394.lean`

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

10 declarations, 6 contain sorry, 0 declared as axioms

Erdős Problem 394

References:

erdosproblems.com/394
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[ErHa78] Erdős, P. and Hall, R. R., On some unconventional problems on the divisors of integers. J. Austral. Math. Soc. Ser. A (1978), 479--485.

Imports (1)

FormalConjecturesUtil

def Erdos394.t Let $t_k(n)$ denote the least $m$ such that $n\mid m(m+1)(m+2)\cdots (m+k-1).$ line 41 source
Source documentation
Let $t_k(n)$ denote the least $m$ such that $n\mid m(m+1)(m+2)\cdots (m+k-1).$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, And, LT.lt, OfNat.ofNat, Dvd.dvd, Finset.prod, Finset.range, HAdd.hAdd
7 instances, coercions and auxiliary constants left out.
```
noncomputable def t (k n : ℕ) : ℕ :=
…
```
theorem Erdos394.t_eq_of t k n = v when v works and nothing positive below it does line 46 source
Source documentation
t k n = v when v works and nothing positive below it does.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Dvd.dvd, Finset.prod, Finset.range, HAdd.hAdd, Membership.mem, Finset, Not, Eq, Erdos394.t
Proof uses: le_antisymm, Nat.sInf_le, Set.ofPred, And, And.intro, Decidable.byContradiction, LE.le, Set.Nonempty, Exists.intro, Set, InfSet.sInf, False, Nat.sInf_mem, Iff.mpr, Finset.mem_range, Eq.mp
27 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
theorem t_eq_of {n k v : ℕ} (hv : 0 < v)
    (hdvd : n ∣ ∏ i ∈ range k, (v + i))
    (hlt : ∀ m ∈ range v, 0 < m → ¬ (n ∣ ∏ i ∈ range k, (m + i))) :
    t k n = v := by
…
```
Mentions: Erdos394.t
theorem Erdos394.t_one The least positive multiple of n is n, so t 1 n = n line 58 source
Source documentation
The least positive multiple of n is n, so t 1 n = n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Eq, Erdos394.t
Proof uses: le_antisymm, Nat.sInf_le, Set.ofPred, And, Dvd.dvd, Finset.prod, Finset.range, HAdd.hAdd, And.intro, of_eq_true, Eq.trans, True, congrArg, Singleton.singleton, Finset, Finset.prod_singleton, add_zero, Set.Nonempty, Exists.intro, Membership.mem and 7 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem t_one {n : ℕ} (hn : 0 < n) : t 1 n = n := by
…
```
Mentions: Erdos394.t
theorem Erdos394.erdos_394.parts.i Is it true that $\sum_{n\leq x}t_2(n)\ll \frac{x^2}{(\log x)^c}$ for some $c>0$? sorry line 69 source
Source documentation
Is it true that $\sum_{n\leq x}t_2(n)\ll \frac{x^2}{(\log x)^c}$ for some $c>0$?

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Asymptotics.IsBigO, Nat, Filter.atTop, Finset.sum, Finset.Icc, Nat.floor, Erdos394.t, HDiv.hDiv, HPow.hPow, Real.log
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_394.parts.i :
    answer(True) ↔
      ∃ c > (0 : ℝ), (fun x ↦ ∑ n ∈ Icc 1 ⌊x⌋₊,
      (t 2 n : ℝ)) ≪ (fun x ↦ x ^ 2 / (Real.log x) ^ c) := by
…
```
Mentions: c, Erdos394.t
theorem Erdos394.erdos_394.parts.ii Is it true that, for $k\geq 2$, $\sum_{n\leq x}t_{k+1}(n) =o\left(\sum_{n\leq x}t_k(n)\right)?$ sorry line 79 source
Source documentation
Is it true that, for $k\geq 2$, $\sum_{n\leq x}t_{k+1}(n) =o\left(\sum_{n\leq x}t_k(n)\right)?$

depends on sorry what this means
Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Asymptotics.IsLittleO, Real, Filter.atTop, Finset.sum, Finset.Icc, Nat.floor, Erdos394.t, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_394.parts.ii :
    answer(True) ↔
      ∀ k ≥ 2, (fun (x : ℝ) ↦ ∑ n ∈ Icc 1 ⌊x⌋₊,
      (t (k + 1) n : ℝ)) =o[atTop]
      (fun (x : ℝ) ↦ ∑ n ∈ Icc 1 ⌊x⌋₊,
      (t k n : ℝ)) := by
…
```
Mentions: Erdos394.t
theorem Erdos394.erdos_394.variants.hall_bound In [ErGr80] they mention a conjecture of Erdős that the sum is $o(x^2)$ sorry line 93 source
Source documentation
In [ErGr80] they mention a conjecture of Erdős that the sum is $o(x^2)$. This was proved by Erdős and Hall [ErHa78], who proved that in fact $\sum_{n\leq x}t_2(n)\ll \frac{\log\log\log x}{\log\log x}x^2.$

