FormalConjectures/OEIS/372761.lean

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Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL
Lean4 Domain Explanation Language

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

Unique realization of odd primes $p \notin \{3, 5\}$ by continued fraction values

$a(n)$ is the denominator of the finite continued fraction $$\frac{1}{2 - \frac{3}{3 - \frac{4}{4 - \frac{5}{\dots - \frac{n-1}{(n-1) - \frac{n}{n+4}}}}}}$$

References:

Imports (1)
  1. def OeisA372761.continuedFractionDenominator Recursive helper computing the continued fraction denominator line 44 source

    Source documentation

    Recursive helper computing the continued fraction denominator $R_k(n)$ for $2 \le k \le n-1$, where $R_k(n) = k - \frac{k+1}{R_{k+1}(n)}$ with base case $R_{n-1}(n) = (n-1) - \frac{n}{n+4}$.

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Nat, Rat

    Proof uses: WellFounded.Nat.fix, HSub.hSub, InvImage, LT.lt, dite, LE.le, OfNat.ofNat, Not, And, Eq, HDiv.hDiv, HAdd.hAdd

    19 instances, coercions and auxiliary constants left out.

    def continuedFractionDenominator (n k : ℕ) : ℚ :=
…
  2. def OeisA372761.a Denominator of the continued fraction $$ \frac{1}{2 - \frac{3}{3 - \frac{4}{4 - \frac{5}{\dots - \frac{n-1}{(n-1) - \frac{n}{n+4}}}}}} $$ line 60 source

    Source documentation

    Denominator of the continued fraction $$ \frac{1}{2 - \frac{3}{3 - \frac{4}{4 - \frac{5}{\dots - \frac{n-1}{(n-1) - \frac{n}{n+4}}}}}} $$

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Nat

    Proof uses: ite, LT.lt, OfNat.ofNat, Rat.den, HDiv.hDiv, Rat, OeisA372761.continuedFractionDenominator

    6 instances, coercions and auxiliary constants left out.

    def a (n : ℕ) : ℕ :=
…
  3. lemma OeisA372761.a_3 line 66 source

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Eq, Nat, OeisA372761.a, OfNat.ofNat

    Proof uses: id, Eq.mpr, ite, LT.lt, Rat.den, HDiv.hDiv, Rat, OeisA372761.continuedFractionDenominator, LE.le, And, HSub.hSub, HAdd.hAdd, congrArg, OeisA372761.continuedFractionDenominator.eq_1, of_eq_true, Eq.trans, True, congrFun', ite_congr, False and 39 more

    61 instances, coercions and auxiliary constants left out.

    Tactic trace: 4 steps

    Open to load the checked before-and-after goal states.

  4. lemma OeisA372761.a_4 line 70 source

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Eq, Nat, OeisA372761.a, OfNat.ofNat

    Proof uses: id, Eq.mpr, ite, LT.lt, Rat.den, HDiv.hDiv, Rat, OeisA372761.continuedFractionDenominator, LE.le, And, HSub.hSub, HAdd.hAdd, congrArg, OeisA372761.continuedFractionDenominator.eq_1, Not, Int.natAbs, Rat.num, congrFun', ite_congr, False and 53 more

    75 instances, coercions and auxiliary constants left out.

    Tactic trace: 4 steps

    Open to load the checked before-and-after goal states.

  5. lemma OeisA372761.a_5 line 74 source

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Eq, Nat, OeisA372761.a, OfNat.ofNat

    Proof uses: id, Eq.mpr, ite, LT.lt, Rat.den, HDiv.hDiv, Rat, OeisA372761.continuedFractionDenominator, LE.le, And, HSub.hSub, HAdd.hAdd, congrArg, OeisA372761.continuedFractionDenominator.eq_1, Not, Int.natAbs, Rat.num, congrFun', ite_congr, False and 53 more

    77 instances, coercions and auxiliary constants left out.

    Tactic trace: 4 steps

    Open to load the checked before-and-after goal states.

  6. lemma OeisA372761.a_6 line 78 source

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Eq, Nat, OeisA372761.a, OfNat.ofNat

    Proof uses: id, Eq.mpr, ite, LT.lt, Rat.den, HDiv.hDiv, Rat, OeisA372761.continuedFractionDenominator, LE.le, And, HSub.hSub, HAdd.hAdd, congrArg, OeisA372761.continuedFractionDenominator.eq_1, Not, Int.natAbs, Rat.num, congrFun', ite_congr, False and 53 more

    77 instances, coercions and auxiliary constants left out.

    Tactic trace: 4 steps

    Open to load the checked before-and-after goal states.

  7. lemma OeisA372761.a_7 line 82 source

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Eq, Nat, OeisA372761.a, OfNat.ofNat

    Proof uses: id, Eq.mpr, ite, LT.lt, Rat.den, HDiv.hDiv, Rat, OeisA372761.continuedFractionDenominator, LE.le, And, HSub.hSub, HAdd.hAdd, congrArg, OeisA372761.continuedFractionDenominator.eq_1, Not, Int.natAbs, Rat.num, congrFun', ite_congr, False and 53 more

    77 instances, coercions and auxiliary constants left out.

    Tactic trace: 4 steps

    Open to load the checked before-and-after goal states.

  8. theorem OeisA372761.exists_unique_a_eq_prime sorry line 94 source

    depends on sorry what this means

    Statement uses: Nat, And, Nat.Prime, Eq, HMod.hMod, OfNat.ofNat, Ne, ExistsUnique, GE.ge, OeisA372761.a

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous

    8 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.