32.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 32

References:

erdosproblems.com/32
[Erd54] Erdős, Paul, Some results on additive number theory. Proc. Amer. Math. Soc. (1954), 847-853.
[Guy04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437
[Ru98c] Ruzsa, Imre Z., On the additive completion of primes. Acta Arith. (1998), 269-275.

Imports (1)

FormalConjecturesUtil

def Erdos32.IsAdditiveComplementToPrimes A set $A \subseteq \mathbb{N}$ is an additive complement to the primes if every sufficiently large natural number can be written as $p + a$ for some prime $p$ and $a \in A$ line 41 source
Source documentation
A set $A \subseteq \mathbb{N}$ is an additive complement to the primes if every sufficiently large natural number can be written as $p + a$ for some prime $p$ and $a \in A$.

standard axioms: Quot.sound, propext what this means
Statement uses: Set, Nat
Proof uses: Filter.Eventually, Exists, And, Nat.Prime, Membership.mem, Eq, HAdd.hAdd, Filter.atTop
4 instances, coercions and auxiliary constants left out.
```
def IsAdditiveComplementToPrimes (A : Set ℕ) : Prop :=
…
```
theorem Erdos32.erdos_32.variants.log_squared Erdős proved in [Erd54] that there exists an additive complement $A$ to the primes with $|A \cap \{1, \ldots, N\}| = O((\log N)^2)$ sorry line 50 source
Source documentation
Erdős proved in [Erd54] that there exists an additive complement $A$ to the primes with $|A \cap \{1, \ldots, N\}| = O((\log N)^2)$.

depends on sorry what this means
Statement uses: Exists, Set, Nat, And, Erdos32.IsAdditiveComplementToPrimes, Asymptotics.IsBigO, Real, Filter.atTop, Finset.card, Finset.filter, Membership.mem, Classical.propDecidable, Finset.Icc, OfNat.ofNat, HPow.hPow, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_32.variants.log_squared : ∃ A : Set ℕ,
    IsAdditiveComplementToPrimes A ∧
    (fun N => (((Finset.Icc 1 N).filter (· ∈ A)).card : ℝ)) =O[atTop]
      fun N => (Real.log N) ^ 2 := by
…
```
Mentions: Erdos32.IsAdditiveComplementToPrimes
theorem Erdos32.erdos_32.variants.liminf_gt_one Must every additive complement $A$ to the primes satisfy $\liminf_{N \to \infty} \frac{|A \cap \{1, \ldots, N\}|}{\log N} > 1$? sorry line 62 source
Source documentation
Must every additive complement $A$ to the primes satisfy $\liminf_{N \to \infty} \frac{|A \cap \{1, \ldots, N\}|}{\log N} > 1$?

depends on sorry what this means
Statement uses: Set, Nat, Erdos32.IsAdditiveComplementToPrimes, LT.lt, EReal, OfNat.ofNat, Filter.liminf, HDiv.hDiv, Finset.card, Finset.filter, Membership.mem, Classical.propDecidable, Finset.Icc, Real.log, Real, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_32.variants.liminf_gt_one : ∀ A : Set ℕ,
    IsAdditiveComplementToPrimes A →
    (1 : EReal) < liminf (fun N => (((Finset.Icc 1 N).filter (· ∈ A)).card / Real.log N : EReal))
      atTop := by
…
```
Mentions: Erdos32.IsAdditiveComplementToPrimes
theorem Erdos32.erdos_32 Does there exist a set sorry line 74 source
Source documentation
Does there exist a set $A \subseteq \mathbb{N}$ such that $|A \cap \{1, \ldots, N\}| = o((\log N)^2)$ and every sufficiently large integer can be written as $p + a$ for some prime $p$ and $a \in A$?

depends on sorry what this means
Statement uses: Iff, True, Exists, Set, Nat, And, Erdos32.IsAdditiveComplementToPrimes, Asymptotics.IsLittleO, Real, Filter.atTop, Finset.card, Finset.filter, Membership.mem, Classical.propDecidable, Finset.Icc, OfNat.ofNat, HPow.hPow, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_32 : answer(sorry) ↔ ∃ A : Set ℕ,
    IsAdditiveComplementToPrimes A ∧
    (fun N => (((Finset.Icc 1 N).filter (· ∈ A)).card : ℝ)) =o[atTop]
      fun N => (Real.log N) ^ 2 := by
…
```
Mentions: Erdos32.IsAdditiveComplementToPrimes
theorem Erdos32.erdos_32.variants.log_bound Can the bound $O(\log N)$ be achieved for an additive complement to the primes? sorry line 86 source
Source documentation
Can the bound $O(\log N)$ be achieved for an additive complement to the primes? [Guy04] writes that Erdős offered $50 for the solution.

depends on sorry what this means
Statement uses: Iff, True, Exists, Set, Nat, And, Erdos32.IsAdditiveComplementToPrimes, Asymptotics.IsBigO, Real, Filter.atTop, Finset.card, Finset.filter, Membership.mem, Classical.propDecidable, Finset.Icc, OfNat.ofNat, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_32.variants.log_bound : answer(sorry) ↔ ∃ A : Set ℕ,
    IsAdditiveComplementToPrimes A ∧
    (fun N => (((Finset.Icc 1 N).filter (· ∈ A)).card : ℝ)) =O[atTop]
      fun N => Real.log N := by
…
```
Mentions: Erdos32.IsAdditiveComplementToPrimes
theorem Erdos32.erdos_32.variants.ruzsa Ruzsa proved that any additive complement sorry line 99 source
Source documentation
Ruzsa proved that any additive complement $A$ to the primes must satisfy $\liminf_{N \to \infty} \frac{|A \cap \{1, \ldots, N\}|}{\log N} \geq e^\gamma$, where $\gamma$ is the Euler-Mascheroni constant.

depends on sorry what this means
Statement uses: Set, Nat, Erdos32.IsAdditiveComplementToPrimes, LE.le, EReal, Real.exp, Real.eulerMascheroniConstant, Filter.liminf, HDiv.hDiv, Finset.card, Finset.filter, Membership.mem, Classical.propDecidable, Finset.Icc, OfNat.ofNat, Real.log, Real, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_32.variants.ruzsa : ∀ A : Set ℕ,
    IsAdditiveComplementToPrimes A →
    (Real.exp Real.eulerMascheroniConstant : EReal) ≤
      liminf (fun N => (((Finset.Icc 1 N).filter (· ∈ A)).card / Real.log N : EReal)) atTop := by
…
```
Mentions: Erdos32.IsAdditiveComplementToPrimes