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, Finset.sum, Finset.Icc, OfNat.ofNat, Nat.floor, Erdos394.t, HMul.hMul, HPow.hPow, HDiv.hDiv, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_394.variants.hall_bound :
    (fun x ↦ ∑ n ∈ Icc 1 ⌊x⌋₊, (t 2 n : ℝ)) ≪
    (fun x ↦ x ^ 2 * (Real.log (Real.log (Real.log x)) / Real.log (Real.log x))) := by
…
```
Mentions: Erdos394.t
theorem Erdos394.erdos_394.variants.hall_conjecture Erdős and Hall conjecture that the sum is $o(x^2/(\log x)^c)$ for any $c<\log 2$ sorry line 102 source
Source documentation
Erdős and Hall conjecture that the sum is $o(x^2/(\log x)^c)$ for any $c<\log 2$.

depends on sorry what this means
Statement uses: Real, LT.lt, Real.log, OfNat.ofNat, Nat, Asymptotics.IsLittleO, Filter.atTop, Finset.sum, Finset.Icc, Nat.floor, Erdos394.t, HDiv.hDiv, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
52 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_394.variants.hall_conjecture :
    ∀ c < Real.log 2, (fun x ↦ ∑ n ∈ Icc 1 ⌊x⌋₊,
    (t 2 n : ℝ)) =o[atTop]
    (fun x ↦ x ^ 2 / (Real.log x) ^ c) := by
…
```
Mentions: c, Erdos394.t
theorem Erdos394.erdos_394.variants.lower_bound Since $t_2(p)=p-1$ for prime $p$ it is trivial that $\sum_{n\leq x}t_2(n)\gg \frac{x^2}{\log x}$ sorry line 112 source
Source documentation
Since $t_2(p)=p-1$ for prime $p$ it is trivial that $\sum_{n\leq x}t_2(n)\gg \frac{x^2}{\log x}$.

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, HDiv.hDiv, HPow.hPow, OfNat.ofNat, Real.log, Finset.sum, Finset.Icc, Nat.floor, Erdos394.t
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_394.variants.lower_bound :
    (fun x ↦ x ^ 2 / Real.log x) ≪
    (fun x ↦ ∑ n ∈ Icc 1 ⌊x⌋₊, (t 2 n : ℝ)) := by
…
```
Mentions: Erdos394.t
theorem Erdos394.erdos_394.variants.factorial_gap_conjecture They ask about the behaviour of $t_{n-3}(n!)$ and also ask whether, for infinitely many $n$, $t_k(n!)< t_{k-1}(n!)-1$ for all $1\leq k < n$ sorry line 122 source
Source documentation
They ask about the behaviour of $t_{n-3}(n!)$ and also ask whether, for infinitely many $n$, $t_k(n!)< t_{k-1}(n!)-1$ for all $1\leq k < n$.

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Nat, Set.ofPred, LE.le, OfNat.ofNat, LT.lt, Erdos394.t, Nat.factorial, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_394.variants.factorial_gap_conjecture :
    answer(sorry) ↔
      Set.Infinite { n : ℕ | ∀ k, 2 ≤ k → k < n →
      t k (n !) < t (k - 1) (n !) - 1 } := by
…
```
Mentions: Erdos394.t
theorem Erdos394.erdos_394.variants.factorial_gap_10 They proved (with Selfridge) that this holds for $n=10$ line 133 source
Source documentation
They proved (with Selfridge) that this holds for $n=10$.

extra axioms: _private.0.Erdos394.erdos_394.variants.factorial_gap_10._native.native_decide.ax_1, _private.0.Erdos394.erdos_394.variants.factorial_gap_10._native.native_decide.ax_2, _private.0.Erdos394.erdos_394.variants.factorial_gap_10._native.native_decide.ax_3, _private.0.Erdos394.erdos_394.variants.factorial_gap_10._native.native_decide.ax_4, _private.0.Erdos394.erdos_394.variants.factorial_gap_10._native.native_decide.ax_5, _private.0.Erdos394.erdos_394.variants.factorial_gap_10._native.native_decide.ax_6 what this means
Statement uses: Nat, LE.le, OfNat.ofNat, LT.lt, Erdos394.t, Nat.factorial, HSub.hSub
Proof uses: Eq, Eq.mpr, id, congrArg, Erdos394.t_one, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_factorial, Bool, Bool.false, Mathlib.Meta.NormNum.isNat_eq_true, congrFun', Mathlib.Meta.NormNum.IsNat.to_eq, Erdos394.t_eq_of, of_decide_eq_true, Dvd.dvd, Finset.prod, Finset.range, HAdd.hAdd and 18 more
38 instances, coercions and auxiliary constants left out.

Tactic trace: 54 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_394.variants.factorial_gap_10 :
    ∀ (k : ℕ), 2 ≤ k → k < 10 →
    t k (10 !) <
    t (k - 1) (10 !) - 1 := by
…
```
Mentions: Erdos394.t