FormalConjectures/OEIS/55487.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

9 declarations, 1 contain sorry, 0 declared as axioms

Least $m$ such that $\phi(m) = n!$

The smallest positive integer $m$ whose Euler totient equals $n!$.

References:

Imports (1)
  1. def OeisA55487.a Least $m$ such that $\phi(m) = n!$ line 33 source

    Source documentation

    Least $m$ such that $\phi(m) = n!$.

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

    Statement uses: Nat

    Proof uses: InfSet.sInf, Set.ofPred, And, LT.lt, OfNat.ofNat, Eq, Nat.totient, Nat.factorial

    3 instances, coercions and auxiliary constants left out.

    noncomputable def a (n : ℕ) : ℕ :=
…
  2. theorem OeisA55487.a_1 line 37 source

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

    Statement uses: Eq, Nat, OeisA55487.a, OfNat.ofNat

    Proof uses: IsLeast.csInf_eq, Set.ofPred, And, LT.lt, Nat.totient, Nat.factorial, of_decide_eq_true, IsLeast, id, Bool, Decidable.decide, Bool.true, Eq.refl

    15 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

  3. theorem OeisA55487.a_2 line 40 source

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

    Statement uses: Eq, Nat, OeisA55487.a, OfNat.ofNat

    Proof uses: IsLeast.csInf_eq, Set.ofPred, And, LT.lt, Nat.totient, Nat.factorial, of_decide_eq_true, IsLeast, id, Bool, Decidable.decide, Bool.true, Eq.refl

    15 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

  4. theorem OeisA55487.a_3 line 43 source

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

    Statement uses: Eq, Nat, OeisA55487.a, OfNat.ofNat

    Proof uses: IsLeast.csInf_eq, Set.ofPred, And, LT.lt, Nat.totient, Nat.factorial, of_decide_eq_true, IsLeast, id, Bool, Decidable.decide, Bool.true, Eq.refl

    15 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

  5. theorem OeisA55487.a_4 line 46 source

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

    Statement uses: Eq, Nat, OeisA55487.a, OfNat.ofNat

    Proof uses: IsLeast.csInf_eq, Set.ofPred, And, LT.lt, Nat.totient, Nat.factorial, of_decide_eq_true, IsLeast, id, Bool, Decidable.decide, Bool.true, Eq.refl

    15 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

  6. def OeisA55487.isFactorialPrime Factorial primes: $n$ such that $n! + 1$ is prime (A002981) line 49 source

    Source documentation

    Factorial primes: $n$ such that $n! + 1$ is prime (A002981).

    standard axioms: propext what this means

    Statement uses: Nat

    Proof uses: Nat.Prime, HAdd.hAdd, Nat.factorial, OfNat.ofNat

    3 instances, coercions and auxiliary constants left out.

    def isFactorialPrime (n : ℕ) : Prop :=
…
  7. def OeisA55487.p The least prime $p > \sqrt{n!}$ such that $(p - 1) \mid n!$ and $q = \frac{n!}{p - 1} + 1$ is prime line 55 source

    Source documentation

    The least prime $p > \sqrt{n!}$ such that $(p - 1) \mid n!$ and $q = \frac{n!}{p - 1} + 1$ is prime.

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

    Statement uses: Nat

    Proof uses: InfSet.sInf, Set.ofPred, And, Nat.Prime, LT.lt, Nat.sqrt, Nat.factorial, Dvd.dvd, HSub.hSub, OfNat.ofNat, HAdd.hAdd, HDiv.hDiv

    10 instances, coercions and auxiliary constants left out.

    noncomputable def p (n : ℕ) : ℕ :=
…
  8. def OeisA55487.q The complementary prime factor $q = \frac{n!}{p - 1} + 1$ line 60 source

    Source documentation

    The complementary prime factor $q = \frac{n!}{p - 1} + 1$.

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

    Statement uses: Nat

    Proof uses: HAdd.hAdd, HDiv.hDiv, Nat.factorial, HSub.hSub, OeisA55487.p, OfNat.ofNat

    7 instances, coercions and auxiliary constants left out.

    noncomputable def q (n : ℕ) : ℕ :=
…
  9. theorem OeisA55487.conjecture Conjecture: unless sorry line 72 source

    Source documentation

    Conjecture: unless $n! + 1$ is prime (i.e., $n \in \text{A002981}$), $a(n) = p q$ where $p$ is the least prime $> \sqrt{n!}$ such that $(p - 1) \mid n!$ and $q = \frac{n!}{p - 1} + 1$ is prime.

    • M. F. Hasler, Oct 04 2009

    The conclusion $(p(n)).\text{Prime}$ asserts that such a prime $p$ exists: if no prime satisfies the search conditions then p n is sInf ∅ = 0, which is not prime.

    depends on sorry what this means

    Statement uses: Nat, LE.le, OfNat.ofNat, Not, OeisA55487.isFactorialPrime, And, Nat.Prime, OeisA55487.p, Eq, OeisA55487.a, HMul.hMul, OeisA55487.q

    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.

    theorem conjecture (n : ℕ) (hn : 1 ≤ n) (h_not_prime : ¬ isFactorialPrime n) :
    (p n).Prime ∧ a n = p n * q n := by
…

    Mentions: OeisA55487.isFactorialPrime, OeisA55487.p, OeisA55487.a, OeisA55487.q