FormalConjectures.ErdosProblems.«372»

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import FormalConjecturesUtil

Erdős Problem 372

Reference: erdosproblems.com/372

Conjectured by Erdős and Pomerance. Proved by Balog, who showed the stronger quantitative result that this holds for $\gg \sqrt{x}$ many $n\leq x$, for all large $x$.

namespace Erdos372

Let $P(n)$ denote the largest prime factor of $n$. There are infinitely many $n$ such that $P(n)>P(n+1)>P(n+2)$.

@[category research solved, AMS 11]
theorem erdos_372 :
    {n : ℕ | Nat.maxPrimeFac n > Nat.maxPrimeFac (n + 1) ∧
      Nat.maxPrimeFac (n + 1) > Nat.maxPrimeFac (n + 2)}.Infinite := by⊢ {n | n.maxPrimeFac > (n + 1).maxPrimeFac ∧ (n + 1).maxPrimeFac > (n + 2).maxPrimeFac}.Infinite
  sorryAll goals completed! 🐙

end Erdos372