244.lean`

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L4DEL: Lean4 Domain Explanation Language

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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 244

Reference: erdosproblems.com/244

Imports (1)

FormalConjecturesUtil

theorem Erdos244.erdos_244 Let $C > 1$ sorry line 33 source
Source documentation
Let $C > 1$. Does the set of integers of the form $p + \lfloor C^k \rfloor$, for some prime $p$ and $k\geq 0$, have density $>0$?

depends on sorry what this means
Statement uses: Iff, True, Real, GT.gt, OfNat.ofNat, LT.lt, Set.lowerDensity, Nat, Set.ofPred, Exists, Nat.Prime, Eq, HAdd.hAdd, Nat.floor, HPow.hPow, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
43 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_244 : answer(sorry) ↔
    ∀ C > (1 : ℝ), 0 < { p + ⌊C ^ k⌋₊ | (p) (k) (_ : p.Prime) }.lowerDensity := by
…
```
theorem Erdos244.erdos_244.variants.Romanoff Romanoff [Ro34] proved that the answer is yes if $C$ is an integer sorry line 42 source
Source documentation
Romanoff [Ro34] proved that the answer is yes if $C$ is an integer.
[Ro34] Romanoff, N. P., Über einige Sätze der additiven Zahlentheorie. Math. Ann. (1934), 668-678.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Real, Set.lowerDensity, Set.ofPred, Exists, Nat.Prime, Eq, HAdd.hAdd, Nat.floor, HPow.hPow, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_244.variants.Romanoff {C : ℕ} (hC : 1 < C) :
    0 < { p + ⌊C ^ k⌋₊ | (p) (k) (_ : p.Prime) }.lowerDensity := by
…
```