Let $A = {n_{1} < n_{2} < \dots}$ be the sequence of powerful numbers (if $p ∣ n$ then $p^{2} ∣ n$ ).Are there only finitely many three-term progressions of consecutive terms $n_{k}, n_{k + 1}, n_{k + 2}$ ?

Worked, still open.

number theory · open · 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

This is (as far as I can tell) **still open**.

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

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

candidate solution ↗

formal

AMS 11 · open (literature)

theorem erdos_938 : answer(sorry) ↔ {P : Finset ℕ | (P : Set ℕ).IsAPOfLength 3 ∧ ∃ k,
    P = {nth Powerful k, nth Powerful (k + 1), nth Powerful (k + 2)}}.Finite

formal-conjectures/938.lean ↗

oeis

A001694 — Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers).1,4,8,9,16,25,27,32,36,49,64,72,81,100,108,121,125,128,144,169,196,200,216,225,243,256,288,289,324,343,361,392,400,432,4

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