7468.lean`

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

6 declarations, 1 contain sorry, 0 declared as axioms

Sum of next $n$ primes

The sum of the primes in the $n$-th row of the prime number triangle: $$a(n) = \sum_{i = 1 + n(n-1)/2}^{n + n(n-1)/2} p_i$$ with $a(0) = 0$.

References:

A007468

Imports (1)

FormalConjecturesUtil

def OeisA7468.a Sum of the next $n$ primes, with $a(0) = 0$ line 36 source
Source documentation
Sum of the next $n$ primes, with $a(0) = 0$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: HDiv.hDiv, HMul.hMul, HSub.hSub, OfNat.ofNat, Finset.sum, Finset.range, Nat.nth, Nat.Prime, HAdd.hAdd
10 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA7468.a_0 Value of the sequence a at 0 line 42 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA7468.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: OeisA7468.a
theorem OeisA7468.a_1 Value of the sequence a at 1 line 46 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA7468.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Finset.sum, Singleton.singleton, Finset, Nat.nth, Nat.Prime, HAdd.hAdd, congrArg, Finset.sum_singleton, add_zero, Nat.nth_prime_zero_eq_two, Eq.refl
12 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 = 2 := by
…
```
Mentions: OeisA7468.a
theorem OeisA7468.a_2 Value of the sequence a at 2 line 52 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA7468.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Finset.sum, Finset.range, Nat.nth, Nat.Prime, HAdd.hAdd, congrArg, Finset.sum_range_succ, Finset.sum_range_one, add_zero, Nat.nth_prime_one_eq_three, Eq.refl, Nat.nth_prime_two_eq_five
12 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 8 := by
…
```
Mentions: OeisA7468.a
theorem OeisA7468.a_3 Value of the sequence a at 3 line 59 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA7468.a, OfNat.ofNat
Proof uses: id, Nat.nth, Nat.Prime, Nat.nth_count, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, Finset.sum, Finset.range, HAdd.hAdd, congrArg, Finset.sum_range_succ, Finset.sum_range_one, add_zero, Nat.nth_prime_three_eq_seven, Nat.nth_prime_four_eq_eleven
13 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 31 := by
…
```
Mentions: OeisA7468.a
theorem OeisA7468.conjecture The only positive integer $n$ such that $a(n)$ is a perfect square is $n=38$ sorry line 71 source
Source documentation
The only positive integer $n$ such that $a(n)$ is a perfect square is $n=38$.
- Carlos Eduardo Olivieri, Mar 09 2015
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Iff, IsSquare, OeisA7468.a, Eq
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 conjecture (n : ℕ) (hn : 0 < n) : IsSquare (a n) ↔ n = 38 := by
…
```
Mentions: OeisA7468.a