69004.lean`

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

10 declarations, 4 contain sorry, 0 declared as axioms

Number of times $n^2 + s^2$ is prime for positive integers $s < n$

The sequence $a(n)$ counts the number of integers $s \in \{1, \dots, n-1\}$ such that $n^2 + s^2$ is prime: $$a(n) = \sum_{s=1}^{n-1} [\text{Prime}(n^2 + s^2)]$$

References:

A069004

Imports (1)

FormalConjecturesUtil

def OeisA69004.a Number of times $n^2 + s^2$ is prime for positive integers $s < n$ line 37 source
Source documentation
Number of times $n^2 + s^2$ is prime for positive integers $s < n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.Ico, OfNat.ofNat, ite, Nat.Prime, HAdd.hAdd, HPow.hPow
11 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA69004.a_1 Value of the sequence a at 1 line 42 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA69004.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, 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 a_1 : a 1 = 0 := by
…
```
Mentions: OeisA69004.a
theorem OeisA69004.a_2 Value of the sequence a at 2 line 47 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA69004.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, 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 a_2 : a 2 = 1 := by
…
```
Mentions: OeisA69004.a
theorem OeisA69004.a_3 Value of the sequence a at 3 line 52 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA69004.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, 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 a_3 : a 3 = 1 := by
…
```
Mentions: OeisA69004.a
theorem OeisA69004.a_4 Value of the sequence a at 4 line 57 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA69004.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, 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 a_4 : a 4 = 1 := by
…
```
Mentions: OeisA69004.a
theorem OeisA69004.a_5 Value of the sequence a at 5 line 62 source
Source documentation
Value of the sequence a at 5.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA69004.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, 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 a_5 : a 5 = 2 := by
…
```
Mentions: OeisA69004.a
theorem OeisA69004.conjecture1 Conjecture: $a(n) > 0$ for all $n > 1$ sorry line 68 source
Source documentation
Conjecture: $a(n) > 0$ for all $n > 1$.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, OeisA69004.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 conjecture1 (n : ℕ) (hn : 1 < n) : 0 < a n := by
…
```
Mentions: OeisA69004.a
theorem OeisA69004.conjecture2 Stronger conjecture: Let $\pi(n)$ be the prime counting function (A000720) sorry line 81 source
Source documentation
Stronger conjecture: Let $\pi(n)$ be the prime counting function (A000720). Then $\pi(n) \ge a(n) \ge \pi(n)/5$ for $n > 1$, with the following equalities: $\pi(2) = a(2)$, $\pi(10) = a(10)$ and $a(12) = \pi(12)/5$.
This conjunction is false: its upper bound $\pi(n) \ge a(n)$ fails at $n = 512720 = 2^4 \cdot 5 \cdot 13 \cdot 17 \cdot 29$, where $a(n) = 42666$ and $\pi(n) = 42493$. The OEIS entry tabulates $a$ only up to $n = 10^4$. The lower bound and the three equalities are unaffected; see conjecture2.variants.without_upper_bound.

depends on sorry what this means
Statement uses: Not, And, Nat, LT.lt, OfNat.ofNat, GE.ge, Nat.primeCounting, OeisA69004.a, HMul.hMul, Eq
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture2 : ¬ (
    (∀ n : ℕ, 1 < n → Nat.primeCounting n ≥ a n) ∧
    (∀ n : ℕ, 1 < n → 5 * a n ≥ Nat.primeCounting n) ∧
    Nat.primeCounting 2 = a 2 ∧
    Nat.primeCounting 10 = a 10 ∧
    5 * a 12 = Nat.primeCounting 12) := by
…
```
Mentions: OeisA69004.a
theorem OeisA69004.conjecture2.variants.upper_bound_false The upper bound of the stronger conjecture on its own sorry line 93 source
Source documentation
The upper bound of the stronger conjecture on its own. It is false: $a(512720) = 42666$ while $\pi(512720) = 42493$.

depends on sorry what this means
Statement uses: Not, Nat, LT.lt, OfNat.ofNat, GE.ge, Nat.primeCounting, OeisA69004.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture2.variants.upper_bound_false :
    ¬ ∀ n : ℕ, 1 < n → Nat.primeCounting n ≥ a n := by
…
```
Mentions: OeisA69004.a
theorem OeisA69004.conjecture2.variants.without_upper_bound The part of the stronger conjecture that survives the counterexample to its upper bound sorry line 102 source
Source documentation
The part of the stronger conjecture that survives the counterexample to its upper bound: $5 a(n) \ge \pi(n)$ for $n > 1$, together with the three claimed equalities $\pi(2) = a(2)$, $\pi(10) = a(10)$ and $a(12) = \pi(12)/5$.

depends on sorry what this means
Statement uses: And, Nat, LT.lt, OfNat.ofNat, GE.ge, HMul.hMul, OeisA69004.a, Nat.primeCounting, Eq
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture2.variants.without_upper_bound :
    (∀ n : ℕ, 1 < n → 5 * a n ≥ Nat.primeCounting n) ∧
    Nat.primeCounting 2 = a 2 ∧
    Nat.primeCounting 10 = a 10 ∧
    5 * a 12 = Nat.primeCounting 12 := by
…
```
Mentions: OeisA69004.a