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import FormalConjecturesUtilLeast $m$ such that $\phi(m) = n!$
The smallest positive integer $m$ whose Euler totient equals $n!$.
References:
namespace OeisA55487Least $m$ such that $\phi(m) = n!$.
noncomputable def a (n : ℕ) : ℕ :=
sInf {m : ℕ | 0 < m ∧ m.totient = n.factorial}@[category test, AMS 11]
theorem a_1 : a 1 = 1 := IsLeast.csInf_eq <| ⊢ IsLeast {m | 0 < m ∧ m.totient = Nat.factorial 1} 1 All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 3 := IsLeast.csInf_eq <| ⊢ IsLeast {m | 0 < m ∧ m.totient = Nat.factorial 2} 3 All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 7 := IsLeast.csInf_eq <| ⊢ IsLeast {m | 0 < m ∧ m.totient = Nat.factorial 3} 7 All goals completed! 🐙@[category test, AMS 11]
theorem a_4 : a 4 = 35 := IsLeast.csInf_eq <| ⊢ IsLeast {m | 0 < m ∧ m.totient = Nat.factorial 4} 35 All goals completed! 🐙Factorial primes: $n$ such that $n! + 1$ is prime (A002981).
def isFactorialPrime (n : ℕ) : Prop :=
Nat.Prime (n.factorial + 1)The least prime $p > \sqrt{n!}$ such that $(p - 1) \mid n!$ and $q = \frac{n!}{p - 1} + 1$ is prime.
noncomputable def p (n : ℕ) : ℕ :=
sInf {p : ℕ | Nat.Prime p ∧ Nat.sqrt n.factorial < p ∧ (p - 1) ∣ n.factorial ∧
Nat.Prime (n.factorial / (p - 1) + 1)}The complementary prime factor $q = \frac{n!}{p - 1} + 1$.
noncomputable def q (n : ℕ) : ℕ :=
n.factorial / (p n - 1) + 1Conjecture: 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
We assume $a(n) \ne 0$ and $(p(n)).\text{Prime}$ to ensure the sInf searches are non-empty
and do not collapse to $0 = 0$.
@[category research open, AMS 11]
theorem conjecture (n : ℕ) (hn : 1 ≤ n) (h_not_prime : ¬ isFactorialPrime n)
(ha : a n ≠ 0) (hp : (p n).Prime) :
a n = p n * q n := n:ℕhn:1 ≤ nh_not_prime:¬isFactorialPrime nha:a n ≠ 0hp:Nat.Prime (p n)⊢ a n = p n * q n
All goals completed! 🐙end OeisA55487