1003.lean`

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

3 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 1003

Reference: erdosproblems.com/1003

Imports (1)

FormalConjecturesUtil

theorem Erdos1003.erdos_1003 Are there infinitely many solutions to $\phi(n) = \phi(n+1)$, where $\phi$ is the Euler totient function? sorry line 38 source
Source documentation
Are there infinitely many solutions to $\phi(n) = \phi(n+1)$, where $\phi$ is the Euler totient function?

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Nat, Set.ofPred, Eq, Nat.totient, HAdd.hAdd, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1003 : answer(sorry) ↔ Set.Infinite {n | φ n = φ (n + 1)} := by
…
```
theorem Erdos1003.erdos_1003.variants.Icc Erdős [Er85e] says that, presumably, for every $k \geq 1$ the equation $$\phi(n) = \phi(n+1) = \cdots = \phi (n+k)$$ has infinitely many solutions sorry line 48 source
Source documentation
Erdős [Er85e] says that, presumably, for every $k \geq 1$ the equation $$\phi(n) = \phi(n+1) = \cdots = \phi (n+k)$$ has infinitely many solutions.
[Er85e] Erdős, P., Some problems and results in number theory. Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) (1985), 65-87.

depends on sorry what this means
Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Set.Infinite, Set.ofPred, Membership.mem, Set, Set.Icc, Eq, Nat.totient, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1003.variants.Icc :
    answer(sorry) ↔ ∀ k ≥ 1, {n | ∀ i ∈ Set.Icc 1 k, φ n = φ (n + i)}.Infinite := by
…
```
theorem Erdos1003.erdos_1003.variants.eps87 Erdős, Pomerance, and Sárközy [EPS87] proved that for all large $x$, the number of $n \leq x$ with $\phi(n) = \phi(n+1)$ is at most $$\frac{x}{\exp((\log x)^{1/3})}$$ sorry line 59 source
Source documentation
Erdős, Pomerance, and Sárközy [EPS87] proved that for all large $x$, the number of $n \leq x$ with $\phi(n) = \phi(n+1)$ is at most $$\frac{x}{\exp((\log x)^{1/3})}$$.
[EPS87] Erd\H os, Paul and Pomerance, Carl and S'ark"ozy, Andr'as, On locally repeated values of certain arithmetic functions. {II}. Proc. Amer. Math. Soc. (1987), 1--7.

depends on sorry what this means
Statement uses: Filter.Eventually, Real, LE.le, Set.ncard, Nat, Set.ofPred, Erdos1003.erdos_1003.variants.eps87.match_1, And, Eq, Nat.totient, HAdd.hAdd, OfNat.ofNat, HDiv.hDiv, Real.exp, HPow.hPow, Real.log, Filter.atTop
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_1003.variants.eps87 :  ∀ᶠ x in atTop,
    {(n : ℕ) | (n ≤ x) ∧ φ n = φ (n + 1)}.ncard ≤
      x / Real.exp ((x.log) ^ ((1 : ℝ) / 3)) := by
…
```