91591.lean`

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

7 declarations, 1 contain sorry, 0 declared as axioms

Number of pairs of twin primes between $n^2$ and $(n+1)^2$

$a(n)$ is the number of pairs of twin primes between $n^2$ and $(n+1)^2$. This counts the number of primes $p$ such that $p$ and $p+2$ are both prime, and the entire twin prime pair $(p, p+2)$ lies strictly between $n^2$ and $(n+1)^2$. That is, $n^2 < p$ and $p + 2 < (n+1)^2$.

References:

A091591

Imports (1)

FormalConjecturesUtil

def OeisA91591.a Number of pairs of twin primes $(p, p+2)$ strictly between $n^2$ and $(n+1)^2$ line 36 source
Source documentation
Number of pairs of twin primes $(p, p+2)$ strictly between $n^2$ and $(n+1)^2$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Finset.filter, And, Nat.Prime, HAdd.hAdd, OfNat.ofNat, LT.lt, HPow.hPow, Finset.Ioo
13 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA91591.a_0 Value of the sequence a at 0 line 42 source
Source documentation
Value of the sequence a at 0.

extra axioms: _private.0.OeisA91591.a_0._native.decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA91591.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 0 := by decide +native
…
```
Mentions: OeisA91591.a
theorem OeisA91591.a_1 Value of the sequence a at 1 line 46 source
Source documentation
Value of the sequence a at 1.

extra axioms: _private.0.OeisA91591.a_1._native.decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA91591.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 0 := by decide +native
…
```
Mentions: OeisA91591.a
theorem OeisA91591.a_2 Value of the sequence a at 2 line 50 source
Source documentation
Value of the sequence a at 2.

extra axioms: _private.0.OeisA91591.a_2._native.decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA91591.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 1 := by decide +native
…
```
Mentions: OeisA91591.a
theorem OeisA91591.a_3 Value of the sequence a at 3 line 54 source
Source documentation
Value of the sequence a at 3.

extra axioms: _private.0.OeisA91591.a_3._native.decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA91591.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 1 := by decide +native
…
```
Mentions: OeisA91591.a
theorem OeisA91591.a_4 Value of the sequence a at 4 line 58 source
Source documentation
Value of the sequence a at 4.

extra axioms: _private.0.OeisA91591.a_4._native.decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA91591.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 1 := by decide +native
…
```
Mentions: OeisA91591.a
theorem OeisA91591.conjecture It is conjectured that $a(n)>0$ for all $n>122$ sorry line 65 source
Source documentation
It is conjectured that $a(n)>0$ for all $n>122$. Proving this would also prove Legendre's conjecture that there is a prime between $n^2$ and $(n+1)^2$. - T. D. Noe, Feb 28 2007

depends on sorry what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, OeisA91591.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 :
    ∀ n : ℕ, n > 122 → a n > 0 := by
…
```
Mentions: OeisA91591.a