FormalConjectures.ErdosProblems.«850»

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import FormalConjecturesUtil

Erdős Problem 850

Reference: erdosproblems.com/850


namespace Erdos850

Can there exist two distinct integers $x$ and $y$ such that $x,y$ have the same prime factors, $x+1,y+1$ have the same prime factors, and $x+2,y+2$ also have the same prime factors?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_850 :
    answer(sorry) ↔ ∃ x y : ℕ, x ≠ y ∧ x.primeFactors = y.primeFactors
      ∧ (x + 1).primeFactors = (y + 1).primeFactors
      ∧ (x + 2).primeFactors = (y + 2).primeFactors := by⊢ True ↔
  ∃ x y,
    x ≠ y ∧
      x.primeFactors = y.primeFactors ∧
        (x + 1).primeFactors = (y + 1).primeFactors ∧ (x + 2).primeFactors = (y + 2).primeFactors
    sorryAll goals completed! 🐙


-- TODO(Paul-Lez): formalise remaining problems

end Erdos850