159829.lean`

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

7 declarations, 2 contain sorry, 0 declared as axioms

Smallest $m$ such that $n^3 + m^3 + 1$ is prime

$a(n)$ is the smallest natural number $m \ge 1$ such that $n^3 + m^3 + 1$ is prime.

References:

A159829

Imports (1)

FormalConjecturesUtil

def OeisA159829.a $a(n)$ is the smallest natural number $m \ge 1$ such that $n^3 + m^3 + 1$ is prime, or none if no such $m$ exists line 35 source
Source documentation
$a(n)$ is the smallest natural number $m \ge 1$ such that $n^3 + m^3 + 1$ is prime, or none if no such $m$ exists.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Option
Proof uses: ite, Exists, And, LE.le, OfNat.ofNat, Nat.Prime, HAdd.hAdd, HPow.hPow, Classical.propDecidable, Option.some, InfSet.sInf, Set.ofPred, Option.none
9 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : Option ℕ :=
…
```
theorem OeisA159829.a_1 Value of the sequence a at 1 line 43 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Option, Nat, OeisA159829.a, OfNat.ofNat, Option.some
Proof uses: IsLeast, Set.ofPred, And, LE.le, Nat.Prime, HAdd.hAdd, HPow.hPow, And.intro, Membership.mem, Set, lowerBounds, le_rfl, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, And.left, Exists and 10 more
19 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = some 1 := by
…
```
Mentions: OeisA159829.a
theorem OeisA159829.a_2 Value of the sequence a at 2 line 51 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Option, Nat, OeisA159829.a, OfNat.ofNat, Option.some
Proof uses: IsLeast, Set.ofPred, And, LE.le, Nat.Prime, HAdd.hAdd, HPow.hPow, And.intro, Membership.mem, Set, lowerBounds, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Decidable.byContradiction, Not, And.left and 23 more
31 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = some 2 := by
…
```
Mentions: OeisA159829.a
theorem OeisA159829.a_3 Value of the sequence a at 3 line 66 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Option, Nat, OeisA159829.a, OfNat.ofNat, Option.some
Proof uses: IsLeast, Set.ofPred, And, LE.le, Nat.Prime, HAdd.hAdd, HPow.hPow, And.intro, Membership.mem, Set, lowerBounds, le_rfl, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, And.left, Exists and 10 more
19 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = some 1 := by
…
```
Mentions: OeisA159829.a
theorem OeisA159829.a_4 Value of the sequence a at 4 line 74 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Option, Nat, OeisA159829.a, OfNat.ofNat, Option.some
Proof uses: IsLeast, Set.ofPred, And, LE.le, Nat.Prime, HAdd.hAdd, HPow.hPow, And.intro, Membership.mem, Set, lowerBounds, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Decidable.byContradiction, Not, And.left and 23 more
31 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = some 2 := by
…
```
Mentions: OeisA159829.a
theorem OeisA159829.conjecture1 sorry line 98 source
depends on sorry what this means
Statement uses: Iff, False, Nat, LE.le, OfNat.ofNat, Set.Infinite, Set.ofPred, Exists, And, Nat.Prime, Eq, HAdd.hAdd, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA159829.conjecture2 Conjecture 2: For any $k \ge 3$, there are infinitely many primes of the form $n^k + m^k + 1$ for $n, m \ge 1$ sorry line 110 source
Source documentation
Conjecture 2: For any $k \ge 3$, there are infinitely many primes of the form $n^k + m^k + 1$ for $n, m \ge 1$.
- Ulrich Krug, 2009
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Set.Infinite, Set.ofPred, Exists, And, Nat.Prime, Eq, HAdd.hAdd, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture2 (k : ℕ) (hk : 3 ≤ k) :
    Set.Infinite {p : ℕ | ∃ n m : ℕ, 1 ≤ n ∧ 1 ≤ m ∧ p.Prime ∧ p = n ^ k + m ^ k + 1} := by
…
```