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import FormalConjecturesUtilErdős Problem 236
open Filter Asymptotics
namespace Erdos236
$f(n)$ counts the number of solutions to $n=p+2^k$ for prime $p$ and $k\geq 0$.
def f (n : ℕ) : ℕ :=
((List.range (Nat.log2 n + 1)).filter (fun k => Nat.Prime (n - 2^k))).length
Let $f(n)$ count the number of solutions to $n=p+2^k$ for prime $p$ and $k\geq 0$. Show that $f(n)=o(\log n)$.
@[category research open, AMS 5 11]
theorem erdos_236: (fun n => (f n : ℝ)) =o[atTop] (fun n => Real.log (n : ℝ)) := ⊢ (fun n => ↑(f n)) =o[atTop] fun n => Real.log ↑n
All goals completed! 🐙
end Erdos236