205.lean`

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

6 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 205

References:

erdosproblems.com/205
[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
[Ro34] Romanoff, N. P., Über einige Sätze der additiven Zahlentheorie. Math. Ann. (1934), 668-678.

Imports (1)

FormalConjecturesUtil

def Erdos205.IsRepresentable IsRepresentable f n states that n can be written as 2 ^ k + m for some k ≥ 0 where the number of prime divisors of m, counted with multiplicity, is less than f m line 43 source
Source documentation
IsRepresentable f n states that n can be written as 2 ^ k + m for some k ≥ 0 where the number of prime divisors of m, counted with multiplicity, is less than f m.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: Exists, And, Eq, HAdd.hAdd, HPow.hPow, OfNat.ofNat, LT.lt, ArithmeticFunction, ArithmeticFunction.cardFactors
14 instances, coercions and auxiliary constants left out.
```
def IsRepresentable (f : ℕ → ℝ) (n : ℕ) : Prop :=
…
```
theorem Erdos205.erdos_205.parts.i Is it true that all sufficiently large $n$ can be written as $2^k+m$ for some $k\geq 0$, where $\Omega(m)<\log\log m$? sorry line 58 source
Source documentation
Is it true that all sufficiently large $n$ can be written as $2^k+m$ for some $k\geq 0$, where $\Omega(m)<\log\log m$? (Here $\Omega(m)$ is the number of prime divisors of $m$ counted with multiplicity.)
Barreto and Leeham, using ChatGPT and Aristotle, have proved a negative answer, which was quantified by Tao and Alexeev (see the comments): in fact there are infinitely many $n$ such that, for all $k$ with $2^k<n$, $n-2^k$ has at least $$\gg \left(\frac{\log n}{\log\log n}\right)^{1/2}$$ many prime factors.

depends on sorry what this means
Statement uses: Iff, False, Filter.Eventually, Nat, Erdos205.IsRepresentable, Real.log, Real, Filter.atTop
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_205.parts.i : answer(False) ↔
    ∀ᶠ n : ℕ in atTop, IsRepresentable (fun m => Real.log (Real.log m)) n := by
…
```
Mentions: Erdos205.IsRepresentable
theorem Erdos205.erdos_205.parts.ii Is it true that all sufficiently large $n$ can be written as $2^k+m$ for some $k\geq 0$, where $\Omega(m)<\log\log m$? sorry line 71 source
Source documentation
Is it true that all sufficiently large $n$ can be written as $2^k+m$ for some $k\geq 0$, where $\Omega(m)<\log\log m$? (Here $\Omega(m)$ is the number of prime divisors of $m$ counted with multiplicity.) What about $<\epsilon \log\log m$?
Barreto and Leeham, using ChatGPT and Aristotle, have proved a negative answer, which was quantified by Tao and Alexeev (see the comments).

depends on sorry what this means
Statement uses: Iff, False, Real, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, Erdos205.IsRepresentable, HMul.hMul, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_205.parts.ii : answer(False) ↔
    ∀ ε > (0 : ℝ), ∀ᶠ n : ℕ in atTop,
      IsRepresentable (fun m => ε * Real.log (Real.log m)) n := by
…
```
Mentions: Erdos205.IsRepresentable
theorem Erdos205.erdos_205.parts.iii Is it true that all sufficiently large $n$ can be written as $2^k+m$ for some $k\geq 0$, where $\Omega(m)<\log\log m$? sorry line 85 source
Source documentation
Is it true that all sufficiently large $n$ can be written as $2^k+m$ for some $k\geq 0$, where $\Omega(m)<\log\log m$? (Here $\Omega(m)$ is the number of prime divisors of $m$ counted with multiplicity.) Or some more slowly growing function?
Barreto and Leeham, using ChatGPT and Aristotle, have proved a negative answer, which was quantified by Tao and Alexeev (see the comments).

depends on sorry what this means
Statement uses: Iff, False, Exists, Nat, Real, And, Asymptotics.IsLittleO, Filter.atTop, Real.log, Filter.Eventually, Erdos205.IsRepresentable
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_205.parts.iii : answer(False) ↔
    ∃ f : ℕ → ℝ, f =o[atTop] (fun m : ℕ => Real.log (Real.log m)) ∧
      ∀ᶠ n : ℕ in atTop, IsRepresentable f n := by
…
```
Mentions: Erdos205.IsRepresentable
theorem Erdos205.erdos_205.variants.many_prime_factors In fact there are infinitely many $n$ such that, for all $k$ with $2^k<n$, $n-2^k$ has at least $$\gg \left(\frac{\log n}{\log\log n}\right)^{1/2}$$ many prime factors sorry line 96 source
Source documentation
In fact there are infinitely many $n$ such that, for all $k$ with $2^k<n$, $n-2^k$ has at least $$\gg \left(\frac{\log n}{\log\log n}\right)^{1/2}$$ many prime factors.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Set.Infinite, Nat, Set.ofPred, LT.lt, HPow.hPow, LE.le, HMul.hMul, Real.sqrt, HDiv.hDiv, Real.log, ArithmeticFunction, ArithmeticFunction.cardFactors, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
46 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_205.variants.many_prime_factors : ∃ c > (0 : ℝ),
    {n : ℕ | ∀ k : ℕ, 2 ^ k < n →
      c * Real.sqrt (Real.log n / Real.log (Real.log n)) ≤ (Ω (n - 2 ^ k) : ℝ)}.Infinite := by
…
```
Mentions: c
theorem Erdos205.erdos_205.variants.odd_counterexamples The $n$ constructed in this way are divisible by a large power of $2$ sorry line 106 source
Source documentation
The $n$ constructed in this way are divisible by a large power of $2$. It remains open whether there exist arbitrarily large odd counterexamples.

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Nat, Set.ofPred, And, Odd, Not, Erdos205.IsRepresentable, Real.log, Real
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_205.variants.odd_counterexamples : answer(sorry) ↔
    {n : ℕ | Odd n ∧ ¬ IsRepresentable (fun m => Real.log (Real.log m)) n}.Infinite := by
…
```
Mentions: Erdos205.IsRepresentable