821.lean`

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

5 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 821

References:

erdosproblems.com/821
[BaHa98] Baker, R. C. and Harman, G., Shifted primes without large prime factors. Acta Arith. (1998), 331--361.
[Er35b] Erdős, P., On the normal number of prime factors of $p-1$ and some related problems concerning Euler's $\varphi$-function. Quart. J. Math. (1935), 205-213.
[Er74b] Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202.
[Li22] J. D. Lichtman, Primes in arithmetic progressions to large moduli and shifted primes without large prime factors. arXiv:2211.09641 (2022).
[LuPo11] Luca, Florian and Pollack, Paul, An arithmetic function arising from {C}armichael's conjecture. J. Théor. Nombres Bordeaux (2011), 697--714.

Imports (1)

FormalConjecturesUtil

def Erdos821.g Let $g(n)$ count the number of $m$ such that $\phi(m)=n$ line 45 source
Source documentation
Let $g(n)$ count the number of $m$ such that $\phi(m)=n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Set.ncard, Set.ofPred, Eq, Nat.totient
```
noncomputable def g (n : ℕ) : ℕ :=
…
```
theorem Erdos821.erdos_821 Is it true that, for every $\epsilon>0$, there exist infinitely many $n$ such that $g(n) > n^{1-\epsilon}$? sorry line 53 source
Source documentation
Is it true that, for every $\epsilon>0$, there exist infinitely many $n$ such that $g(n) > n^{1-\epsilon}$?

depends on sorry what this means
Statement uses: Iff, True, Real, GT.gt, OfNat.ofNat, Set.Infinite, Nat, Set.ofPred, Erdos821.g, HPow.hPow, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_821 :
    answer(sorry) ↔ ∀ ε > (0 : ℝ), { n : ℕ | (g n : ℝ) > (n : ℝ) ^ (1 - ε) }.Infinite := by
…
```
Mentions: Erdos821.g
theorem Erdos821.erdos_821.variants.pillai Pillai proved that $\limsup g(n)=\infty$ sorry line 61 source
Source documentation
Pillai proved that $\limsup g(n)=\infty$.

depends on sorry what this means
Statement uses: Eq, EReal, Filter.limsup, Nat, Erdos821.g, Filter.atTop, Top.top
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
21 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_821.variants.pillai :
    atTop.limsup (fun n : ℕ ↦ (g n : EReal)) = ⊤ := by
…
```
Mentions: Erdos821.g
theorem Erdos821.erdos_821.variants.erdos Erdős [Er35b] proved that there exists some constant $c>0$ such that $g(n) > n^c$ for infinitely many $n$ sorry line 70 source
Source documentation
Erdős [Er35b] proved that there exists some constant $c>0$ such that $g(n) > n^c$ for infinitely many $n$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Set.Infinite, Nat, Set.ofPred, LT.lt, HPow.hPow, Erdos821.g
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_821.variants.erdos :
    ∃ c > (0 : ℝ), { n : ℕ | (n : ℝ) ^ c < (g n : ℝ) }.Infinite := by
…
```
Mentions: c, Erdos821.g
theorem Erdos821.erdos_821.variants.lichtman The best known bound is that there are infinitely many sorry line 82 source
Source documentation
The best known bound is that there are infinitely many $n$ such that $g(n) > n^{0.71568\cdots}$, obtained by Lichtman [Li22] as a consequence of proving that there are $\geq \frac{x}{(\log x)^{O(1)}}$ many primes $p\leq x$ such that all prime factors of $p-1$ are $\leq x^{0.2843\cdots}$ (which improves a number of previous exponents, most recently Baker and Harman [BaHa98]).

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfScientific.ofScientific, Bool.true, Set.Infinite, Nat, Set.ofPred, LT.lt, HPow.hPow, Erdos821.g
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_821.variants.lichtman :
    ∃ c > (0.71568 : ℝ), { n : ℕ | (n : ℝ) ^ c < (g n : ℝ) }.Infinite := by
…
```
Mentions: c, Erdos821.g