185150.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 odd primes between $n^2$ and $(n+1)^2$ with $(n/p) = 1$

$a(n)$ is the number of odd primes $p$ between $n^2$ and $(n+1)^2$ such that the Legendre symbol $\left(\frac{n}{p}\right) = 1$.

References:

A185150
Z.-W. Sun, "Conjectures involving primes and quadratic forms", arXiv preprint arXiv:1211.1588 [math.NT], 2012.

Imports (1)

FormalConjecturesUtil

def OeisA185150.a Number of odd primes $p \in (n^2, (n+1)^2)$ with $(n/p) = 1$ line 37 source
Source documentation
Number of odd primes $p \in (n^2, (n+1)^2)$ with $(n/p) = 1$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.Ioo, HPow.hPow, OfNat.ofNat, HAdd.hAdd, ite, And, Nat.Prime, Ne, Eq, Int, jacobiSym
18 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA185150.a_0 Value of the sequence a at 0 line 43 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA185150.a, OfNat.ofNat
Proof uses: Eq.refl
2 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 rfl
…
```
Mentions: OeisA185150.a
theorem OeisA185150.a_1 Value of the sequence a at 1 line 47 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA185150.a, OfNat.ofNat
Proof uses: id, of_eq_true, HAdd.hAdd, ite, And, Nat.Prime, Ne, Int, jacobiSym, Eq.trans, True, congrFun', congrArg, congr, ite_congr, False, eq_true, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl and 28 more
19 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 1 := by
…
```
Mentions: OeisA185150.a
theorem OeisA185150.a_2 Value of the sequence a at 2 line 54 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA185150.a, OfNat.ofNat
Proof uses: id, of_eq_true, HAdd.hAdd, ite, And, Nat.Prime, Ne, Int, jacobiSym, Eq.trans, True, congrFun', congrArg, congr, ite_congr, False, eq_true, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl and 31 more
20 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 1 := by
…
```
Mentions: OeisA185150.a
theorem OeisA185150.a_3 Value of the sequence a at 3 line 63 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA185150.a, OfNat.ofNat
2 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 = 2 := by decide +kernel
…
```
Mentions: OeisA185150.a
theorem OeisA185150.a_4 Value of the sequence a at 4 line 67 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA185150.a, OfNat.ofNat
2 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 = 3 := by decide +kernel
…
```
Mentions: OeisA185150.a
theorem OeisA185150.conjecture Conjecture: $a(n) > 0$ for all $n > 0$ sorry line 74 source
Source documentation
Conjecture: $a(n) > 0$ for all $n > 0$.
- Zhi-Wei Sun, Dec 29 2012
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, OeisA185150.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) (hn : 0 < n) : 0 < a n := by
…
```
Mentions: OeisA185150.a