Let $r \geq 3$ . A number $n$ is $r$ -powerful if for every prime $p$ which divides $n$ we have $p^{r} ∣ n$ .Are there infinitely many integers which are not the sum of at most $r$ many $r$ -powerful numbers? Does the set of integers which are the sum of at most $r$ $r$ -powerful numbers have density $0$ ?

Worked, still open.

number theory · open · possible · formalized (Lean) · 0 attempts

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

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unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

As of **January 2026**, for **every fixed (r\ge 3)** both questions you ask are **open** in general.

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

AMS 11 · open (literature)

theorem erdos_940 :
    answer(sorry) ↔ ∀ r ≥ 3,
      {n : ℕ | ∃ (S : Multiset ℕ), S.card ≤ r ∧ (∀ s ∈ S, r.Full s) ∧ n = S.sum}.HasDensity 0

open