117027.lean`

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

16 declarations, 1 contain sorry, 0 declared as axioms

Determinants of 2 X 2 matrices of non-overlapping blocks of 4 consecutive primes

$a(n) = p_{4n-3}p_{4n} - p_{4n-2}p_{4n-1}$ where $p_k$ is the k-th prime number (1-indexed).

References:

A117027

Imports (1)

FormalConjecturesUtil

def OeisA117027.a a n is the determinant of a 2x2 matrix of non-overlapping blocks of 4 consecutive primes line 36 source
Source documentation
a n is the determinant of a 2x2 matrix of non-overlapping blocks of 4 consecutive primes.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int
Proof uses: ite, LT.lt, OfNat.ofNat, HMul.hMul, Nat.nth, Nat.Prime, HSub.hSub
12 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℤ :=
…
```
lemma OeisA117027.nth_prime_five line 51 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA117027.nth_prime_six line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA117027.nth_prime_seven line 61 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA117027.nth_prime_eight line 66 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA117027.nth_prime_nine line 71 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA117027.nth_prime_ten line 76 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA117027.nth_prime_eleven line 81 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA117027.a_0 line 86 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA117027.a, OfNat.ofNat, Nat
Proof uses: Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA117027.a_1 line 90 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA117027.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: id, Eq.mpr, HSub.hSub, HMul.hMul, Nat.nth, Nat.Prime, congrArg, Nat.nth_prime_zero_eq_two, Nat.nth_prime_one_eq_three, Nat.nth_prime_two_eq_five, Nat.nth_prime_three_eq_seven, Mathlib.Meta.NormNum.isInt_eq_true, Int.negOfNat, Mathlib.Meta.NormNum.isInt_sub, Int.ofNat, Eq.refl, Mathlib.Meta.NormNum.IsNat.to_isInt, Mathlib.Meta.NormNum.isNat_mul, Mathlib.Meta.NormNum.isNat_natCast, Mathlib.Meta.NormNum.isNat_ofNat and 1 more
28 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA117027.a_2 line 97 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA117027.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: id, Eq.mpr, HSub.hSub, HMul.hMul, Nat.nth, Nat.Prime, congrArg, Nat.nth_prime_four_eq_eleven, OeisA117027.nth_prime_seven, OeisA117027.nth_prime_five, OeisA117027.nth_prime_six, Mathlib.Meta.NormNum.isInt_eq_true, Int.negOfNat, Mathlib.Meta.NormNum.isInt_sub, Int.ofNat, Eq.refl, Mathlib.Meta.NormNum.IsNat.to_isInt, Mathlib.Meta.NormNum.isNat_mul, Mathlib.Meta.NormNum.isNat_natCast, Mathlib.Meta.NormNum.isNat_ofNat and 1 more
28 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA117027.a_3 line 103 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA117027.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: id, Eq.mpr, HSub.hSub, HMul.hMul, Nat.nth, Nat.Prime, congrArg, OeisA117027.nth_prime_eight, OeisA117027.nth_prime_eleven, OeisA117027.nth_prime_nine, OeisA117027.nth_prime_ten, Mathlib.Meta.NormNum.isInt_eq_true, Int.negOfNat, Mathlib.Meta.NormNum.isInt_sub, Int.ofNat, Eq.refl, Mathlib.Meta.NormNum.IsNat.to_isInt, Mathlib.Meta.NormNum.isNat_mul, Mathlib.Meta.NormNum.isNat_natCast, Mathlib.Meta.NormNum.isNat_ofNat and 1 more
28 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
def OeisA117027.positiveCount The count of positive terms among $a(1)$, ..., $a(N)$ line 109 source
Source documentation
The count of positive terms among $a(1)$, ..., $a(N)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: List.countP, Decidable.decide, LT.lt, Int, OfNat.ofNat, OeisA117027.a, HAdd.hAdd, List.range
6 instances, coercions and auxiliary constants left out.
```
noncomputable def positiveCount (N : ℕ) : ℕ :=
…
```
def OeisA117027.negativeCount The count of negative terms among $a(1)$, ..., $a(N)$ line 113 source
Source documentation
The count of negative terms among $a(1)$, ..., $a(N)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: List.countP, Decidable.decide, LT.lt, Int, OeisA117027.a, HAdd.hAdd, OfNat.ofNat, List.range
6 instances, coercions and auxiliary constants left out.
```
noncomputable def negativeCount (N : ℕ) : ℕ :=
…
```
def OeisA117027.ratioSeq The sequence of ratios $P(N)/Neg(N)$ as a sequence of real numbers line 117 source
Source documentation
The sequence of ratios $P(N)/Neg(N)$ as a sequence of real numbers.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: ite, Eq, OeisA117027.negativeCount, OfNat.ofNat, HDiv.hDiv, OeisA117027.positiveCount
9 instances, coercions and auxiliary constants left out.
```
noncomputable def ratioSeq (N : ℕ) : ℝ :=
…
```
theorem OeisA117027.conjecture This suggests the ratio is approaching a limit close to 0.87 sorry line 130 source
Source documentation
This suggests the ratio is approaching a limit close to 0.87.
Formalized as: The sequence of ratios $P(N)/Neg(N)$ converges to a limit L, and L is in the interval (0.8, 0.9).

depends on sorry what this means
Statement uses: Exists, Real, And, Filter.Tendsto, Nat, OeisA117027.ratioSeq, Filter.atTop, nhds, LT.lt, OfScientific.ofScientific, Bool.true
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture :
  ∃ L : ℝ, Tendsto ratioSeq atTop (nhds L) ∧ 0.8 < L ∧ L < 0.9 :=
…
```
Mentions: OeisA117027.ratioSeq