Kurepa.lean`

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

8 declarations, 3 contain sorry, 0 declared as axioms

Kurepa's conjecture

Reference: On the left factorial function !N, by Đuro Kurepa Math. Balkanica 1, p. 147-153, 1971

Imports (1)

FormalConjecturesUtil

def Kurepa.left_factorial Left factorial of n $$!n = 0! + 1! + 2! + \dots + (n-1)!$$ line 36 source
Source documentation
Left factorial of n $$!n = 0! + 1! + 2! + \dots + (n-1)!$$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.range, Nat.factorial
1 instances, coercions and auxiliary constants left out.
```
def left_factorial (n : ℕ) := ∑ m ∈ Finset.range n, m !
…
```
theorem Kurepa.kurepa_conjecture Kurepa's conjecture For all $n$, $$!n\not\equiv 0 \mod n$$ This appears as B44 "Sums of factorials." in Unsolved Problems in Number Theory by *Richard K sorry line 50 source
Source documentation
Kurepa's conjecture
For all $n$, $$!n\not\equiv 0 \mod n$$
This appears as B44 "Sums of factorials." in Unsolved Problems in Number Theory by Richard K. Guy

L4DEL explanation
## Kurepa's conjecture For all $n$, $$!n\not\equiv 0 \mod n$$ This appears as B44 "Sums of factorials." in Unsolved Problems in Number Theory by Richard K. Guy
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Ne, HMod.hMod, Kurepa.left_factorial
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 kurepa_conjecture (n : ℕ) (h_n : 2 < n) : (!n : ℕ) % n ≠ 0 := by
…
```
theorem Kurepa.kurepa_conjecture.variants.prime This statement can be reduced to the prime case only sorry line 57 source
Source documentation
This statement can be reduced to the prime case only.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Nat.Prime, Ne, HMod.hMod, Kurepa.left_factorial
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 kurepa_conjecture.variants.prime (p : ℕ) (h_p : 2 < p) :
    p.Prime → (!p : ℕ) % p ≠ 0 := by
…
```
theorem Kurepa.kurepa_conjecture.prime_reduction Kurepa's conjecture for all integers greater than 2 is equivalent to the conjecture restricted to primes greater than 2 line 65 source
Source documentation
Kurepa's conjecture for all integers greater than 2 is equivalent to the conjecture restricted to primes greater than 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Iff, Nat, LT.lt, OfNat.ofNat, Ne, HMod.hMod, Kurepa.left_factorial, Nat.Prime
Proof uses: Iff.intro, Eq, List.prod, Nat.primeFactorsList, Membership.mem, List, Nat.prime_of_mem_primeFactorsList, Or.resolve_right, Nat.Prime.eq_two_or_odd, Dvd.dvd, Finset.sum, Finset.range, Nat.factorial, Dvd.dvd.trans, of_eq_true, Nat.dvd_of_mem_primeFactorsList, Nat.dvd_of_mod_eq_zero, LE.le, Finset, Eq.mpr and 113 more
152 instances, coercions and auxiliary constants left out.

Tactic trace: 22 steps
Open to load the checked before-and-after goal states.
```
theorem kurepa_conjecture.prime_reduction : (∀ n, 2 < n → (!n : ℕ) % n ≠ 0)
    ↔ (∀ p, 2 < p → p.Prime → (!p : ℕ) % p ≠ 0) := by
…
```
theorem Kurepa.kurepa_conjecture.variants.gcd An equivalent formulation in terms of the gcd of $n!$ and $!n$ sorry line 96 source
Source documentation
An equivalent formulation in terms of the gcd of $n!$ and $!n$.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Eq, Nat.gcd, Nat.factorial, Kurepa.left_factorial
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem kurepa_conjecture.variants.gcd (n : ℕ) : 2 < n → (n !).gcd (! n) = 2 := by
…
```
theorem Kurepa.kurepa_conjecture.gcd_reduction Kurepa's conjecture for all integers greater than 2 is equivalent to the statement that $\gcd(n!, !n) = 2$ for all integers greater than 2 line 103 source
Source documentation
Kurepa's conjecture for all integers greater than 2 is equivalent to the statement that $\gcd(n!, !n) = 2$ for all integers greater than 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Iff, Nat, LT.lt, OfNat.ofNat, Ne, HMod.hMod, Kurepa.left_factorial, Eq, Nat.gcd, Nat.factorial
Proof uses: Iff.intro, HAdd.hAdd, Iff.mpr, And, Dvd.dvd, Nat.gcd_eq_iff, And.intro, Nat.factorial_dvd_factorial, LT.lt.le, of_eq_true, Eq.trans, Finset.sum, Finset.range, True, congrArg, Finset.sum_range_succ', congrFun', zero_add, Nat.add_mod, congr and 51 more
91 instances, coercions and auxiliary constants left out.

Tactic trace: 22 steps
Open to load the checked before-and-after goal states.
```
theorem kurepa_conjecture.gcd_reduction : (∀ n, 2 < n → (!n : ℕ) % n ≠ 0)
    ↔ (∀ n, 2 < n → (n)!.gcd (!n) = 2) := by
…
```
theorem Kurepa.kurepa_conjecture.variants.first_cases Sanity check: for small values we can just compute that the conjecture is true line 133 source
Source documentation
Sanity check: for small values we can just compute that the conjecture is true

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Ne, HMod.hMod, Kurepa.left_factorial
Proof uses: dite, LE.le, of_decide_eq_true, Eq, id, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.symm, le_antisymm, Nat.ge_of_not_lt, Mathlib.Tactic.IntervalCases.of_lt_right, Nat.rawCast, Mathlib.Meta.NormNum.IsNat.to_raw_eq, Mathlib.Meta.NormNum.isNat_ofNat, Not, Nat.gt_of_not_le, Mathlib.Tactic.IntervalCases.of_lt_left
16 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem kurepa_conjecture.variants.first_cases (n : ℕ) (h_n : 2 < n) (h_n_upper : n < 50) :
    (!n : ℕ) % n ≠ 0 := by
…
```
theorem Kurepa.kurepa_conjecture.variants.gcd.first_cases Sanity check: for small values we can just compute that the conjecture is true line 141 source
Source documentation
Sanity check: for small values we can just compute that the conjecture is true.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Eq, Nat.gcd, Nat.factorial, Kurepa.left_factorial
Proof uses: dite, LE.le, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.symm, le_antisymm, Nat.ge_of_not_lt, Mathlib.Tactic.IntervalCases.of_lt_right, Nat.rawCast, Mathlib.Meta.NormNum.IsNat.to_raw_eq, Mathlib.Meta.NormNum.isNat_ofNat, Not, Nat.gt_of_not_le, Mathlib.Tactic.IntervalCases.of_lt_left
11 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem kurepa_conjecture.variants.gcd.first_cases (n : ℕ) (h_n : 2 < n) (h_n_upper : n < 50) :
    (n !).gcd (! n) = 2 := by
…
```
Mentions: Kurepa.kurepa_conjecture.variants.gcd