83753.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

8 declarations, 1 contain sorry, 0 declared as axioms

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

Imports (1)

FormalConjecturesUtil

def OeisA83753.IsDecimalPalindrome A natural number $m$ is a decimal palindrome if its base-$10$ digits read the same forwards and backwards line 36 source
Source documentation
A natural number $m$ is a decimal palindrome if its base-$10$ digits read the same forwards and backwards.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Eq, List, Nat.digits, OfNat.ofNat, List.reverse
1 instances, coercions and auxiliary constants left out.
```
def IsDecimalPalindrome (m : ℕ) : Prop :=
…
```
instance OeisA83753.(instance at line 39) line 39 source
def OeisA83753.a Smallest positive palindrome with exactly $n$ divisors, or $0$ if no such number exists line 44 source
Source documentation
Smallest positive palindrome with exactly $n$ divisors, or $0$ if no such number exists.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: dite, Exists, And, LT.lt, OfNat.ofNat, OeisA83753.IsDecimalPalindrome, Eq, Finset.card, Nat.divisors, Classical.propDecidable, Nat.find, Not
6 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA83753.a_1 Value of the sequence a at 1 line 52 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA83753.a, OfNat.ofNat
Proof uses: id, dite, Exists, And, LT.lt, OeisA83753.IsDecimalPalindrome, Finset.card, Nat.divisors, Classical.propDecidable, Nat.find, Not, Eq.mpr, congrFun', congrArg, dif_pos, propext, Nat.find_eq_iff, of_decide_eq_true, Bool, Decidable.decide and 10 more
9 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 1 := by
…
```
Mentions: OeisA83753.a
theorem OeisA83753.a_2 Value of the sequence a at 2 line 62 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA83753.a, OfNat.ofNat
Proof uses: id, dite, Exists, And, LT.lt, OeisA83753.IsDecimalPalindrome, Finset.card, Nat.divisors, Classical.propDecidable, Nat.find, Not, Eq.mpr, congrFun', congrArg, dif_pos, propext, Nat.find_eq_iff, of_decide_eq_true, Bool, Decidable.decide and 10 more
9 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 2 := by
…
```
Mentions: OeisA83753.a
theorem OeisA83753.a_3 Value of the sequence a at 3 line 72 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA83753.a, OfNat.ofNat
Proof uses: id, dite, Exists, And, LT.lt, OeisA83753.IsDecimalPalindrome, Finset.card, Nat.divisors, Classical.propDecidable, Nat.find, Not, Eq.mpr, congrFun', congrArg, dif_pos, propext, Nat.find_eq_iff, of_decide_eq_true, Bool, Decidable.decide and 10 more
9 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 4 := by
…
```
Mentions: OeisA83753.a
theorem OeisA83753.a_4 Value of the sequence a at 4 line 82 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA83753.a, OfNat.ofNat
Proof uses: id, dite, Exists, And, LT.lt, OeisA83753.IsDecimalPalindrome, Finset.card, Nat.divisors, Classical.propDecidable, Nat.find, Not, Eq.mpr, congrFun', congrArg, dif_pos, propext, Nat.find_eq_iff, of_decide_eq_true, Bool, Decidable.decide and 10 more
9 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 6 := by
…
```
Mentions: OeisA83753.a
theorem OeisA83753.conjecture There are no palindromic numbers greater than 1 which are the fifth or higher power of a natural number sorry line 94 source
Source documentation
There are no palindromic numbers greater than 1 which are the fifth or higher power of a natural number.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, OeisA83753.IsDecimalPalindrome, LE.le, Not, Exists, Eq, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture (m k : ℕ) (hm : 1 < m) (hpal : IsDecimalPalindrome m) (hk : 5 ≤ k) :
    ¬ ∃ x : ℕ, m = x ^ k := by
…
```
Mentions: OeisA83753.IsDecimalPalindrome