FormalConjectures.OEIS.«83753»

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import FormalConjecturesUtil

Smallest palindrome with exactly $n$ divisors

The sequence $a(n)$ is the smallest palindromic number with exactly $n$ divisors, or $0$ if no such number exists.

References:

A083753

namespace OeisA83753

A natural number $m$ is a decimal palindrome if its base-$10$ digits read the same forwards and backwards.

def IsDecimalPalindrome (m : ℕ) : Prop :=
  Nat.digits 10 m = (Nat.digits 10 m).reverse

instance (m : ℕ) : Decidable (IsDecimalPalindrome m) :=
  inferInstanceAs (Decidable (Nat.digits 10 m = (Nat.digits 10 m).reverse))

open Classical in

Smallest positive palindrome with exactly $n$ divisors, or $0$ if no such number exists.

noncomputable def a (n : ℕ) : ℕ :=
  if h : ∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ (Nat.divisors m).card = n then
    Nat.find h
  else
    0

Value of the sequence a at 1.


@[category test, AMS 11]
theorem a_1 : a 1 = 1 := by⊢ a 1 = 1
  classical
  dsimp [a]⊢ (if h : ∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 1 then Nat.find h else 0) = 1
  split_ifs with hposh:∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 1⊢ Nat.find h = 1negh:¬∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 1⊢ False
  ·posh:∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 1⊢ Nat.find h = 1 rw [Nat.find_eq_iffposh:∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 1⊢ (0 < 1 ∧ IsDecimalPalindrome 1 ∧ (Nat.divisors 1).card = 1) ∧
  ∀ n < 1, ¬(0 < n ∧ IsDecimalPalindrome n ∧ n.divisors.card = 1)posh:∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 1⊢ (0 < 1 ∧ IsDecimalPalindrome 1 ∧ (Nat.divisors 1).card = 1) ∧
  ∀ n < 1, ¬(0 < n ∧ IsDecimalPalindrome n ∧ n.divisors.card = 1)]posh:∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 1⊢ (0 < 1 ∧ IsDecimalPalindrome 1 ∧ (Nat.divisors 1).card = 1) ∧
  ∀ n < 1, ¬(0 < n ∧ IsDecimalPalindrome n ∧ n.divisors.card = 1)
    decide +nativeAll goals completed! 🐙
  ·negh:¬∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 1⊢ False exact (h ⟨1, byh:¬∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 1⊢ 0 < 1 ∧ IsDecimalPalindrome 1 ∧ (Nat.divisors 1).card = 1 decide +nativeAll goals completed! 🐙⟩).elim

Value of the sequence a at 2.


@[category test, AMS 11]
theorem a_2 : a 2 = 2 := by⊢ a 2 = 2
  classical
  dsimp [a]⊢ (if h : ∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 2 then Nat.find h else 0) = 2
  split_ifs with hposh:∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 2⊢ Nat.find h = 2negh:¬∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 2⊢ False
  ·posh:∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 2⊢ Nat.find h = 2 rw [Nat.find_eq_iffposh:∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 2⊢ (0 < 2 ∧ IsDecimalPalindrome 2 ∧ (Nat.divisors 2).card = 2) ∧
  ∀ n < 2, ¬(0 < n ∧ IsDecimalPalindrome n ∧ n.divisors.card = 2)posh:∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 2⊢ (0 < 2 ∧ IsDecimalPalindrome 2 ∧ (Nat.divisors 2).card = 2) ∧
  ∀ n < 2, ¬(0 < n ∧ IsDecimalPalindrome n ∧ n.divisors.card = 2)]posh:∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 2⊢ (0 < 2 ∧ IsDecimalPalindrome 2 ∧ (Nat.divisors 2).card = 2) ∧
  ∀ n < 2, ¬(0 < n ∧ IsDecimalPalindrome n ∧ n.divisors.card = 2)
    decide +nativeAll goals completed! 🐙
  ·negh:¬∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 2⊢ False exact (h ⟨2, byh:¬∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 2⊢ 0 < 2 ∧ IsDecimalPalindrome 2 ∧ (Nat.divisors 2).card = 2 decide +nativeAll goals completed! 🐙⟩).elim

Value of the sequence a at 3.


