167604.lean`

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

8 declarations, 1 contain sorry, 0 declared as axioms

Chua's Euclidean prime sequence

For a product $n$ of the preceding terms, Chua's sequence chooses the least prime dividing $d + n / d$ for some divisor $d$ of $n$. The open question is whether every prime occurs.

References:

OEIS A167604
Andrew R. Booker, A variant of the Euclid--Mullin sequence containing every prime

Imports (1)

FormalConjecturesUtil

def OeisA167604.next The least prime occurring among the sums $d + n / d$ for divisors $d$ of $n$ line 41 source
Source documentation
The least prime occurring among the sums $d + n / d$ for divisors $d$ of $n$.
Taking the least prime factor of the product gives the same minimum and makes the definition directly executable.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.minFac, Finset.prod, Nat.divisors, HAdd.hAdd, HDiv.hDiv
5 instances, coercions and auxiliary constants left out.
```
def next (n : ℕ) : ℕ :=
…
```
def OeisA167604.product Product of the first $n$ terms of Chua's sequence line 45 source
Source documentation
Product of the first $n$ terms of Chua's sequence.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
2 instances, coercions and auxiliary constants left out.
```
def product : ℕ → ℕ
  | 0 => 1
  | n + 1 => product n * next (product n)

/-- Chua's sequence, extended by $a(0) = 1$. -/
```
Mentions: OeisA167604.next, OeisA167604.a
def OeisA167604.a Chua's sequence, extended by $a(0) = 1$ line 50 source
Source documentation
Chua's sequence, extended by $a(0) = 1$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: OeisA167604.product.match_1, Unit, OfNat.ofNat, OeisA167604.next, OeisA167604.product
1 instances, coercions and auxiliary constants left out.
```
def a : ℕ → ℕ
  | 0 => 1
  | n + 1 => next (product n)

@[category test, AMS 11]
```
Mentions: OeisA167604.next, OeisA167604.product, test, AMS
theorem OeisA167604.a_1 line 55 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167604.a, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Nat.minFac, Finset.prod, Nat.divisors, OeisA167604.product, HAdd.hAdd, HDiv.hDiv, Singleton.singleton, Finset, Nat.divisors_one, Finset.prod_singleton, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_add, Eq.refl, Mathlib.Meta.NormNum.isNat_ofNat, Mathlib.Meta.NormNum.isNat_natDiv and 9 more
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA167604.a_2 line 59 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167604.a, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Nat.minFac, Finset.prod, Nat.divisors, OeisA167604.product, HAdd.hAdd, HDiv.hDiv, HMul.hMul, Finset.prod_congr, Finset, OeisA167604.next, Singleton.singleton, Nat.divisors_one, Finset.prod_singleton, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_add and 18 more
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA167604.a_3 line 63 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167604.a, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Nat.minFac, Finset.prod, Nat.divisors, OeisA167604.product, HAdd.hAdd, HDiv.hDiv, HMul.hMul, Insert.insert, Finset, Singleton.singleton, Finset.prod_congr, OeisA167604.next, congr, Nat.divisors_one, Finset.prod_singleton and 36 more
34 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA167604.a_4 line 68 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167604.a, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Nat.minFac, Finset.prod, Nat.divisors, OeisA167604.product, HAdd.hAdd, HDiv.hDiv, HMul.hMul, Insert.insert, Finset, Singleton.singleton, Finset.prod_congr, OeisA167604.next, congr, Nat.divisors_one, Finset.prod_singleton and 38 more
34 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA167604.conjecture Does Chua's sequence contain every prime? sorry line 75 source
Source documentation
Does Chua's sequence contain every prime?

depends on sorry what this means
Statement uses: Iff, True, Nat, Nat.Prime, Exists, And, GE.ge, OfNat.ofNat, Eq, OeisA167604.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture :
    answer(sorry) ↔ ∀ p : ℕ, p.Prime → ∃ n ≥ 1, a n = p := by
…
```
Mentions: OeisA167604.a