factor is p: if $p_1<p_2<\cdots$ are the primes dividing $n$ then $p_k=p$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set
Proof uses: Set.ofPred, And, LT.lt, OfNat.ofNat, Membership.mem, Finset, Nat.primeFactors, Eq, Finset.card, Finset.filter, HSub.hSub
7 instances, coercions and auxiliary constants left out.
```
def kthPrimeFactorSet (k p : ℕ) : Set ℕ :=
…
```
def Erdos690.IsUnimodalOnPrimes A function on the primes is unimodular if it first increases in $p$ until its maximum then decreases line 42 source
Source documentation
A function on the primes is unimodular if it first increases in $p$ until its maximum then decreases.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: Exists, And, Nat.Prime, LE.le
2 instances, coercions and auxiliary constants left out.
```
def IsUnimodalOnPrimes (f : ℕ → ℝ) : Prop :=
…
```
theorem Erdos690.erdos_690 sorry line 61 source
depends on sorry what this means
Statement uses: Iff, False, Nat, GE.ge, OfNat.ofNat, Real, Nat.Prime, Set.HasDensity, Erdos690.kthPrimeFactorSet, Set.univ, Erdos690.IsUnimodalOnPrimes
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 Erdos690.erdos_690.variants.hasDensity sorry line 70 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Nat.Prime, Exists, Real, Set.HasDensity, Erdos690.kthPrimeFactorSet, Set.univ
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 Erdos690.erdos_690.variants.cambie_unimodal sorry line 77 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Real, Nat.Prime, Set.HasDensity, Erdos690.kthPrimeFactorSet, Set.univ, Erdos690.IsUnimodalOnPrimes
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 Erdos690.erdos_690.variants.cambie_not_unimodal sorry line 85 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Real, Nat.Prime, Set.HasDensity, Erdos690.kthPrimeFactorSet, Set.univ, Not, Erdos690.IsUnimodalOnPrimes
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 Erdos690.erdos_690.variants.large_k Is $d_k(p)$ unimodular for any $k\geq 21$? sorry line 92 source
Source documentation
Is $d_k(p)$ unimodular for any $k\geq 21$?

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, And, GE.ge, OfNat.ofNat, Real, Nat.Prime, Set.HasDensity, Erdos690.kthPrimeFactorSet, Set.univ, Erdos690.IsUnimodalOnPrimes
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_690.variants.large_k : answer(sorry) ↔
    ∃ k ≥ 21, ∀ d : ℕ → ℝ,
      (∀ p, p.Prime → (kthPrimeFactorSet k p).HasDensity (d p)) → IsUnimodalOnPrimes d := by
…
```
Mentions: Erdos690.kthPrimeFactorSet, Erdos690.IsUnimodalOnPrimes