FormalConjectures.ErdosProblems.«972»

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import FormalConjecturesUtil

Erdős Problem 972

Reference: erdosproblems.com/972


namespace Erdos972

The set of primes p such that Nat.floor (α * p) is also prime.

def primeSet (α : ℝ) : Set ℕ :=
  {p : ℕ | Nat.Prime p ∧ Nat.Prime ⌊ (α * p) ⌋₊}

Erdős problem 972. Let $\alpha > 1$ be irrational. Are there infinitely many primes $p$ such that $\lfloor p\alpha \rfloor$ is also prime?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_972 : answer(sorry) ↔ ∀ α > 1, Irrational α → (primeSet α).Infinite := by⊢ True ↔ ∀ α > 1, Irrational α → (primeSet α).Infinite
  sorryAll goals completed! 🐙


end Erdos972