38771.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

8 declarations, 2 contain sorry, 0 declared as axioms

Smallest composite $c$ such that $\textrm{primorial}(n) + c$ is prime

References:

A038771

Imports (1)

FormalConjecturesUtil

def OeisA38771.a $a(n)$ is the smallest composite number $c$ such that $\textrm{primorial}(n) + c$ is prime line 37 source
Source documentation
$a(n)$ is the smallest composite number $c$ such that $\textrm{primorial}(n) + c$ is prime.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.prod, Finset.range, Nat.nth, Nat.Prime, And, GT.gt, OfNat.ofNat, Not, InfSet.sInf, Set.ofPred, HAdd.hAdd
6 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA38771.a_0 line 43 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA38771.a, OfNat.ofNat
Proof uses: id, Finset.prod, Finset.range, Nat.nth, Nat.Prime, Eq.refl, Eq.mpr, InfSet.sInf, Set.ofPred, And, GT.gt, Not, HAdd.hAdd, congrArg, IsLeast, And.intro, Membership.mem, Set, lowerBounds, of_decide_eq_true and 18 more
30 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
theorem OeisA38771.a_1 line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA38771.a, OfNat.ofNat
Proof uses: id, Finset.prod, Finset.range, Nat.nth, Nat.Prime, Eq.mpr, congrArg, Finset.prod_range_one, Nat.nth_prime_zero_eq_two, Eq.refl, InfSet.sInf, Set.ofPred, And, GT.gt, Not, HAdd.hAdd, IsLeast, And.intro, Membership.mem, Set and 20 more
30 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
theorem OeisA38771.a_2 line 70 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA38771.a, OfNat.ofNat
Proof uses: id, Finset.prod, Finset.range, Nat.nth, Nat.Prime, Eq.mpr, HMul.hMul, congrArg, HAdd.hAdd, Finset.prod_range_succ, Finset.prod_range_one, Nat.nth_prime_zero_eq_two, Nat.nth_prime_one_eq_three, Eq.refl, InfSet.sInf, Set.ofPred, And, GT.gt, Not, IsLeast and 23 more
35 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
theorem OeisA38771.a_3 line 86 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA38771.a, OfNat.ofNat
Proof uses: id, Finset.prod, Finset.range, Nat.nth, Nat.Prime, Eq.mpr, HMul.hMul, congrArg, HAdd.hAdd, Finset.prod_range_succ, Finset.prod_range_one, Nat.nth_prime_zero_eq_two, Nat.nth_prime_one_eq_three, Nat.nth_prime_two_eq_five, Eq.refl, InfSet.sInf, Set.ofPred, And, GT.gt, Not and 24 more
35 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
theorem OeisA38771.a_n_exists $a(n) \ne 0$ for all $n$ (i.e., a suitable composite $c$ always exists) line 111 source
Source documentation
$a(n) \ne 0$ for all $n$ (i.e., a suitable composite $c$ always exists). The following more general statement follows from Dirichlet's theorem on primes in arithmetic progressions: there doesn't exist a > 0 natural number such that p - a is prime for every prime p > a.
Choose q prime such that q is coprime with a, and p > a + q prime such that q | p - a (such a p exists from Dirichlet's theorem). Then p - a is composite, a contradiction.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Ne, OeisA38771.a, OfNat.ofNat
Proof uses: id, Finset.prod, Finset.range, Nat.nth, Nat.Prime, Eq.mpr, And, GT.gt, Not, InfSet.sInf, Set.ofPred, HAdd.hAdd, Eq, congrArg, rfl, LT.lt, Finset.prod_pos, Membership.mem, Finset, Nat.Prime.pos and 41 more
45 instances, coercions and auxiliary constants left out.

Tactic trace: 25 steps
Open to load the checked before-and-after goal states.
```
theorem a_n_exists (n : ℕ) : a n ≠ 0 := by
…
```
Mentions: OeisA38771.a
theorem OeisA38771.conjecture1 Conjecture: $\liminf_{n \to \infty} \frac{a(n)}{p_{n+1}^2} = 1 <$ $\limsup_{n \to \infty} \frac{a(n)}{p_{n+1}^2} = 2$ sorry line 139 source
Source documentation
Conjecture: $\liminf_{n \to \infty} \frac{a(n)}{p_{n+1}^2} = 1 <$ $\limsup_{n \to \infty} \frac{a(n)}{p_{n+1}^2} = 2$.
- Charles R Greathouse IV and Thomas Ordowski, Apr 24 2015
depends on sorry what this means
Statement uses: Nat, Real, HPow.hPow, Nat.nth, Nat.Prime, OfNat.ofNat, HDiv.hDiv, OeisA38771.a, And, Eq, Filter.liminf, Filter.atTop, Filter.limsup
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture1 :
    let p_next_sq (n : ℕ) : ℝ := ((Nat.nth Nat.Prime n : ℝ)) ^ 2
…
```
theorem OeisA38771.conjecture2 All the terms in this sequence have exactly two prime factors sorry line 151 source
Source documentation
All the terms in this sequence have exactly two prime factors. This conjecture is true for the first 133 terms.
- Dmitry Kamenetsky, Jan 06 2019
L4DEL explanation
All the terms in this sequence have exactly two prime factors. This conjecture is true for the first 133 terms. - Dmitry Kamenetsky, Jan 06 2019
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Nat, Nat.IsSemiprime, OeisA38771.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 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) : (a n).IsSemiprime := by
…
```
Mentions: OeisA38771.a