961.lean`

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

8 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 961

References:

erdosproblems.com/961
[Ju74] Jutila, Matti, On numbers with a large prime factor. {II}. J. Indian Math. Soc. (N.S.) (1974), 125--130.
RaSh73 Ramachandra, K. and Shorey, T. N., On gaps between numbers with a large prime factor. Acta Arith. (1973), 99--111.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/ErdosProblems/683.lean

def Erdos961.Erdos961Prop line 35 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: GE.ge, HAdd.hAdd, OfNat.ofNat, Exists, And, Membership.mem, Set, Set.Ico, Not, Nat.smoothNumbers
6 instances, coercions and auxiliary constants left out.
theorem Erdos961.erdos_961.sylvester_schur Sylvester and Schur [Er34] proved that every set of $k$ consecutive integers greater than $k$ contains an integer divisible by a prime greater than $k$, i.e. not $(k+1)$-smooth sorry line 43 source
Source documentation
Sylvester and Schur [Er34] proved that every set of $k$ consecutive integers greater than $k$ contains an integer divisible by a prime greater than $k$, i.e. not $(k+1)$-smooth.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Erdos961.Erdos961Prop
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 erdos_961.sylvester_schur (k : ℕ) (hk : 0 < k) : Erdos961Prop k k := by
…
```
Mentions: Erdos961.Erdos961Prop
theorem Erdos961.erdos_961.variants.sylvester_schur_1_1 line 47 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Erdos961.Erdos961Prop, OfNat.ofNat, Nat
Proof uses: GE.ge, HAdd.hAdd, Exists.intro, And, Membership.mem, Set, Set.Ico, Not, Nat.smoothNumbers, And.intro, of_eq_true, Eq.trans, LE.le, LT.lt, True, congr, congrArg, Eq, eq_self, and_self and 19 more
52 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
theorem Erdos961.erdos_961.variants.well_defined There exists $n$ such that Erdos961Prop k n holds line 60 source
Source documentation
There exists $n$ such that Erdos961Prop k n holds.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Exists, Erdos961.Erdos961Prop
Proof uses: Exists.intro, Erdos961.erdos_961.sylvester_schur
4 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_961.variants.well_defined (k : ℕ) (hk : 0 < k): ∃ n, Erdos961Prop k n := by
…
```
Mentions: Erdos961.Erdos961Prop
def Erdos961.f For $k$, let $f(k)$ be the minimal $n$ such that every set of $n$ consecutive integers $>k$ contains an integer divisible by a prime $>k$, i.e. not $(k+1)$-smooth line 68 source
Source documentation
For $k$, let $f(k)$ be the minimal $n$ such that every set of $n$ consecutive integers $>k$ contains an integer divisible by a prime $>k$, i.e. not $(k+1)$-smooth.

depends on sorry what this means
Statement uses: Nat
Proof uses: dite, LT.lt, OfNat.ofNat, Nat.find, Erdos961.Erdos961Prop, Classical.propDecidable, Erdos961.erdos_961.variants.well_defined, Not
3 instances, coercions and auxiliary constants left out.
```
noncomputable def f (k : ℕ) : ℕ :=
…
```
theorem Erdos961.erdos_961 It is conjectured that $f(k) \ll (\log k)^O(1)$ sorry line 76 source
Source documentation
It is conjectured that $f(k) \ll (\log k)^O(1)$.

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, Filter.Eventually, LT.lt, Real, Erdos961.f, HPow.hPow, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_961 : answer(sorry) ↔ ∃ C : ℕ, ∀ᶠ k : ℕ in atTop, f k < log k ^ C := by
…
```
Mentions: Erdos961.f
theorem Erdos961.erdos_961.variants.erdos_upper_bound Erdos [Er55d] proved $f(k) < 3 \frac{k}{\log k}$ for sufficiently large $k$ sorry line 83 source
Source documentation
Erdos [Er55d] proved $f(k) < 3 \frac{k}{\log k}$ for sufficiently large $k$.

depends on sorry what this means
Statement uses: Filter.Eventually, Nat, LT.lt, Real, Erdos961.f, HDiv.hDiv, HMul.hMul, OfNat.ofNat, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_961.variants.erdos_upper_bound :
    ∀ᶠ k in atTop, f k < 3 * k / log k := by
…
```
Mentions: Erdos961.f
theorem Erdos961.erdos_961.variants.jutila_ramachandra_shorey_upper_bound Jutila [Ju74], and Ramachandra--Shorey [RaSh73] proved a stronger upper bound $f(k) \ll \frac{\log \log \log k}{\log \log k} \frac{k}{\log k}$ sorry line 92 source
Source documentation
Jutila [Ju74], and Ramachandra--Shorey [RaSh73] proved a stronger upper bound $f(k) \ll \frac{\log \log \log k}{\log \log k} \frac{k}{\log k}$.

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, Erdos961.f, HMul.hMul, HDiv.hDiv, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
19 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_961.variants.jutila_ramachandra_shorey_upper_bound :
    (fun k => (f k : ℝ)) =O[atTop] fun k => log (log (log k)) / log (log k) * (k / log k) := by
…
```
Mentions: Erdos961.f