7918.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

8 declarations, 2 contain sorry, 0 declared as axioms

Least prime $\ge n$

Least prime $\ge n$ (version 1 of the "next prime" function).

References:

A007918

Imports (1)

FormalConjecturesUtil

def OeisA7918.a Least prime $\ge n$ (version 1 of the "next prime" function) line 34 source
Source documentation
Least prime $\ge n$ (version 1 of the "next prime" function).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, And, Nat.Prime, LE.le
2 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA7918.a_0 Value of the sequence a at 0 line 39 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA7918.a, OfNat.ofNat
Proof uses: id, IsLeast, Set.ofPred, And, Nat.Prime, LE.le, And.intro, Membership.mem, Set, lowerBounds, Nat.prime_two, Nat.zero_le, Nat.Prime.two_le, And.left, IsLeast.csInf_eq
9 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 2 := by
…
```
Mentions: OeisA7918.a
theorem OeisA7918.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, OeisA7918.a, OfNat.ofNat
Proof uses: id, IsLeast, Set.ofPred, And, Nat.Prime, LE.le, And.intro, Membership.mem, Set, lowerBounds, Nat.prime_two, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.Prime.two_le, And.left, IsLeast.csInf_eq
10 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 2 := by
…
```
Mentions: OeisA7918.a
theorem OeisA7918.a_2 Value of the sequence a at 2 line 55 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA7918.a, OfNat.ofNat
Proof uses: id, IsLeast, Set.ofPred, And, Nat.Prime, LE.le, And.intro, Membership.mem, Set, lowerBounds, Nat.prime_two, le_rfl, Nat.Prime.two_le, And.left, IsLeast.csInf_eq
10 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 = 2 := by
…
```
Mentions: OeisA7918.a
theorem OeisA7918.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, OeisA7918.a, OfNat.ofNat
Proof uses: id, IsLeast, Set.ofPred, And, Nat.Prime, LE.le, And.intro, Membership.mem, Set, lowerBounds, Nat.prime_three, le_rfl, And.right, IsLeast.csInf_eq
10 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 3 := by
…
```
Mentions: OeisA7918.a
def OeisA7918.isApOfNPrimes The initial term $p_0$ and common difference $d$ form an arithmetic progression of length $n$ consisting entirely of prime numbers with $d > 0$ line 73 source
Source documentation
The initial term $p_0$ and common difference $d$ form an arithmetic progression of length $n$ consisting entirely of prime numbers with $d > 0$.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: And, GT.gt, OfNat.ofNat, LT.lt, Nat.Prime, HAdd.hAdd, HMul.hMul
6 instances, coercions and auxiliary constants left out.
```
def isApOfNPrimes (n p0 d : ℕ) : Prop :=
…
```
theorem OeisA7918.conjecture1 According to the "k-tuple" conjecture sorry line 82 source
Source documentation
According to the "k-tuple" conjecture, $a(n)$ is the initial term of the lexicographically earliest increasing arithmetic progression of $n$ primes; the corresponding common differences are given by A061558.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Eq, OeisA7918.a, InfSet.sInf, Set.ofPred, Exists, OeisA7918.isApOfNPrimes
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 conjecture1 (n : ℕ) (hn : 0 < n) :
    a n = sInf { p0 : ℕ | ∃ d : ℕ, isApOfNPrimes n p0 d } := by
…
```
Mentions: OeisA7918.a, OeisA7918.isApOfNPrimes
theorem OeisA7918.conjecture2 If $n > 1$, then $a(n) < n^{n^{1/n}}$ sorry line 91 source
Source documentation
If $n > 1$, then $a(n) < n^{n^{1/n}}$.
- Thomas Ordowski, Feb 23 2023
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Real, OeisA7918.a, HPow.hPow, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 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) (hn : 1 < n) :
    (a n : ℝ) < (n : ℝ) ^ ((n : ℝ) ^ (1 / (n : ℝ))) := by
…
```
Mentions: OeisA7918.a