1146.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, 1 contain sorry, 0 declared as axioms

$a(n) = 2^(2^n)$

References:

A001146

Imports (1)

FormalConjecturesUtil

def OeisA1146.a The primary defining sequence a line 35 source
Source documentation
The primary defining sequence a. $a(n) = 2^{2^n}$.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: HPow.hPow, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ := 2 ^ (2 ^ n)
…
```
theorem OeisA1146.a_0 line 38 source
standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA1146.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 OeisA1146.a_1 line 41 source
standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA1146.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 OeisA1146.a_2 line 44 source
standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA1146.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 OeisA1146.a_3 line 47 source
standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA1146.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.
lemma OeisA1146.n_add_two_le_two_pow line 50 source
standard axioms: Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, HAdd.hAdd, HPow.hPow
Proof uses: Nat.le_induction, of_decide_eq_true, id, Eq, Bool, Decidable.decide, Bool.true, Eq.refl, Trans.trans, Decidable.byContradiction, Not, Nat.add_le_add_right, Nat.add_le_add_left, Nat.one_le_pow', Mathlib.Tactic.Ring.of_eq, HMul.hMul, Nat.rawCast, Mathlib.Tactic.Ring.Common.add_congr, Mathlib.Tactic.Ring.Common.pow_congr, Mathlib.Tactic.Ring.cast_pos and 27 more
37 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
theorem OeisA1146.divisibility_fact The forward direction: if $k = a(n)$ for $n \ge 2$, then $k^4 - 1$ divides $2^k - 1$ line 65 source
Source documentation
The forward direction: if $k = a(n)$ for $n \ge 2$, then $k^4 - 1$ divides $2^k - 1$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Dvd.dvd, HSub.hSub, HPow.hPow, OeisA1146.a
Proof uses: HAdd.hAdd, Eq, HMul.hMul, id, Trans.trans, Eq.refl, Eq.mpr, congrArg, Eq.symm, pow_add, HPow, Eq.rec, Nat.add_sub_cancel', OeisA1146.n_add_two_le_two_pow, pow_mul, mul_assoc, Nat.sub_one_dvd_pow_sub_one
28 instances, coercions and auxiliary constants left out.

Tactic trace: 18 steps
Open to load the checked before-and-after goal states.
```
theorem divisibility_fact (n : ℕ) (hn : 2 ≤ n) : (((a n)^4 - 1) : ℕ) ∣ (2^(a n) - 1 : ℕ) := by
…
```
Mentions: OeisA1146.a
theorem OeisA1146.conjecture I conjecture that { $a(n)$ ; $n>1$ } are the numbers such that $n^4-1$ divides $2^n-1$, intersection of A247219 and A247165 sorry line 91 source
Source documentation
I conjecture that { $a(n)$ ; $n>1$ } are the numbers such that $n^4-1$ divides $2^n-1$, intersection of A247219 and A247165. - M. F. Hasler, Jul 25 2015 This formalizes the reverse direction.

depends on sorry what this means
Statement uses: Nat, Dvd.dvd, HSub.hSub, HPow.hPow, OfNat.ofNat, GT.gt, Exists, And, LE.le, Eq, OeisA1146.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture :
  ∀ k : ℕ, ((k^4 - 1) : ℕ) ∣ (2^k - 1 : ℕ) → k > 1 → ∃ n : ℕ, 2 ≤ n ∧ k = a n := by
…
```
Mentions: OeisA1146.a