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erdős #1107

← #1106 · #1108 → (packet.json; erdosproblems.com)

Let $r \geq 2$ . A number $n$ is $r$ -powerful if for every prime $p$ which divides $n$ we have $p^{r} ∣ n$ . Is every large integer the sum of at most $r + 1$ many $r$ -powerful numbers?

Worked, still open.

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

use this record

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 $r=2$ the answer is **yes** (and this is a theorem):

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_1107 : ∀ r ≥ 2, ∀ᶠ n in atTop, SumOfRPowerful r n

formal-conjectures/1107.lean ↗

oeis

A056828 — Numbers that are not the sum of at most three powerful numbers (A001694).7,15,23,87,111,119 A392342 — Numbers that are not the sum of at most four cubefull numbers.5,6,7,12,13,14,15,20,21,22,23,31,38,39,46,47,53,58,69,77,79,85,95,101,103,111,175,196,212,228,231,247,327,444,458,490,60 A392343 — Numbers that are not the sum of at most five 4-full numbers.6,7,8,9,10,11,12,13,14,15,21,22,23,24,25,26,27,28,29,30,31,37,38,39,40,41,42,43,44,45,46,47,52,53,54,55,56,57,58,59,60,6

links

#941Are all large integers the sum of at most three powerful numbers (i.e. if

p ∣ n

p^{2} ∣ n

status

open

finding.noted · reviewer:will-blair · 1 day

renders the record as of vev_d199cb2e · 1,338 events · hub