1110.lean`

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

2 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 1110

Reference: Erdős Problem 1110

Imports (2)

def Erdos1110.Representable $n$ is representable with respect to $p$ and $q$ if it is the sum of a finite divisibility antichain of terms of the form $p^kq^l$ line 35 source
Source documentation
$n$ is representable with respect to $p$ and $q$ if it is the sum of a finite divisibility antichain of terms of the form $p^kq^l$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Exists, Finset, And, LE.le, Set, Erdos246.Gamma, IsAntichain, Dvd.dvd, Eq, Finset.sum, id
5 instances, coercions and auxiliary constants left out.
```
def Representable (p q n : ℕ) : Prop :=
…
```
theorem Erdos1110.erdos_1110 Let $p>q\geq 2$ be two coprime integers sorry line 50 source
Source documentation
Let $p>q\geq 2$ be two coprime integers. We call $n$ representable if it is the sum of integers of the form $p^kq^l$, none of which divide each other.
If $\{p,q\}\neq \{2,3\}$ then what can be said about the density of non-representable numbers? Are there infinitely many coprime non-representable numbers, that is, an infinite family of pairwise coprime non-representable integers?

depends on sorry what this means
Statement uses: Iff, True, Nat, LT.lt, LE.le, OfNat.ofNat, Nat.Coprime, Not, And, Eq, Exists, Set, Set.Infinite, Set.Pairwise, Membership.mem, Erdos1110.Representable
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1110 :
    answer(sorry) ↔ ∀ (p q : ℕ), q < p → 2 ≤ q →
      Nat.Coprime p q → ¬(p = 3 ∧ q = 2) →
      ∃ A : Set ℕ, A.Infinite ∧ A.Pairwise Nat.Coprime ∧
        ∀ n ∈ A, ¬Representable p q n := by
…
```
Mentions: Erdos1110.Representable