885.lean`

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

7 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 885

References:

erdosproblems.com/885
[ErRo97] Erdős, P. and Rosenfeld, M., The factor-difference set of integers. (1997)
[Ji99] Jiménez-Urroz, J., A note on a conjecture of Erdős and {R}osenfeld. (1999)
[Br19] Bremner, A., On a problem of Erdős related to common factor differences. (2019)

Imports (1)

FormalConjecturesUtil

def Erdos885.factorDifferenceSet For integer $n \geq 1$ we define the factor difference set of $n$ by $D(n) = \{|a-b| : n=ab\}$ line 40 source
Source documentation
For integer $n \geq 1$ we define the factor difference set of $n$ by $D(n) = \{|a-b| : n=ab\}$.

standard axioms: propext what this means
Statement uses: Nat, Set
Proof uses: Set.ofPred, Exists, And, Eq, HMul.hMul, Int, abs, HSub.hSub
8 instances, coercions and auxiliary constants left out.
```
def factorDifferenceSet (n : ℕ) : Set ℕ :=
…
```
lemma Erdos885.mem_factorDifferenceSet_of_eq line 44 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, HMul.hMul, HAdd.hAdd, Membership.mem, Set, Erdos885.factorDifferenceSet
Proof uses: Exists.intro, Exists, And, Int, abs, HSub.hSub, And.intro, Eq.mpr, id, congrArg, Nat.cast_add, Neg.neg, Mathlib.Tactic.Ring.of_eq, HPow.hPow, Nat.rawCast, Int.rawCast, inferInstance, Ring, Int.negOfNat, OfNat.ofNat and 26 more
56 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma Erdos885.factorDifferenceSet_finite line 50 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Set.Finite, Erdos885.factorDifferenceSet
Proof uses: Set.Finite.subset, Set.Iic, Set.finite_Iic, Membership.mem, Set, Exists, And, Eq, HMul.hMul, Int, abs, HSub.hSub, Nat.pos_of_ne_zero, False, False.elim, Eq.mp, Eq.trans, congrArg, MulZeroClass.zero_mul, Eq.symm and 72 more
103 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
theorem Erdos885.erdos_885 Is it true that, for every $k \geq 1$, there exist integers $N_1 < \dots < N_k$ such that $|\cap_i D(N_i)| \geq k$? sorry line 65 source
Source documentation
Is it true that, for every $k \geq 1$, there exist integers $N_1 < \dots < N_k$ such that $|\cap_i D(N_i)| \geq k$?

depends on sorry what this means
Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Exists, Finset, And, Membership.mem, LE.le, Eq, Finset.card, Set.ncard, Set.iInter, Erdos885.factorDifferenceSet
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_885 : answer(sorry) ↔ ∀ k ≥ 1,
    ∃ Ns : Finset ℕ,
      (∀ n ∈ Ns, 1 ≤ n) ∧
      Ns.card = k ∧
      (⋂ n ∈ Ns, factorDifferenceSet n).ncard ≥ k := by
…
```
Mentions: Erdos885.factorDifferenceSet
theorem Erdos885.erdos_885.variants.k_eq_2 Erdős and Rosenfeld [ErRo97] proved this is true for $k=2$ sorry line 76 source
Source documentation
Erdős and Rosenfeld [ErRo97] proved this is true for $k=2$.

depends on sorry what this means
Statement uses: Exists, Finset, Nat, And, Membership.mem, LE.le, OfNat.ofNat, Eq, Finset.card, GE.ge, Set.ncard, Set.iInter, Erdos885.factorDifferenceSet
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_885.variants.k_eq_2 :
    ∃ Ns : Finset ℕ,
      (∀ n ∈ Ns, 1 ≤ n) ∧
      Ns.card = 2 ∧
      (⋂ n ∈ Ns, factorDifferenceSet n).ncard ≥ 2 := by
…
```
Mentions: Erdos885.factorDifferenceSet
theorem Erdos885.erdos_885.variants.k_eq_3 Jiménez-Urroz [Ji99] proved this for $k=3$ sorry line 87 source
Source documentation
Jiménez-Urroz [Ji99] proved this for $k=3$.

depends on sorry what this means
Statement uses: Exists, Finset, Nat, And, Membership.mem, LE.le, OfNat.ofNat, Eq, Finset.card, GE.ge, Set.ncard, Set.iInter, Erdos885.factorDifferenceSet
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_885.variants.k_eq_3 :
    ∃ Ns : Finset ℕ,
      (∀ n ∈ Ns, 1 ≤ n) ∧
      Ns.card = 3 ∧
      (⋂ n ∈ Ns, factorDifferenceSet n).ncard ≥ 3 := by
…
```
Mentions: Erdos885.factorDifferenceSet
theorem Erdos885.erdos_885.variants.k_eq_4 Bremner [Br19] proved this for $k=4$ line 98 source
Source documentation
Bremner [Br19] proved this for $k=4$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Exists, Finset, Nat, And, Membership.mem, LE.le, OfNat.ofNat, Eq, Finset.card, GE.ge, Set.ncard, Set.iInter, Erdos885.factorDifferenceSet
Proof uses: Exists.intro, Insert.insert, Singleton.singleton, And.intro, of_eq_true, Eq.trans, Or, True, forall_congr, implies_congr, congrArg, Eq.refl, congr, and_self, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Set and 30 more
53 instances, coercions and auxiliary constants left out.

Tactic trace: 32 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_885.variants.k_eq_4 :
    ∃ Ns : Finset ℕ,
      (∀ n ∈ Ns, 1 ≤ n) ∧
      Ns.card = 4 ∧
      (⋂ n ∈ Ns, factorDifferenceSet n).ncard ≥ 4 := by
…
```
Mentions: Erdos885.factorDifferenceSet