34693.lean`

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

11 declarations, 4 contain sorry, 0 declared as axioms

Smallest number $k$ such that $kn + 1$ is prime

References:

A34693

Imports (1)

FormalConjecturesUtil

def OeisA34693.a Smallest number $k$ such that $kn + 1$ is prime line 34 source
Source documentation
Smallest number $k$ such that $kn + 1$ is prime.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.nth, Nat.Prime, HAdd.hAdd, HMul.hMul, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ := Nat.nth (fun k ↦ (k * n + 1).Prime) 0
…
```
theorem OeisA34693.a_0 line 37 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA34693.a, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Nat.nth, Nat.Prime, HAdd.hAdd, HMul.hMul, False, funext, MulZeroClass.mul_zero, zero_add, Nat.nth_false, eq_self
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA34693.a_1 line 40 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA34693.a, OfNat.ofNat
Proof uses: Eq.mpr, Nat.nth, Nat.Prime, HAdd.hAdd, Nat.count, id, Eq.rec, Eq.refl, Eq.trans, congrArg, Eq.symm, Nat.nth_count, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_ofNat, Bool, Bool.true, Mathlib.Meta.NormNum.isNat_minFac_2, Mathlib.Meta.NormNum.IsNat.raw_refl, congrFun' and 3 more
12 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA34693.a_2 line 45 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA34693.a, OfNat.ofNat
Proof uses: Eq.mpr, Nat.nth, Nat.Prime, HAdd.hAdd, HMul.hMul, Nat.count, id, Eq.rec, Eq.refl, Eq.trans, congrArg, Eq.symm, Nat.nth_count, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_mul, Mathlib.Meta.NormNum.isNat_ofNat, Bool, Bool.true, Mathlib.Meta.NormNum.isNat_minFac_4 and 3 more
14 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA34693.a_3 line 50 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA34693.a, OfNat.ofNat
Proof uses: Eq.mpr, Nat.nth, Nat.Prime, HAdd.hAdd, HMul.hMul, Nat.count, id, Eq.rec, Eq.refl, Eq.trans, congrArg, Eq.symm, Nat.nth_count, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_mul, Mathlib.Meta.NormNum.isNat_ofNat, Bool, Bool.true, Mathlib.Meta.NormNum.isNat_minFac_4 and 3 more
14 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA34693.a_7 line 55 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA34693.a, OfNat.ofNat
Proof uses: Eq.mpr, Nat.nth, Nat.Prime, HAdd.hAdd, HMul.hMul, Nat.count, id, Eq.rec, Eq.refl, Eq.trans, congrArg, Eq.symm, Nat.nth_count, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_mul, Mathlib.Meta.NormNum.isNat_ofNat, Bool, Bool.true, Mathlib.Meta.NormNum.isNat_minFac_4 and 4 more
14 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA34693.exists_k Conjecture: for every $n > 1$ there exists a number $k < n$ such that $nk + 1$ is a prime sorry line 61 source
Source documentation
Conjecture: for every $n > 1$ there exists a number $k < n$ such that $nk + 1$ is a prime.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Exists, And, Nat.Prime, HAdd.hAdd, HMul.hMul
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 exists_k {n : ℕ} (hn : 1 < n) : ∃ k < n, (n * k + 1).Prime := by
…
```
theorem OeisA34693.exists_k_stronger A stronger conjecture: for every n there exists a number $k < 1 + n^{0.75}$ such that $nk + 1$ is a prime sorry line 67 source
Source documentation
A stronger conjecture: for every n there exists a number $k < 1 + n^{0.75}$ such that $nk + 1$ is a prime.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Exists, And, Real, HAdd.hAdd, HPow.hPow, Real.nthRoot, Nat.Prime, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem exists_k_stronger {n : ℕ} (hn : 0 < n) : ∃ k : ℕ,
    k < 1 + (Real.nthRoot 4 n) ^ 3 ∧ (n * k + 1).Prime := by
…
```
Mentions: Real.nthRoot
theorem OeisA34693.exists_k_best_possible The expression $1 + n^{0.74}$ does not work as an upper bound line 73 source
Source documentation
The expression $1 + n^{0.74}$ does not work as an upper bound.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Exists, Nat, And, GT.gt, OfNat.ofNat, LT.lt, Real, HAdd.hAdd, HPow.hPow, Real.nthRoot, Not, Nat.Prime, HMul.hMul
Proof uses: Exists.intro, And.intro, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false, Eq, Inv.inv, id, Eq.mpr, ite, Even, OrderHom, SignType, SignType.sign, abs, congrArg, if_pos, of_decide_eq_true and 100 more
135 instances, coercions and auxiliary constants left out.

Tactic trace: 27 steps
Open to load the checked before-and-after goal states.
```
theorem exists_k_best_possible : ∃ n > (0 : ℕ), ∀ (k : ℕ),
    k < 1 + (Real.nthRoot 100 n) ^ 74 → ¬(n * k + 1).Prime := by
…
```
Mentions: Real.nthRoot
theorem OeisA34693.a_isBigO Conjecture: $a(n) = O(\log(n)\log(\log(n)))$ sorry line 96 source
Source documentation
Conjecture: $a(n) = O(\log(n)\log(\log(n)))$.

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, OeisA34693.a, HMul.hMul, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_isBigO : (fun n ↦ (a n : ℝ)) =O[atTop] (fun n ↦ Real.log n * Real.log (Real.log n)) := by
…
```
Mentions: OeisA34693.a
theorem OeisA34693.a_unbounded Counter-conjecture to a_isBigO: $a(n) / (\log n \log \log n)$ is unbounded sorry line 101 source
Source documentation
Counter-conjecture to a_isBigO: $a(n) / (\log n \log \log n)$ is unbounded.

depends on sorry what this means
Statement uses: Not, BddAbove, Real, Set.range, Nat, HDiv.hDiv, OeisA34693.a, HMul.hMul, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_unbounded : ¬BddAbove (Set.range fun n ↦ a n / (Real.log n * Real.log (Real.log n))) := by
…
```
Mentions: OeisA34693.a