182126.lean`

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

11 declarations, 3 contain sorry, 0 declared as axioms

Product of two consecutive primes modulo the next prime

The sequence is defined by $$a(n) = \mathrm{prime}(n) \cdot \mathrm{prime}(n+1) \bmod \mathrm{prime}(n+2),$$ where $\mathrm{prime}(k)$ is the $k$-th prime number ($\mathrm{prime}(1)=2$).

References:

A182126

Imports (1)

FormalConjecturesUtil

def OeisA182126.prime $\mathrm{prime}(k)$ is the $k$-th prime number ($\mathrm{prime}(1) = 2$) line 35 source
Source documentation
$\mathrm{prime}(k)$ is the $k$-th prime number ($\mathrm{prime}(1) = 2$).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.nth, Nat.Prime, HSub.hSub, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.
```
noncomputable def prime (k : ℕ) : ℕ := Nat.nth Nat.Prime (k - 1)
```
def OeisA182126.a $a(n) = \mathrm{prime}(n) \cdot \mathrm{prime}(n+1) \bmod \mathrm{prime}(n+2)$ line 38 source
Source documentation
$a(n) = \mathrm{prime}(n) \cdot \mathrm{prime}(n+1) \bmod \mathrm{prime}(n+2)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, Eq, OfNat.ofNat, HMod.hMod, HMul.hMul, OeisA182126.prime, HAdd.hAdd
8 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA182126.a_1 line 43 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA182126.a, OfNat.ofNat
Proof uses: Nat.nth, Nat.Prime, Nat.nth_count, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, HMod.hMod, HMul.hMul, congrArg
7 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
theorem OeisA182126.a_2 line 51 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA182126.a, OfNat.ofNat
Proof uses: Nat.nth, Nat.Prime, Nat.nth_count, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, HMod.hMod, HMul.hMul, congrArg
7 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
theorem OeisA182126.a_3 line 59 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA182126.a, OfNat.ofNat
Proof uses: Nat.nth, Nat.Prime, Nat.nth_count, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, HMod.hMod, HMul.hMul, congrArg
7 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
theorem OeisA182126.a_4 line 67 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA182126.a, OfNat.ofNat
Proof uses: Nat.nth, Nat.Prime, Nat.nth_count, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, HMod.hMod, HMul.hMul, congrArg
7 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
def OeisA182126.countA Count of occurrences of value $v$ among $a(1), \dots, a(x)$ line 75 source
Source documentation
Count of occurrences of value $v$ among $a(1), \dots, a(x)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Finset.filter, And, LE.le, OfNat.ofNat, Eq, OeisA182126.a, Finset.range, HAdd.hAdd
7 instances, coercions and auxiliary constants left out.
```
noncomputable def countA (x v : ℕ) : ℕ :=
…
```
def OeisA182126.IsMostFrequent $v_0$ is a most frequent value among $a(1), \dots, a(x)$ line 79 source
Source documentation
$v_0$ is a most frequent value among $a(1), \dots, a(x)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: LE.le, OeisA182126.countA
1 instances, coercions and auxiliary constants left out.
```
def IsMostFrequent (x v₀ : ℕ) : Prop :=
…
```
theorem OeisA182126.conjecture1 Conjecture: For $x > 10^9$, the most frequent value in $a(n)$, $n=1\dots x$, has form $120k$ sorry line 86 source
Source documentation
Conjecture: For $x > 10^9$, the most frequent value in $a(n)$, $n=1\dots x$, has form $120k$.

depends on sorry what this means
Statement uses: Nat, LT.lt, HPow.hPow, OfNat.ofNat, OeisA182126.IsMostFrequent, Dvd.dvd
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 conjecture1 (x : ℕ) (hx : 10 ^ 9 < x) (v₀ : ℕ) (hv : IsMostFrequent x v₀) :
    120 ∣ v₀ := by
…
```
Mentions: OeisA182126.IsMostFrequent
theorem OeisA182126.conjecture2 Let $b = \mathrm{prime}(n+2) - \mathrm{prime}(n)$ and $c = \mathrm{prime}(n+2) - \mathrm{prime}(n+1)$ sorry line 96 source
Source documentation
Let $b = \mathrm{prime}(n+2) - \mathrm{prime}(n)$ and $c = \mathrm{prime}(n+2) - \mathrm{prime}(n+1)$. Conjecture: for $n > 61$, $a(n) = b \cdot c$.
- Charles R Greathouse IV, May 11 2012
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, HSub.hSub, OeisA182126.prime, HAdd.hAdd, Eq, OeisA182126.a, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 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 : 61 < n) :
    let b := prime (n + 2) - prime n
…
```
Mentions: OeisA182126.prime
theorem OeisA182126.conjecture3 Are 2, 7, 11, 13, 29 the only primes in this sequence? sorry line 107 source
Source documentation
Are 2, 7, 11, 13, 29 the only primes in this sequence?
- Hugo Pfoertner, Sep 22 2025
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Iff, Nat.Prime, OeisA182126.a, Membership.mem, List, List.cons, List.nil
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 conjecture3 (n : ℕ) (hn : 0 < n) :
    (a n).Prime ↔ a n ∈ ([2, 7, 11, 13, 29] : List ℕ) := by
…
```
Mentions: OeisA182126.a