Let $r \geq 2$ . An $r$ -powerful number $n$ is one such that if $p ∣ n$ then $p^{r} ∣ n$ .If $r \geq 4$ then can the sum of $r - 2$ coprime $r$ -powerful numbers ever be itself $r$ -powerful? Are there at most finitely many such solutions?Are there infinitely many triples of coprime $3$ -powerful numbers $a, b, c$ such that $a + b = c$ ?

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

evidence

unverified AI candidates (2)

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

### For (r\ge 4): can a sum of $r-2$ coprime $r$-powerful numbers be $r$-powerful?

llm-hunter · gpt pro 5.2 · unverified

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

AMS 11 · open (literature)

theorem erdos_939 : answer(sorry) ↔ ∀ r ≥ 4, (Erdos939Sums r).Nonempty

open