418.lean`

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7 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 418

References:

erdosproblems.com/418
Wikipedia: Noncototient
[BaLu05] Banks, William D. and Luca, Florian, Nonaliquots and {R}obbins numbers. Colloq. Math. (2005), 27--32.
[BrSc95] Browkin, J. and Schinzel, A., On integers not of the form {$n-\phi(n)$}. Colloq. Math. (1995), 55-58.
[ChZh11] Chen, Yong-Gao and Zhao, Qing-Qing, Nonaliquot numbers. Publ. Math. Debrecen (2011), 439--442.
[Er73b] Erdős, P., "Über die Zahlen der Form $\sigma (n)-n$ und $n-\phi(n)$. Elem. Math. (1973), 83-86.
[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.
[PoPo16] Pollack, Paul and Pomerance, Carl, Some problems of Erdős on the sum-of-divisors function. Trans. Amer. Math. Soc. Ser. B (2016), 1-26.

Imports (1)

FormalConjecturesUtil

theorem Erdos418.erdos_418 Are there infinitely many integers not of the form $n - \phi(n)$? sorry line 58 source
Source documentation
Are there infinitely many integers not of the form $n - \phi(n)$?
Asked by Erdős and Sierpiński. Numbers not of the form we call non-cototients.
Browkin and Schinzel [BrSc95] provided an affirmative answer to this question, proving that any integer of the shape $2^{k}\cdot 509203$ for $k\geq 1$ is a non-cototient.
This is discussed in problem B36 of Guy's collection [Gu04].
This was formalized in Lean by Alexeev using Aristotle.

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

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_418 : answer(True) ↔ { (n - n.totient : ℕ) | n }ᶜ.Infinite := by
…
```
theorem Erdos418.erdos_418.variants.conditional It follows from a slight strengthening of the Goldbach conjecture that every odd number can be written as $n - \phi(n)$ line 69 source
Source documentation
It follows from a slight strengthening of the Goldbach conjecture that every odd number can be written as $n - \phi(n)$. In particular, we assume that every even number greater than 6 can be written as the sum of two distinct primes, in contrast to the usual Goldbach conjecture that every even number greater than 2 can be written as the sum of two primes.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Even, Exists, And, Ne, Nat.Prime, Eq, HAdd.hAdd, Odd, Nat.totient
Proof uses: Or, LE.le, HMul.hMul, Decidable.byContradiction, Not, Eq.symm, Exists.intro, rfl, Eq.mpr, id, Eq.trans, Nat.Prime.two_le, congrArg, Nat.totient_mul, Nat.Coprime, propext, Nat.coprime_primes, HSub.hSub, Nat.totient_prime, Iff.mp and 27 more
68 instances, coercions and auxiliary constants left out.

Tactic trace: 22 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_418.variants.conditional
    (goldbach : ∀ (n : ℕ), 6 < n → Even n → ∃ p q, p ≠ q ∧ p.Prime ∧ q.Prime ∧ n = p + q)
    (m : ℕ) (h : Odd m) :
    ∃ n, m + n.totient = n := by
…
```
theorem Erdos418.erdos_418.variants.odd_noncototient The Odd Noncototient Conjecture: every non-cototient is even sorry line 98 source
Source documentation
The Odd Noncototient Conjecture: every non-cototient is even. Equivalently, every odd natural number is of the form $n - \phi(n)$ for some $n$.
This is the unconditional form of erdos_418.variants.conditional, which derives the odd case from a strengthening of the Goldbach conjecture. See [Wikipedia: Noncototient].

depends on sorry what this means
Statement uses: LE.le, Set, Nat, Compl.compl, Set.ofPred, Exists, Eq, HSub.hSub, Nat.totient, Even
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_418.variants.odd_noncototient :
    { (n - n.totient : ℕ) | n }ᶜ ⊆ { k | Even k } := by
…
```
theorem Erdos418.erdos_418.variants.seven_mem_cototient A sanity check for the definition: $7$ is a cototient, since $7 = 15 - \phi(15)$ and $\phi(15) = 8$ line 107 source
Source documentation
A sanity check for the definition: $7$ is a cototient, since $7 = 15 - \phi(15)$ and $\phi(15) = 8$. In particular $7$ is not a non-cototient.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Membership.mem, Nat, Set, Set.ofPred, Exists, Eq, HSub.hSub, Nat.totient, OfNat.ofNat
Proof uses: Exists.intro, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_418.variants.seven_mem_cototient :
    7 ∈ { (n - n.totient : ℕ) | n } :=
…
```
theorem Erdos418.erdos_418.variants.sigma Documented declaration: sigma sorry line 121 source
Source documentation
Erdős [Er73b] has shown that a positive density set of natural numbers cannot be written as $\sigma(n)-n$ (numbers not of this form are called nonaliquot, or sometimes untouchable).
Here "positive density" means positive lower density: Banks and Luca [BaLu05] proved that the set of nonaliquots has lower density at least $1/48$, Chen and Zhao [ChZh11] improved this to $0.06$, and it is not known whether the natural density of the nonaliquots exists (Pollack and Pomerance [PoPo16] give a heuristic predicting its value).

depends on sorry what this means
Statement uses: LT.lt, Real, OfNat.ofNat, Set.lowerDensity, Nat, Compl.compl, Set, Set.ofPred, Exists, Eq, HSub.hSub, ArithmeticFunction, ArithmeticFunction.sigma, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_418.variants.sigma :
    0 < { (σ 1 n - n : ℕ) | n }ᶜ.lowerDensity := by
…
```
theorem Erdos418.erdos_418.variants.soln A solution to erdos_418 was shown by Browkin and Schinzel [BrSc95] by showing that any integer of the form $2^(k + 1)\cdot 509203$ is not of the form $n - \phi(n)$ sorry line 130 source
Source documentation
A solution to erdos_418 was shown by Browkin and Schinzel [BrSc95] by showing that any integer of the form $2^(k + 1)\cdot 509203$ is not of the form $n - \phi(n)$.

depends on sorry what this means
Statement uses: LE.le, Set, Nat, Set.ofPred, Exists, Eq, HMul.hMul, HPow.hPow, OfNat.ofNat, HAdd.hAdd, Compl.compl, HSub.hSub, Nat.totient
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_418.variants.soln :
    { 2 ^ (k + 1) * 509203 | k } ⊆ { (n - n.totient : ℕ) | n }ᶜ := by
…
```
theorem Erdos418.erdos_418.variants.density It is open whether the set of non-cototients has positive (lower) density sorry line 138 source
Source documentation
It is open whether the set of non-cototients has positive (lower) density.

depends on sorry what this means
Statement uses: Iff, True, LT.lt, Real, OfNat.ofNat, Set.lowerDensity, Nat, Compl.compl, Set, Set.ofPred, Exists, Eq, HSub.hSub, Nat.totient, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
21 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_418.variants.density :
    answer(sorry) ↔ 0 < { (n - n.totient : ℕ) | n }ᶜ.lowerDensity := by
…
```