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import FormalConjecturesUtilAndrica's conjecture
open Filternamespace Andrica
Andrica's conjecture The inequality $\sqrt{p_{n+1}}-\sqrt{p_n} < 1$ holds for all $n$, where $p_n$ is the $n$-th prime number.
@[category research open, AMS 11]
theorem andrica_conjecture (n : ℕ) :
Real.sqrt ((n+1).nth Nat.Prime) - Real.sqrt (n.nth Nat.Prime) < 1 := n:ℕ⊢ √↑(Nat.nth Nat.Prime (n + 1)) - √↑(Nat.nth Nat.Prime n) < 1
All goals completed! 🐙
Ferreira proved that Andrica's conjecture is true for sufficiently large n.
@[category research solved, AMS 11]
theorem andrica_conjecture.ferreira_large_n :
∀ᶠ n in atTop, Real.sqrt ((n+1).nth Nat.Prime) - Real.sqrt (n.nth Nat.Prime) < 1 := ⊢ ∀ᶠ (n : ℕ) in atTop, √↑(Nat.nth Nat.Prime (n + 1)) - √↑(Nat.nth Nat.Prime n) < 1
All goals completed! 🐙
end Andrica