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import FormalConjecturesUtilErdős Problem 376
namespace Erdos376
Are there infinitely many $n$ such that ${2n\choose n}$ is coprime to $105$?
@[category research open, AMS 11]
theorem erdos_376 : answer(sorry) ↔ { (n : ℕ) | n.centralBinom.Coprime 105 }.Infinite := ⊢ True ↔ {n | n.centralBinom.Coprime 105}.Infinite
All goals completed! 🐙
Erdős, Graham, Ruzsa, and Straus [EGRS75] have shown that, for any two odd primes $p$ and $q$, there are infinite many $n$ such that ${2n\choose n}$ is coprime to $pq$.
@[category research solved, AMS 11]
theorem erdos_376.variants.prime {p q : ℕ} (h₁ : p.Prime)
(h₂ : Odd p) (h₃ : q.Prime) (h₄ : Odd q) :
{ (n : ℕ) | n.centralBinom.Coprime (p * q) }.Infinite := p:ℕq:ℕh₁:Nat.Prime ph₂:Odd ph₃:Nat.Prime qh₄:Odd q⊢ {n | n.centralBinom.Coprime (p * q)}.Infinite
All goals completed! 🐙
end Erdos376