945.lean`

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

11 declarations, 1 contain sorry, 0 declared as axioms

Euclid-Mullin sequence

The Euclid-Mullin sequence starts with $a(1) = 2$. Each subsequent term is the smallest prime factor of one plus the product of all preceding terms. We extend the sequence by $a(0) = 1$ and write $b(n)$ for the product of the first $n$ official terms.

References:

A000945
[Mullin63] A. A. Mullin, "Research Problem 8 (ii)", Bull. Amer. Math. Soc. 69 (1963), p. 737.
[Wagstaff93] S. S. Wagstaff, Jr., "Computing Euclid's primes", Bull. Institute Combin. Applications 8 (1993), pp. 23-32.
[CrandallPomerance01] R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer (2001), p. 6.
A. R. Booker, "A variant of the Euclid-Mullin sequence containing every prime," arXiv:1605.08929, Journal of Integer Sequences 19 (2016), Article 16.6.4.

Imports (1)

FormalConjecturesUtil

def OeisA945.b b n is the product of the first n terms of the Euclid-Mullin sequence line 48 source
Source documentation
b n is the product of the first n terms of the Euclid-Mullin sequence.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
2 instances, coercions and auxiliary constants left out.
```
def b : ℕ → ℕ
  | 0 => 1
  | n + 1 => b n * Nat.minFac (b n + 1)

/-- The Euclid-Mullin sequence, extended by `a 0 = 1`. -/
```
Mentions: OeisA945.a
def OeisA945.a The Euclid-Mullin sequence, extended by a 0 = 1 line 53 source
Source documentation
The Euclid-Mullin sequence, extended by a 0 = 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: OeisA945.b.match_1, Unit, OfNat.ofNat, Nat.minFac, HAdd.hAdd, OeisA945.b
3 instances, coercions and auxiliary constants left out.
```
def a : ℕ → ℕ
  | 0 => 1
  | n + 1 => Nat.minFac (b n + 1)

@[category test, AMS 11]
```
Mentions: OeisA945.b, test, AMS
theorem OeisA945.a_0 line 58 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA945.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA945.a_1 line 61 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA945.a, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Nat.minFac, HAdd.hAdd, OeisA945.b, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_add, Eq.refl, Mathlib.Meta.NormNum.isNat_ofNat, Nat.Prime.minFac_eq, Mathlib.Meta.NormNum.isNat_prime_2, Bool, Bool.true, Mathlib.Meta.NormNum.isNat_minFac_2, Mathlib.Meta.NormNum.IsNat.raw_refl, eq_self, eq_true and 1 more
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA945.a_2 line 64 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA945.a, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Nat.minFac, HAdd.hAdd, OeisA945.b, HMul.hMul, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_add, Eq.refl, Mathlib.Meta.NormNum.isNat_ofNat, Nat.Prime.minFac_eq, Mathlib.Meta.NormNum.isNat_prime_2, Bool, Bool.true, Mathlib.Meta.NormNum.isNat_minFac_2, Mathlib.Meta.NormNum.IsNat.raw_refl, one_mul and 6 more
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA945.a_3 line 67 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA945.a, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Nat.minFac, HAdd.hAdd, OeisA945.b, HMul.hMul, congr, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_add, Eq.refl, Mathlib.Meta.NormNum.isNat_ofNat, Nat.Prime.minFac_eq, Mathlib.Meta.NormNum.isNat_prime_2, Bool, Bool.true, Mathlib.Meta.NormNum.isNat_minFac_2, Mathlib.Meta.NormNum.IsNat.raw_refl and 8 more
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA945.a_4 line 70 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA945.a, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Nat.minFac, HAdd.hAdd, OeisA945.b, HMul.hMul, congr, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_add, Eq.refl, Mathlib.Meta.NormNum.isNat_ofNat, Nat.Prime.minFac_eq, Mathlib.Meta.NormNum.isNat_prime_2, Bool, Bool.true, Mathlib.Meta.NormNum.isNat_minFac_2, Mathlib.Meta.NormNum.IsNat.raw_refl and 9 more
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA945.a_5 line 73 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA945.a, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Nat.minFac, HAdd.hAdd, OeisA945.b, HMul.hMul, congr, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_add, Eq.refl, Mathlib.Meta.NormNum.isNat_ofNat, Nat.Prime.minFac_eq, Mathlib.Meta.NormNum.isNat_prime_2, Bool, Bool.true, Mathlib.Meta.NormNum.isNat_minFac_2, Mathlib.Meta.NormNum.IsNat.raw_refl and 12 more
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA945.a_6 line 76 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA945.a, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Nat.minFac, HAdd.hAdd, OeisA945.b, HMul.hMul, congr, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_add, Eq.refl, Mathlib.Meta.NormNum.isNat_ofNat, Nat.Prime.minFac_eq, Mathlib.Meta.NormNum.isNat_prime_2, Bool, Bool.true, Mathlib.Meta.NormNum.isNat_minFac_2, Mathlib.Meta.NormNum.IsNat.raw_refl and 12 more
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA945.a_7 line 79 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA945.a, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Nat.minFac, HAdd.hAdd, OeisA945.b, HMul.hMul, congr, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_add, Eq.refl, Mathlib.Meta.NormNum.isNat_ofNat, Nat.Prime.minFac_eq, Mathlib.Meta.NormNum.isNat_prime_2, Bool, Bool.true, Mathlib.Meta.NormNum.isNat_minFac_2, Mathlib.Meta.NormNum.IsNat.raw_refl and 12 more
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA945.every_prime_occurs "Does the sequence sorry line 91 source
Source documentation
"Does the sequence ... contain every prime? ... [It] was considered by Guy and Nowakowski and later by Shanks, [Wagstaff93] computed the sequence through the 43rd term. The computational problem inherent in continuing the sequence further is the enormous size of the numbers that must be factored. Already the number $a(1) \cdots a(43) + 1$ has 180 digits."
- [CrandallPomerance01]
See also [Mullin63].
depends on sorry what this means
Statement uses: Iff, True, Nat, Nat.Prime, Exists, And, GE.ge, OfNat.ofNat, Eq, OeisA945.a
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 every_prime_occurs :
    answer(sorry) ↔ ∀ p, p.Prime → ∃ n ≥ 1, a n = p := by
…
```
Mentions: OeisA945.a