FormalConjectures/OEIS/363347.lean

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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

Realization of primes $p \equiv \pm 1 \pmod{10}$ by continued fraction denominators

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

References:

Imports (1)
  1. def OeisA363347.continuedFractionDenominator Helper function for a, which computes the denominator $R_k(n)$ of the continued fraction expression line 44 source

    Source documentation

    Helper function for a, which computes the denominator $R_k(n)$ of the continued fraction expression. For $2 \le k \le n-1$, $R_k(n)$ is defined recursively: $$R_k(n) = k - \frac{k+1}{R_{k+1}(n)}$$ The base case is $R_{n-1}(n) = (n-1) - \frac{n}{-4}$.

    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, HAdd.hAdd, HDiv.hDiv

    18 instances, coercions and auxiliary constants left out.

    def continuedFractionDenominator (n k : ℕ) : ℚ :=
…
  2. def OeisA363347.a Denominator of the continued fraction line 68 source

    Source documentation

    Denominator of the continued fraction $$\frac{1}{2 - \frac{3}{3 - \frac{4}{4 - \frac{5}{\dots - \frac{n-1}{(n-1) - \frac{n}{-4}}}}}} $$ The value of the continued fraction is $C_n = 1/R_2(n)$. If $R_2(n) = N/D$ in reduced form, $C_n = D/N$. The sequence $a(n)$ is the denominator of the final fraction, which is $\vert N \vert$.

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

    Statement uses: Nat

    Proof uses: ite, LE.le, OfNat.ofNat, Rat, OeisA363347.continuedFractionDenominator, Int.natAbs, Rat.num

    3 instances, coercions and auxiliary constants left out.

    def a (n : ℕ) : ℕ :=
…
  3. lemma OeisA363347.a_3 line 76 source

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

    Statement uses: Eq, Nat, OeisA363347.a, OfNat.ofNat

    Proof uses: id, Eq.mpr, ite, LE.le, Rat, OeisA363347.continuedFractionDenominator, Int.natAbs, Rat.num, And, HSub.hSub, HAdd.hAdd, HDiv.hDiv, congrArg, OeisA363347.continuedFractionDenominator.eq_1, of_eq_true, Eq.trans, True, congrFun', ite_congr, False and 39 more

    54 instances, coercions and auxiliary constants left out.

    Tactic trace: 4 steps

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

  4. lemma OeisA363347.a_4 line 79 source

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

    Statement uses: Eq, Nat, OeisA363347.a, OfNat.ofNat

    Proof uses: id, Eq.mpr, ite, LE.le, Rat, OeisA363347.continuedFractionDenominator, Int.natAbs, Rat.num, And, HSub.hSub, HAdd.hAdd, HDiv.hDiv, congrArg, OeisA363347.continuedFractionDenominator.eq_1, congrFun', ite_congr, False, eq_false, Mathlib.Meta.NormNum.isNat_le_false, Mathlib.Meta.NormNum.isNat_ofNat and 43 more

    58 instances, coercions and auxiliary constants left out.

    Tactic trace: 4 steps

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

  5. lemma OeisA363347.a_5 line 82 source

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

    Statement uses: Eq, Nat, OeisA363347.a, OfNat.ofNat

    Proof uses: id, Eq.mpr, ite, LE.le, Rat, OeisA363347.continuedFractionDenominator, Int.natAbs, Rat.num, And, HSub.hSub, HAdd.hAdd, HDiv.hDiv, congrArg, OeisA363347.continuedFractionDenominator.eq_1, congrFun', ite_congr, False, eq_false, Mathlib.Meta.NormNum.isNat_le_false, Mathlib.Meta.NormNum.isNat_ofNat and 42 more

    59 instances, coercions and auxiliary constants left out.

    Tactic trace: 4 steps

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

  6. lemma OeisA363347.a_6 line 85 source

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

    Statement uses: Eq, Nat, OeisA363347.a, OfNat.ofNat

    Proof uses: id, Eq.mpr, ite, LE.le, Rat, OeisA363347.continuedFractionDenominator, Int.natAbs, Rat.num, And, HSub.hSub, HAdd.hAdd, HDiv.hDiv, congrArg, OeisA363347.continuedFractionDenominator.eq_1, congrFun', ite_congr, False, eq_false, Mathlib.Meta.NormNum.isNat_le_false, Mathlib.Meta.NormNum.isNat_ofNat and 42 more

    59 instances, coercions and auxiliary constants left out.

    Tactic trace: 4 steps

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

  7. lemma OeisA363347.a_7 line 88 source

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

    Statement uses: Eq, Nat, OeisA363347.a, OfNat.ofNat

    Proof uses: id, Eq.mpr, ite, LE.le, Rat, OeisA363347.continuedFractionDenominator, Int.natAbs, Rat.num, And, HSub.hSub, HAdd.hAdd, HDiv.hDiv, congrArg, OeisA363347.continuedFractionDenominator.eq_1, congrFun', ite_congr, False, eq_false, Mathlib.Meta.NormNum.isNat_le_false, Mathlib.Meta.NormNum.isNat_ofNat and 42 more

    59 instances, coercions and auxiliary constants left out.

    Tactic trace: 4 steps

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

  8. theorem OeisA363347.exists_a_eq_prime sorry line 99 source

    depends on sorry what this means

    Statement uses: Nat, And, Nat.Prime, Or, Nat.ModEq, OfNat.ofNat, Exists, Eq, OeisA363347.a

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

    2 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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