@[category test, AMS 11]
theorem a_3 : a 3 = 4 := by⊢ a 3 = 4
  classical
  dsimp [a]⊢ (if h : ∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 3 then Nat.find h else 0) = 4
  split_ifs with hposh:∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 3⊢ Nat.find h = 4negh:¬∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 3⊢ False
  ·posh:∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 3⊢ Nat.find h = 4 rw [Nat.find_eq_iffposh:∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 3⊢ (0 < 4 ∧ IsDecimalPalindrome 4 ∧ (Nat.divisors 4).card = 3) ∧
  ∀ n < 4, ¬(0 < n ∧ IsDecimalPalindrome n ∧ n.divisors.card = 3)posh:∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 3⊢ (0 < 4 ∧ IsDecimalPalindrome 4 ∧ (Nat.divisors 4).card = 3) ∧
  ∀ n < 4, ¬(0 < n ∧ IsDecimalPalindrome n ∧ n.divisors.card = 3)]posh:∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 3⊢ (0 < 4 ∧ IsDecimalPalindrome 4 ∧ (Nat.divisors 4).card = 3) ∧
  ∀ n < 4, ¬(0 < n ∧ IsDecimalPalindrome n ∧ n.divisors.card = 3)
    decide +nativeAll goals completed! 🐙
  ·negh:¬∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 3⊢ False exact (h ⟨4, byh:¬∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 3⊢ 0 < 4 ∧ IsDecimalPalindrome 4 ∧ (Nat.divisors 4).card = 3 decide +nativeAll goals completed! 🐙⟩).elim

Value of the sequence a at 4.


@[category test, AMS 11]
theorem a_4 : a 4 = 6 := by⊢ a 4 = 6
  classical
  dsimp [a]⊢ (if h : ∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 4 then Nat.find h else 0) = 6
  split_ifs with hposh:∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 4⊢ Nat.find h = 6negh:¬∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 4⊢ False
  ·posh:∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 4⊢ Nat.find h = 6 rw [Nat.find_eq_iffposh:∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 4⊢ (0 < 6 ∧ IsDecimalPalindrome 6 ∧ (Nat.divisors 6).card = 4) ∧
  ∀ n < 6, ¬(0 < n ∧ IsDecimalPalindrome n ∧ n.divisors.card = 4)posh:∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 4⊢ (0 < 6 ∧ IsDecimalPalindrome 6 ∧ (Nat.divisors 6).card = 4) ∧
  ∀ n < 6, ¬(0 < n ∧ IsDecimalPalindrome n ∧ n.divisors.card = 4)]posh:∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 4⊢ (0 < 6 ∧ IsDecimalPalindrome 6 ∧ (Nat.divisors 6).card = 4) ∧
  ∀ n < 6, ¬(0 < n ∧ IsDecimalPalindrome n ∧ n.divisors.card = 4)
    decide +nativeAll goals completed! 🐙
  ·negh:¬∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 4⊢ False exact (h ⟨6, byh:¬∃ m, 0 < m ∧ IsDecimalPalindrome m ∧ m.divisors.card = 4⊢ 0 < 6 ∧ IsDecimalPalindrome 6 ∧ (Nat.divisors 6).card = 4 decide +nativeAll goals completed! 🐙⟩).elim

There are no palindromic numbers greater than 1 which are the fifth or higher power of a natural number.

@[category research open, AMS 11]
theorem conjecture (m k : ℕ) (hm : 1 < m) (hpal : IsDecimalPalindrome m) (hk : 5 ≤ k) :
    ¬ ∃ x : ℕ, m = x ^ k := bym:ℕk:ℕhm:1 < mhpal:IsDecimalPalindrome mhk:5 ≤ k⊢ ¬∃ x, m = x ^ k
  sorryAll goals completed! 🐙

end OeisA83753