1057.lean`

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

8 declarations, 7 contain sorry, 0 declared as axioms

Erdős Problem 1057

References:

erdosproblems.com/1057
[AGP94] Alford, W. R. and Granville, Andrew and Pomerance, Carl, There are infinitely many Carmichael numbers. Ann. of Math. (2) (1994), 703--722.
[Er56c] Erdős, P., On pseudoprimes and Carmichael numbers. Publ. Math. Debrecen (1956), 201--206.
[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.
[Ha08] Harman, Glyn, Watt's mean value theorem and Carmichael numbers. Int. J. Number Theory (2008), 241--248.
[Li22] J. D. Lichtman, Primes in arithmetic progressions to large moduli and shifted primes without large prime factors. arXiv:2211.09641 (2022).
[Po89] Pomerance, Carl, Two methods in elementary analytic number theory. (1989), 135--161.

Imports (1)

FormalConjecturesUtil

def Erdos1057.carmichaelCounting Let $C(x)$ count the number of Carmichael numbers in the interval $[1,x]$ line 48 source
Source documentation
Let $C(x)$ count the number of Carmichael numbers in the interval $[1,x]$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: Set.ncard, Nat, Set.ofPred, And, IsCarmichael, LE.le
3 instances, coercions and auxiliary constants left out.
```
noncomputable def carmichaelCounting (x : ℝ) : ℝ :=
…
```
theorem Erdos1057.erdos_1057 Is it true that $C(x)=x^{1-o(1)}$? sorry line 57 source
Source documentation
Is it true that $C(x)=x^{1-o(1)}$?
This is discussed in problem A13 of Guy's collection [Gu04].

depends on sorry what this means
Statement uses: Iff, True, Filter.Tendsto, Real, HDiv.hDiv, Real.log, Erdos1057.carmichaelCounting, Filter.atTop, nhds, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
19 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1057 :
    answer(sorry) ↔ Tendsto (fun x ↦ Real.log (carmichaelCounting x) / Real.log x) atTop (𝓝 1) := by
…
```
Mentions: Erdos1057.carmichaelCounting
theorem Erdos1057.erdos_1057.variants.upper_bound Erdős [Er56c] proved $C(x) < x \exp\left(-c \frac{\log x\log\log\log x}{\log\log x}\right)$ for some constant $c>0$ sorry line 66 source
Source documentation
Erdős [Er56c] proved $C(x) < x \exp\left(-c \frac{\log x\log\log\log x}{\log\log x}\right)$ for some constant $c>0$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, LT.lt, Erdos1057.carmichaelCounting, HMul.hMul, Real.exp, HDiv.hDiv, Neg.neg, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
21 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1057.variants.upper_bound :
    ∃ c > 0, ∀ᶠ x in atTop,
      carmichaelCounting x < x * Real.exp (-c * (Real.log x * Real.log (Real.log (Real.log x))) /
        Real.log (Real.log x)) := by
…
```
Mentions: c, Erdos1057.carmichaelCounting
theorem Erdos1057.erdos_1057.variants.pomerance Pomerance [Po89] gave a heuristic suggesting that this is the true order of growth, and in fact $C(x)= x \exp\left(-(1+o(1))\frac{\log x\log\log\log x}{\log\log x}\right)$ sorry line 77 source
Source documentation
Pomerance [Po89] gave a heuristic suggesting that this is the true order of growth, and in fact $C(x)= x \exp\left(-(1+o(1))\frac{\log x\log\log\log x}{\log\log x}\right)$.

depends on sorry what this means
Statement uses: Iff, True, Filter.Tendsto, Real, HDiv.hDiv, Neg.neg, HMul.hMul, Real.log, Erdos1057.carmichaelCounting, Filter.atTop, nhds, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
25 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1057.variants.pomerance :
    answer(sorry) ↔ Tendsto (fun x ↦
      -(Real.log (carmichaelCounting x / x) * Real.log (Real.log x)) /
      (Real.log x * Real.log (Real.log (Real.log x)))) atTop (𝓝 1) := by
…
```
Mentions: Erdos1057.carmichaelCounting
theorem Erdos1057.erdos_1057.variants.agp_infinite Alford, Granville, and Pomerance [AGP94] proved that $C(x)\to \infty$ sorry line 87 source
Source documentation
Alford, Granville, and Pomerance [AGP94] proved that $C(x)\to \infty$.

depends on sorry what this means
Statement uses: Filter.Tendsto, Real, Erdos1057.carmichaelCounting, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1057.variants.agp_infinite :
    Tendsto carmichaelCounting atTop atTop := by
…
```
Mentions: Erdos1057.carmichaelCounting
theorem Erdos1057.erdos_1057.variants.agp_lower_bound Alford, Granville, and Pomerance [AGP94] proved that $C(x)>x^{2/7}$ for large $x$ sorry line 95 source
Source documentation
Alford, Granville, and Pomerance [AGP94] proved that $C(x)>x^{2/7}$ for large $x$.

depends on sorry what this means
Statement uses: Filter.Eventually, Real, GT.gt, Erdos1057.carmichaelCounting, HPow.hPow, HDiv.hDiv, OfNat.ofNat, Nat, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1057.variants.agp_lower_bound :
    ∀ᶠ x in atTop, carmichaelCounting x > x ^ (2 / 7 : ℝ) := by
…
```
Mentions: Erdos1057.carmichaelCounting
theorem Erdos1057.erdos_1057.variants.harman_lower_bound The lower bound $C(x)> x^{0.33336704}$ was proved by Harman [Ha08] sorry line 103 source
Source documentation
The lower bound $C(x)> x^{0.33336704}$ was proved by Harman [Ha08].

depends on sorry what this means
Statement uses: Filter.Eventually, Real, GT.gt, Erdos1057.carmichaelCounting, HPow.hPow, OfScientific.ofScientific, Bool.true, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1057.variants.harman_lower_bound :
    ∀ᶠ x in atTop, carmichaelCounting x > x ^ (0.33336704 : ℝ) := by
…
```
Mentions: Erdos1057.carmichaelCounting
theorem Erdos1057.erdos_1057.variants.lichtman_lower_bound This exponent was improved to $0.3389$ by Lichtman [Li22] sorry line 111 source
Source documentation
This exponent was improved to $0.3389$ by Lichtman [Li22].

depends on sorry what this means
Statement uses: Filter.Eventually, Real, GT.gt, Erdos1057.carmichaelCounting, HPow.hPow, OfScientific.ofScientific, Bool.true, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1057.variants.lichtman_lower_bound :
    ∀ᶠ x in atTop, carmichaelCounting x > x ^ (0.3389 : ℝ) := by
…
```
Mentions: Erdos1057.carmichaelCounting