375.lean`

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

6 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 375

References:

erdosproblems.com/375
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[RST75] Ramachandra, K. and Shorey, T. N. and Tijdeman, R., On Grimm's problem relating to factorisation of a block of consecutive integers. J. Reine Angew. Math. (1975), 109-124.

Imports (2)

def Erdos375.Erdos375Prop This is a proposition saying that for any n ≥ 1 and any k, if n + 1, ..., n + k are all composite, then there are distinct primes `p₁, line 42 source
Source documentation
This is a proposition saying that for any n ≥ 1 and any k, if n + 1, ..., n + k are all composite, then there are distinct primes p₁, ... pₖ such that pᵢ ∣ n + i for all 1 ≤ i ≤ k.

standard axioms: propext what this means
Proof uses: Nat, GE.ge, OfNat.ofNat, LT.lt, Not, Nat.Prime, HAdd.hAdd, Exists, Fin, And, Function.Injective, Dvd.dvd
7 instances, coercions and auxiliary constants left out.
```
def Erdos375Prop : Prop := ∀ n ≥ 1, ∀ k, (∀ i < k, ¬ (n + i + 1).Prime) →
…
```
theorem Erdos375.erdos_375 Is Erdos375Prop true? sorry line 47 source
Source documentation
Is Erdos375Prop true?

depends on sorry what this means
Statement uses: Iff, True, Erdos375.Erdos375Prop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_375 : answer(sorry) ↔ Erdos375Prop := by
…
```
Mentions: Erdos375.Erdos375Prop
theorem Erdos375.erdos_375.variants.bounded_gap If Erdos375Prop is true, then (n + 1).nth Prime - n.nth Prime < (n.nth Prime) ^ (1 / 2 - c) for some c > 0 sorry line 53 source
Source documentation
If Erdos375Prop is true, then (n + 1).nth Prime - n.nth Prime < (n.nth Prime) ^ (1 / 2 - c) for some c > 0.

depends on sorry what this means
Statement uses: Erdos375.Erdos375Prop, Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, LT.lt, HSub.hSub, Nat.nth, Nat.Prime, HAdd.hAdd, HPow.hPow, HDiv.hDiv, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_375.variants.bounded_gap : Erdos375Prop →
    ∃ c > 0, ∀ᶠ n in atTop, (n + 1).nth Nat.Prime - n.nth Nat.Prime
    < (n.nth Nat.Prime : ℝ) ^ (1 / (2 : ℝ) - c) := by
…
```
Mentions: Erdos375.Erdos375Prop, c
theorem Erdos375.erdos_375.variants.legendre In particular, if Erdos375Prop is true, then Legendre's conjecture is asymptotically true line 60 source
Source documentation
In particular, if Erdos375Prop is true, then Legendre's conjecture is asymptotically true.

depends on sorry what this means
Statement uses: Erdos375.Erdos375Prop, Filter.Eventually, Nat, Exists, And, Membership.mem, Set, Set.Ioo, HPow.hPow, OfNat.ofNat, HAdd.hAdd, Nat.Prime, Filter.atTop
Proof uses: LegendreConjecture.bounded_gap_legendre, Erdos375.erdos_375.variants.bounded_gap
9 instances, coercions and auxiliary constants left out.
```
theorem erdos_375.variants.legendre : Erdos375Prop →
    (∀ᶠ n in atTop, ∃ p ∈ Set.Ioo (n ^ 2) ((n + 1) ^ 2), Nat.Prime p) :=
…
```
Mentions: Erdos375.Erdos375Prop
theorem Erdos375.erdos_375.variants.le_two It is easy to see that for any n ≥ 1 and k ≤ 2, if n + 1, ..., n + k are all composite, then there are distinct primes `p₁, line 67 source
Source documentation
It is easy to see that for any n ≥ 1 and k ≤ 2, if n + 1, ..., n + k are all composite, then there are distinct primes p₁, ... pₖ such that pᵢ ∣ n + i for all 1 ≤ i ≤ k.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, LE.le, LT.lt, Not, Nat.Prime, HAdd.hAdd, Exists, Fin, And, Function.Injective, Dvd.dvd
Proof uses: dite, Exists.intro, And.intro, Eq, Decidable.byContradiction, Or, False, Classical.em, Ne, List, List.Mem, List.finRange, HEq, Membership.mem, Finset, Fintype.elems, Fintype.complete, List.cons, List.Mem.head, Fin.mk and 95 more
132 instances, coercions and auxiliary constants left out.

Tactic trace: 30 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_375.variants.le_two : ∀ n ≥ 1, ∀ k ≤ 2, (∀ i < k, ¬ (n + i + 1).Prime) →
    ∃ p : Fin k → ℕ, p.Injective ∧ ∀ i, (p i).Prime ∧ p i ∣ n + i + 1 := by
…
```
theorem Erdos375.erdos_375.variants.log There exists a constant c > 0 such that for all n, if k < c * (log n / (log (log n))) ^ 3 → (∀ i < k, ¬ (n + i + 1).Prime), then there are distinct primes `p₁, sorry line 91 source
Source documentation
There exists a constant c > 0 such that for all n, if k < c * (log n / (log (log n))) ^ 3 → (∀ i < k, ¬ (n + i + 1).Prime), then there are distinct primes p₁, ... pₖ such that pᵢ ∣ n + i for all 1 ≤ i ≤ k. This is proved in [RST75]. There is no need to only consider sufficiently large n because one can always take c small enough so that k < c * (log n / (log (log n))) ^ 3 implies that k = 0 until n is large.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Nat, LT.lt, HMul.hMul, HPow.hPow, HDiv.hDiv, Real.log, Not, Nat.Prime, HAdd.hAdd, Fin, Function.Injective, Dvd.dvd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_375.variants.log : ∃ c > 0, ∀ n k : ℕ,
    k < c * (Real.log n / (Real.log (Real.log n))) ^ 3 → (∀ i < k, ¬ (n + i + 1).Prime) →
    ∃ p : Fin k → ℕ, p.Injective ∧ ∀ i, (p i).Prime ∧ p i ∣ n + i + 1 := by
…
```
Mentions: c