119591.lean`

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

9 declarations, 1 contain sorry, 0 declared as axioms

Least $k \ge 1$ such that $2 \cdot n^k - 1$ is prime

$a(n) = \min \{k \ge 1 \mid \text{Prime}(2 \cdot n^k - 1)\}$ for $n \ge 2$.

References:

A119591

Imports (1)

FormalConjecturesUtil

def OeisA119591.a Least $k \ge 1$ such that $2 \cdot n^k - 1$ is prime, or $0$ if no such $k$ exists line 33 source
Source documentation
Least $k \ge 1$ such that $2 \cdot n^k - 1$ is prime, or $0$ if no such $k$ exists.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, And, LT.lt, OfNat.ofNat, Nat.Prime, HSub.hSub, HMul.hMul, HPow.hPow
11 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA119591.a_0 Value of the sequence a at 0 line 38 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA119591.a, OfNat.ofNat
Proof uses: id, Set, Set.ofPred, And, LT.lt, Nat.Prime, HSub.hSub, HMul.hMul, HPow.hPow, EmptyCollection.emptyCollection, Set.ext, Eq.mpr, Iff, Membership.mem, Not, Eq.trans, False, congrArg, iff_false, zero_pow and 7 more
25 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 0 := by
…
```
Mentions: OeisA119591.a
theorem OeisA119591.a_1 Value of the sequence a at 1 line 50 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA119591.a, OfNat.ofNat
Proof uses: id, Set, Set.ofPred, And, LT.lt, Nat.Prime, HSub.hSub, HMul.hMul, HPow.hPow, EmptyCollection.emptyCollection, Set.ext, Eq.mpr, Iff, Membership.mem, Not, Eq.trans, False, congrArg, iff_false, one_pow and 6 more
22 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 0 := by
…
```
Mentions: OeisA119591.a
theorem OeisA119591.a_2 Value of the sequence a at 2 line 62 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA119591.a, OfNat.ofNat
Proof uses: IsLeast, Set.ofPred, And, LT.lt, Nat.Prime, HSub.hSub, HMul.hMul, HPow.hPow, And.intro, Membership.mem, Set, lowerBounds, id, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_natSub and 10 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 1 := by
…
```
Mentions: OeisA119591.a
theorem OeisA119591.a_3 Value of the sequence a at 3 line 73 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA119591.a, OfNat.ofNat
Proof uses: IsLeast, Set.ofPred, And, LT.lt, Nat.Prime, HSub.hSub, HMul.hMul, HPow.hPow, And.intro, Membership.mem, Set, lowerBounds, id, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_natSub and 10 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 1 := by
…
```
Mentions: OeisA119591.a
theorem OeisA119591.a_4 Value of the sequence a at 4 line 84 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA119591.a, OfNat.ofNat
Proof uses: IsLeast, Set.ofPred, And, LT.lt, Nat.Prime, HSub.hSub, HMul.hMul, HPow.hPow, And.intro, Membership.mem, Set, lowerBounds, id, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_natSub and 10 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 1 := by
…
```
Mentions: OeisA119591.a
theorem OeisA119591.a_5 Value of the sequence a at 5 line 95 source
Source documentation
Value of the sequence a at 5.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA119591.a, OfNat.ofNat
Proof uses: IsLeast, Set.ofPred, And, LT.lt, Nat.Prime, HSub.hSub, HMul.hMul, HPow.hPow, And.intro, Membership.mem, Set, lowerBounds, id, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_natSub and 42 more
32 instances, coercions and auxiliary constants left out.

Tactic trace: 26 steps
Open to load the checked before-and-after goal states.
```
theorem a_5 : a 5 = 4 := by
…
```
Mentions: OeisA119591.a
theorem OeisA119591.a_6 Value of the sequence a at 6 line 118 source
Source documentation
Value of the sequence a at 6.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA119591.a, OfNat.ofNat
Proof uses: IsLeast, Set.ofPred, And, LT.lt, Nat.Prime, HSub.hSub, HMul.hMul, HPow.hPow, And.intro, Membership.mem, Set, lowerBounds, id, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_natSub and 11 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
theorem a_6 : a 6 = 1 := by
…
```
Mentions: OeisA119591.a
theorem OeisA119591.conjecture Is $a(n)$ defined for all $n \ge 2$? sorry line 131 source
Source documentation
Is $a(n)$ defined for all $n \ge 2$? That is, does there exist $k > 0$ such that $2 \cdot n^k - 1$ is prime?

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, And, GT.gt, Nat.Prime, HSub.hSub, HMul.hMul, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture (n : ℕ) (hn : 2 ≤ n) : ∃ k > 0, (2 * n ^ k - 1).Prime := by
…
```