FortuneConjecture.lean`

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9 declarations, 1 contain sorry, 0 declared as axioms

Fortune's Conjecture

A Fortunate number is the smallest integer $m > 1$ such that $p_n\\# + m$ is prime, where $p_n\\#$ denotes the primorial of the $n$-th prime — equivalently, the product of the first $n$ primes.

Fortune's Conjecture asserts that every Fortunate number is prime — equivalently, that no Fortunate number is composite.

The conjecture is named after the social anthropologist Reo Fortune, who proposed it. The first few Fortunate numbers are $3, 5, 7, 13, 23, 17, 19, 23, 37, 61, \ldots$ (OEIS A005235); all known values are prime.

References:

Imports (1)

FormalConjecturesUtil

lemma FortuneConjecture.exists_one_lt_prime_add For any natural number N there is some m > 1 with N + m prime; an immediate consequence of the infinitude of primes line 50 source
Source documentation
For any natural number N there is some m > 1 with N + m prime; an immediate consequence of the infinitude of primes.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Exists, And, LT.lt, OfNat.ofNat, Nat.Prime, HAdd.hAdd
Proof uses: LE.le, Nat.exists_infinite_primes, Exists.intro, HSub.hSub, And.intro, Decidable.byContradiction, Not, Eq, Eq.mpr, id, congrArg
17 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
lemma exists_one_lt_prime_add (N : ℕ) : ∃ m, 1 < m ∧ Nat.Prime (N + m) := by
…
```
def FortuneConjecture.fortunateNumber The $n$-th Fortunate number (0-indexed): the smallest integer $m > 1$ such that $p_{n+1}\\# + m$ is prime line 63 source
Source documentation
The $n$-th Fortunate number (0-indexed): the smallest integer $m > 1$ such that $p_{n+1}\\# + m$ is prime.
Nat.nth Nat.Prime n is the $(n+1)$-st prime (0-indexed), and primorial p is the product of all primes $\le p$; when $p$ is the $(n+1)$-st prime this equals the product of the first $n+1$ primes. Thus fortunateNumber 0 corresponds to $F_1 = 3$ in the OEIS A005235 indexing.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.find, And, LT.lt, OfNat.ofNat, Nat.Prime, HAdd.hAdd, primorial, Nat.nth
8 instances, coercions and auxiliary constants left out.
```
noncomputable def fortunateNumber (n : ℕ) : ℕ :=
…
```
lemma FortuneConjecture.fortunateNumber_spec fortunateNumber n is greater than $1$, and adding it to the primorial of the $(n+1)$-st prime yields a prime line 69 source
Source documentation
fortunateNumber n is greater than $1$, and adding it to the primorial of the $(n+1)$-st prime yields a prime.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, And, LT.lt, OfNat.ofNat, FortuneConjecture.fortunateNumber, Nat.Prime, HAdd.hAdd, primorial, Nat.nth
Proof uses: Nat.find_spec, FortuneConjecture.exists_one_lt_prime_add
11 instances, coercions and auxiliary constants left out.
```
lemma fortunateNumber_spec (n : ℕ) :
    1 < fortunateNumber n ∧
      Nat.Prime (primorial (Nat.nth Nat.Prime n) + fortunateNumber n) :=
…
```
Mentions: FortuneConjecture.fortunateNumber
lemma FortuneConjecture.fortunateNumber_le Minimality of fortunateNumber n: no smaller integer $m > 1$ makes primorial (Nat.nth Nat.Prime n) + m prime line 77 source
Source documentation
Minimality of fortunateNumber n: no smaller integer $m > 1$ makes primorial (Nat.nth Nat.Prime n) + m prime.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Nat.Prime, HAdd.hAdd, primorial, Nat.nth, LE.le, FortuneConjecture.fortunateNumber
Proof uses: Nat.find_min', And, FortuneConjecture.exists_one_lt_prime_add, And.intro
12 instances, coercions and auxiliary constants left out.
```
lemma fortunateNumber_le (n m : ℕ) (hm : 1 < m)
    (hp : Nat.Prime (primorial (Nat.nth Nat.Prime n) + m)) :
    fortunateNumber n ≤ m :=
…
```
Mentions: FortuneConjecture.fortunateNumber
theorem FortuneConjecture.fortunateNumber_zero line 85 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, FortuneConjecture.fortunateNumber, OfNat.ofNat
Proof uses: primorial, Nat.nth, Nat.Prime, Eq.mpr, id, congrArg, Nat.nth_prime_zero_eq_two, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.find, And, LT.lt, HAdd.hAdd, FortuneConjecture.exists_one_lt_prime_add, Not, propext, Nat.find_eq_iff and 23 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
theorem FortuneConjecture.fortunateNumber_one line 99 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, FortuneConjecture.fortunateNumber, OfNat.ofNat
Proof uses: primorial, Nat.nth, Nat.Prime, Eq.mpr, id, congrArg, Nat.nth_prime_one_eq_three, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.find, And, LT.lt, HAdd.hAdd, FortuneConjecture.exists_one_lt_prime_add, Not, propext, Nat.find_eq_iff and 26 more
25 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
theorem FortuneConjecture.fortunateNumber_two line 112 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, FortuneConjecture.fortunateNumber, OfNat.ofNat
Proof uses: primorial, Nat.nth, Nat.Prime, Eq.mpr, id, congrArg, Nat.nth_prime_two_eq_five, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.find, And, LT.lt, HAdd.hAdd, FortuneConjecture.exists_one_lt_prime_add, Not, propext, Nat.find_eq_iff and 26 more
25 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
theorem FortuneConjecture.fortunateNumber_three line 125 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, FortuneConjecture.fortunateNumber, OfNat.ofNat
Proof uses: primorial, Nat.nth, Nat.Prime, Eq.mpr, id, congrArg, Nat.nth_prime_three_eq_seven, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.find, And, LT.lt, HAdd.hAdd, FortuneConjecture.exists_one_lt_prime_add, Not, propext, Nat.find_eq_iff and 27 more
25 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
theorem FortuneConjecture.fortune_conjecture Fortune's Conjecture: Every Fortunate number is prime sorry line 139 source
Source documentation
Fortune's Conjecture: Every Fortunate number is prime.

depends on sorry what this means
Statement uses: Iff, True, Nat, Nat.Prime, FortuneConjecture.fortunateNumber
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem fortune_conjecture :
    answer(sorry) ↔ (∀ n : ℕ, Nat.Prime (fortunateNumber n)) := by
…
```
Mentions: FortuneConjecture.fortunateNumber