105020.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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

9 declarations, 1 contain sorry, 0 declared as axioms

Array read by upward antidiagonals

Array read by upward antidiagonals: row $n$ ($n \ge 0$) contains the numbers $m^2 - n^2$, $m \ge n+1$.

References:

A105020

Imports (1)

FormalConjecturesUtil

def OeisA105020.triangularNumber Triangular number $T_c = c(c+1)/2$ line 37 source
Source documentation
Triangular number $T_c = c(c+1)/2$.

no axioms what this means
Statement uses: Nat
Proof uses: Nat.choose, HAdd.hAdd, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.
```
def triangularNumber (c : ℕ) : ℕ := Nat.choose (c + 1) 2
```
Mentions: c
def OeisA105020.antidiagonalIndex The index $c$ of the antidiagonal containing $n$, i.e., the largest $c$ such that $T_c \le n$ line 43 source
Source documentation
The index $c$ of the antidiagonal containing $n$, i.e., the largest $c$ such that $T_c \le n$. This is derived from $c = \lfloor (\sqrt{8n+1} - 1) / 2 \rfloor$.

no axioms what this means
Statement uses: Nat
Proof uses: HDiv.hDiv, HSub.hSub, Nat.sqrt, HAdd.hAdd, HMul.hMul, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.
```
def antidiagonalIndex (n : ℕ) : ℕ :=
…
```
def OeisA105020.a The primary defining sequence a line 53 source
Source documentation
The primary defining sequence a. Array read by upward antidiagonals: row $n$ ($n \ge 0$) contains the numbers $m^2 - n^2$, $m \ge n+1$. The array entry $T(i, j)$ is $(i+j+1)^2 - i^2$. The sequence $a(n)$ is generated by reading $T(i, j)$ for $i+j=c$, with $i$ decreasing from $c$ to $0$.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: OeisA105020.antidiagonalIndex, HSub.hSub, OeisA105020.triangularNumber, HAdd.hAdd, OfNat.ofNat, HPow.hPow
9 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA105020.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file line 62 source
Source documentation
Term theorems verifying the first few values of the sequence against the official OEIS b-file

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA105020.a, OfNat.ofNat
Proof uses: id, OeisA105020.antidiagonalIndex, HSub.hSub, OeisA105020.triangularNumber, HAdd.hAdd, HPow.hPow, HDiv.hDiv, Nat.sqrt, HMul.hMul, of_eq_true, Nat.choose, Eq.trans, True, congrFun', congrArg, congr, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_pow, Eq.refl, Mathlib.Meta.NormNum.isNat_add and 17 more
35 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 1 := by
…
```
Mentions: OeisA105020.a
theorem OeisA105020.a_1 line 67 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA105020.a, OfNat.ofNat
Proof uses: id, OeisA105020.antidiagonalIndex, HSub.hSub, OeisA105020.triangularNumber, HAdd.hAdd, HPow.hPow, HDiv.hDiv, Nat.sqrt, HMul.hMul, of_eq_true, Nat.choose, Eq.trans, True, congrFun', congrArg, congr, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_pow, Eq.refl, Mathlib.Meta.NormNum.isNat_add and 17 more
34 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA105020.a_2 line 72 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA105020.a, OfNat.ofNat
Proof uses: id, OeisA105020.antidiagonalIndex, HSub.hSub, OeisA105020.triangularNumber, HAdd.hAdd, HPow.hPow, HDiv.hDiv, Nat.sqrt, HMul.hMul, of_eq_true, Nat.choose, Eq.trans, True, congrFun', congrArg, congr, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_pow, Eq.refl, Mathlib.Meta.NormNum.isNat_add and 19 more
35 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA105020.a_3 line 77 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA105020.a, OfNat.ofNat
Proof uses: id, OeisA105020.antidiagonalIndex, HSub.hSub, OeisA105020.triangularNumber, HAdd.hAdd, HPow.hPow, HDiv.hDiv, Nat.sqrt, HMul.hMul, of_eq_true, Nat.choose, Eq.trans, True, congrFun', congrArg, congr, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_pow, Eq.refl, Mathlib.Meta.NormNum.isNat_add and 16 more
30 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA105020.a_4 line 82 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA105020.a, OfNat.ofNat
Proof uses: id, OeisA105020.antidiagonalIndex, HSub.hSub, OeisA105020.triangularNumber, HAdd.hAdd, HPow.hPow, HDiv.hDiv, Nat.sqrt, HMul.hMul, of_eq_true, Nat.choose, Eq.trans, True, congrFun', congrArg, congr, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_pow, Eq.refl, Mathlib.Meta.NormNum.isNat_add and 15 more
25 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA105020.conjecture A "Goldbach Conjecture" for this sequence: when there are sorry line 94 source
Source documentation
A "Goldbach Conjecture" for this sequence: when there are $n$ terms between consecutive odd integers $2n+1$ and $2n+3$ for $n > 0$, at least one will be the product of 2 primes (not necessarily distinct). Example: $n=3$ for consecutive odd integers $a(7) = 7$ and $a(11) = 9$ and of the 3 sequence entries $a(8) = 12$, $a(9) = 15$ and $a(10) = 16$ between them, one is the product of 2 primes $a(9) = 15=3*5$. - Michael Hiebl, Jul 15 2007

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Eq, OeisA105020.a, HAdd.hAdd, HMul.hMul, Exists, And, LT.lt, Nat.IsSemiprime
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture :
  ∀ (n i j : ℕ), 1 ≤ n →
    a i = 2 * n + 1 →
    a j = 2 * n + 3 →
    j = i + n + 1 →
    ∃ (k : ℕ),
      i < k ∧
      k < j ∧
      (a k).IsSemiprime := by
…
```
Mentions: OeisA105020.a