385.lean`

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5 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 385

Reference: erdosproblems.com/385

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

def Erdos385.F Let $F(n) := \max\{m + p(m) \mid \textrm{$m < n$ composite}\}\}$ where $p(m)$ is the least prime divisor of $m$ line 34 source
Source documentation
Let $F(n) := \max\{m + p(m) \mid \textrm{$m < n$ composite}\}\}$ where $p(m)$ is the least prime divisor of $m$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, And, LT.lt, Nat.Composite, Eq, HAdd.hAdd, Nat.minFac
4 instances, coercions and auxiliary constants left out.
```
noncomputable def F (n : ℕ) : ℕ := sSup {m + m.minFac | (m < n) (_ : m.Composite)}
…
```
theorem Erdos385.trivial_ub Note that trivially $F(n) \leq n + \sqrt{n}$ line 38 source
Source documentation
Note that trivially $F(n) \leq n + \sqrt{n}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, Real, Erdos385.F, HAdd.hAdd, Real.sqrt
Proof uses: Nat.sqrt, csSup_le', Set.ofPred, Exists, And, LT.lt, Nat.Composite, Eq, Nat.minFac, Membership.mem, Set, OfNat.ofNat, And.left, Not, Nat.Prime, Decidable.byContradiction, HPow.hPow, Nat.minFac_sq_le_self, And.right, Iff.mpr and 28 more
79 instances, coercions and auxiliary constants left out.

Tactic trace: 22 steps
Open to load the checked before-and-after goal states.
```
theorem trivial_ub (n : ℕ) : F n ≤ n + √n := by
…
```
Mentions: Erdos385.F
theorem Erdos385.erdos_385.parts.i Let $F(n) := \max\{m + p(m) \mid \textrm{$m < n$ composite}\}\}$ where $p(m)$ is the least prime divisor of $m$ sorry line 58 source
Source documentation
Let $F(n) := \max\{m + p(m) \mid \textrm{$m < n$ composite}\}\}$ where $p(m)$ is the least prime divisor of $m$. Is it true that $F(n)>n$ for all sufficiently large $n$?

depends on sorry what this means
Statement uses: Iff, True, Filter.Eventually, Nat, LT.lt, Erdos385.F, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_385.parts.i : answer(sorry) ↔ ∀ᶠ n in atTop, n < F n := by
…
```
Mentions: Erdos385.F
theorem Erdos385.erdos_385.parts.ii Let $F(n) := \max\{m + p(m) \mid \textrm{$m < n$ composite}\}\}$ where $p(m)$ is the least prime divisor of $m$ sorry line 64 source
Source documentation
Let $F(n) := \max\{m + p(m) \mid \textrm{$m < n$ composite}\}\}$ where $p(m)$ is the least prime divisor of $m$. Does $F(n) - n \to \infty$ as $n\to\infty$?

depends on sorry what this means
Statement uses: Iff, True, Filter.Tendsto, Nat, HSub.hSub, Erdos385.F, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_385.parts.ii : answer(sorry) ↔ atTop.Tendsto (fun n ↦ F n - n) atTop := by
…
```
Mentions: Erdos385.F
theorem Erdos385.erdos_385.variants.lb A question of Erdős, Eggleton, and Selfridge, who write that in fact it is possible that this quantity is always at least $n+(1-o(1))\sqrt{n}$ sorry line 70 source
Source documentation
A question of Erdős, Eggleton, and Selfridge, who write that in fact it is possible that this quantity is always at least $n+(1-o(1))\sqrt{n}$

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, Real, Asymptotics.IsLittleO, Filter.atTop, OfNat.ofNat, LE.le, HAdd.hAdd, HMul.hMul, HSub.hSub, Real.sqrt, Erdos385.F
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
29 instances, coercions and auxiliary constants left out.
```
theorem erdos_385.variants.lb : answer(sorry) ↔ ∃ (e : ℕ → ℝ) (he : e =o[atTop] (1 : ℕ → ℝ)),
    ∀ n, n + (1 - e n) * √n ≤ F n :=
…
```
Mentions: Erdos385.F