1064.lean`

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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 1064

Reference: erdosproblems.com/1064

Imports (1)

FormalConjecturesUtil

theorem Erdos1064.erdos_1064 Let $ϕ(n)$ be the Euler's totient function, then the $n$ satisfies $ϕ(n)>ϕ(n - ϕ(n))$ have asymptotic density 1 sorry line 40 source
Source documentation
Let $ϕ(n)$ be the Euler's totient function, then the $n$ satisfies $ϕ(n)>ϕ(n - ϕ(n))$ have asymptotic density 1. Reference: [LuPo02] Luca, Florian and Pomerance, Carl, On some problems of {M}\polhk akowski-{S}chinzel and {E}rd\H os concerning the arithmetical functions {$\phi$} and {$\sigma$}. Colloq. Math.

depends on sorry what this means
Statement uses: Set.HasDensity, Nat, Set.ofPred, GT.gt, Nat.totient, HSub.hSub, OfNat.ofNat, Real, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
19 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1064 : {n | φ n > φ (n - φ n)}.HasDensity 1 := by
…
```
theorem Erdos1064.erdos_1064.variants.k2 Let $ϕ(n)$ be the Euler's totient function, there exist infinitely many $n$ such that $ϕ(n)< ϕ(n - ϕ(n))$ Reference: [GLW01] Grytczuk, A line 50 source
Source documentation
Let $ϕ(n)$ be the Euler's totient function, there exist infinitely many $n$ such that $ϕ(n)< ϕ(n - ϕ(n))$ Reference: [GLW01] Grytczuk, A. and Luca, F. and W'ojtowicz, M., A conjecture of {E}rdős concerning inequalities for the {E}uler totient function.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set.Infinite, Nat, Set.ofPred, LT.lt, Nat.totient, HSub.hSub
Proof uses: Set.infinite_of_injective_forall_mem, HMul.hMul, OfNat.ofNat, HPow.hPow, Eq, Decidable.byContradiction, Not, Nat.pow_right_injective, le_refl, id, Membership.mem, Set, HAdd.hAdd, Mathlib.Tactic.Ring.of_eq, Nat.rawCast, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Tactic.Ring.Common.pow_congr and 47 more
88 instances, coercions and auxiliary constants left out.

Tactic trace: 31 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1064.variants.k2 : {n | φ n < φ (n - φ n)}.Infinite := by
…
```
theorem Erdos1064.erdos_1064.variants.general_function For any function $f(n)=o(n)$, we have $\phi(n)>\phi(n-\phi(n))+f(n)$ for almost all $n$ sorry line 87 source
Source documentation
For any function $f(n)=o(n)$, we have $\phi(n)>\phi(n-\phi(n))+f(n)$ for almost all $n$. Reference: [LuPo02] Luca, Florian and Pomerance, Carl, On some problems of {M}\polhk akowski-{S}chinzel and {E}rd\H os concerning the arithmetical functions {$\phi$} and {$\sigma$}. Colloq. Math. (2002), 111--130.

depends on sorry what this means
Statement uses: Nat, Asymptotics.IsLittleO, Real, Filter.atTop, Set.HasDensity, Set.ofPred, LT.lt, HAdd.hAdd, Nat.totient, HSub.hSub, OfNat.ofNat, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
29 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1064.variants.general_function (f : ℕ → ℕ)
    (hf : (fun n ↦ (f n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ))) :
    {n : ℕ | φ (n - φ n) + f n < φ n}.HasDensity 1 := by
…
```