FormalConjectures.ErdosProblems.«15»

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import FormalConjecturesUtil

Erdős Problem 15: Convergence of Series with Primes

Reference: erdosproblems.com/15


namespace Erdos15


open Filter Topology

Is it true that $\sum_{n=1}^\infty(-1)^n\frac{n}{p_n}$ converges, where $p_n$ is the sequence of primes?

Note: In the problem statement, $p_n$ is the $n$-th prime, indexed such that $p_1=2, p_2=3, \ldots$. We 0-index here to reflect how Nat.nth works.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_15 : answer(sorry) ↔
    Summable (fun k : ℕ => (-1 : ℚ) ^ (k + 1) * (k + 1) / (k.nth Nat.Prime)) := by⊢ True ↔ Summable fun k => (-1) ^ (k + 1) * (↑k + 1) / ↑(Nat.nth Nat.Prime k)
  sorryAll goals completed! 🐙



-- TODO: add the other statements from the additional material


end Erdos15