11.lean`

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

8 declarations, 7 contain sorry, 0 declared as axioms

Erdős Problem 11

Reference: erdosproblems.com/11

Imports (1)

FormalConjecturesUtil

theorem Erdos11.erdos_11 Is every large odd integer $n$ the sum of a squarefree number and a power of 2? sorry line 36 source
Source documentation
Is every large odd integer $n$ the sum of a squarefree number and a power of 2?

depends on sorry what this means
Statement uses: Iff, True, Filter.Eventually, Nat, Odd, Exists, And, Squarefree, Eq, HAdd.hAdd, HPow.hPow, OfNat.ofNat, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_11 : answer(sorry) ↔
    ∀ᶠ n : ℕ in atTop, Odd n → ∃ k l : ℕ, Squarefree k ∧ n = k + 2 ^ l := by
…
```
theorem Erdos11.erdos_11.variants.all_odd Every odd $n > 1$ is the sum of a squarefree number and a power of 2 sorry line 46 source
Source documentation
Every odd $n > 1$ is the sum of a squarefree number and a power of 2. This is the form in earlier versions of erdosproblems.com/11 and the hypothesis of [GrSo98]. It is stronger than erdos_11, which asks only about large $n$: a single exception would refute it.

depends on sorry what this means
Statement uses: Nat, Odd, LT.lt, OfNat.ofNat, Exists, And, Squarefree, Eq, HAdd.hAdd, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_11.variants.all_odd (n : ℕ) (hn : Odd n) (hn' : 1 < n) :
    ∃ k l : ℕ, Squarefree k ∧ n = k + 2 ^ l := by
…
```
theorem Erdos11.eventually_of_all_odd The form for every odd $n > 1$ implies the form for large odd $n$ line 52 source
Source documentation
The form for every odd $n > 1$ implies the form for large odd $n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Odd, LT.lt, OfNat.ofNat, Exists, And, Squarefree, Eq, HAdd.hAdd, HPow.hPow, Filter.Eventually, Filter.atTop
Proof uses: Iff.mpr, LE.le, Filter.eventually_atTop, Exists.intro, Decidable.byContradiction, Not
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem eventually_of_all_odd (H : type_of% erdos_11.variants.all_odd) :
    ∀ᶠ n : ℕ in atTop, Odd n → ∃ k l : ℕ, Squarefree k ∧ n = k + 2 ^ l :=
…
```
Mentions: Erdos11.erdos_11.variants.all_odd
theorem Erdos11.erdos_11.variants.not_four_dvd Erdős often asked this under the weaker assumption that $n > 1$ is not divisible by 4 sorry line 61 source
Source documentation
Erdős often asked this under the weaker assumption that $n > 1$ is not divisible by 4.

depends on sorry what this means
Statement uses: Nat, Not, Dvd.dvd, OfNat.ofNat, LT.lt, Exists, And, Squarefree, Eq, HAdd.hAdd, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_11.variants.not_four_dvd (n : ℕ) (hn : ¬ 4 ∣ n) (hn' : 1 < n) :
    ∃ k l : ℕ , Squarefree k ∧ n = k + 2^l := by
…
```
theorem Erdos11.erdos_11.variants.two_pow_two Is every odd $n > 1$ the sum of a squarefree number and two powers of 2? sorry line 69 source
Source documentation
Is every odd $n > 1$ the sum of a squarefree number and two powers of 2?

depends on sorry what this means
Statement uses: Nat, Odd, LT.lt, OfNat.ofNat, Exists, And, Squarefree, Eq, HAdd.hAdd, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_11.variants.two_pow_two (n : ℕ) (hn : Odd n) (hn' : 1 < n) :
    ∃ k l m : ℕ , Squarefree k ∧ n = k + 2^l + 2^m := by
…
```
theorem Erdos11.erdos_11.variants.finite_bound1 Every odd $1 < n < 10^7$ is the sum of a squarefree number and a power of 2 sorry line 77 source
Source documentation
Every odd $1 < n < 10^7$ is the sum of a squarefree number and a power of 2.

depends on sorry what this means
Statement uses: Nat, Odd, LT.lt, HPow.hPow, OfNat.ofNat, Exists, And, Squarefree, Eq, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_11.variants.finite_bound1 (n : ℕ) (hn : Odd n) (h : n < 10^7) (hn' : 1 < n) :
    ∃ k l : ℕ , Squarefree k ∧ n = k + 2^l := by
…
```
theorem Erdos11.erdos_11.variants.finite_bound2 Every odd $1 < n < 2^50$ is the sum of a squarefree number and a power of 2 sorry line 85 source
Source documentation
Every odd $1 < n < 2^50$ is the sum of a squarefree number and a power of 2.

depends on sorry what this means
Statement uses: Nat, Odd, LT.lt, HPow.hPow, OfNat.ofNat, Exists, And, Squarefree, Eq, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_11.variants.finite_bound2 (n : ℕ) (hn : Odd n) (h : n < 2^50) (hn' : 1 < n) :
    ∃ k l : ℕ , Squarefree k ∧ n = k + 2^l := by
…
```
theorem Erdos11.erdos_11.variants.granville_soundararajan Suppose that every odd $n > 1$ is the sum of a squarefree number and a power of 2 sorry line 96 source
Source documentation
Suppose that every odd $n > 1$ is the sum of a squarefree number and a power of 2. Then the set of primes $p$ such that $2^p \not\equiv 2 \pmod{p^2}$ (the non-Wieferich primes) is infinite. This follows from Theorem 1 in [GrSo98]. [GrSo98] Granville, A. and Soundararajan, K., A Binary Additive Problem of Erdős and the Order of $2$ mod $p^2$. The Ramanujan Journal (1998), 283-298.

depends on sorry what this means
Statement uses: Nat, Odd, LT.lt, OfNat.ofNat, Exists, And, Squarefree, Eq, HAdd.hAdd, HPow.hPow, Set.Infinite, Set.ofPred, Nat.Prime, Not, Nat.ModEq
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_11.variants.granville_soundararajan (H : type_of% erdos_11.variants.all_odd) :
    {p : ℕ | p.Prime ∧ ¬ 2 ^ p ≡ 2 [MOD p ^ 2]}.Infinite := by
…
```
Mentions: Erdos11.erdos_11.variants.all_odd