erdős #364

← #363 · #365 → (packet.json; erdosproblems.com)

Are there any triples of consecutive positive integers all of which are powerful (i.e. if $p ∣ n$ then $p^{2} ∣ n$ )?

Worked, still open.

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

machinery: abc/kernel-bound,powerful-numbers,consecutive-integer-window,Pell-equation,superelliptic-curve,S-unit-equation,radical-bound,integral-points

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vela registry pull vfr_37aec80d874a0239

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evidence

unverified AI candidates (2)

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

No example is known, and the general question is 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_364 :
    ¬ ∃ (n : ℕ), Powerful n ∧ Powerful (n + 1) ∧ Powerful (n + 2)

formal-conjectures/364.lean ↗

oeis

A060355 — Numbers k such that k and k+1 are powerful numbers.8,288,675,9800,12167,235224,332928,465124,1825200,11309768,384199200,592192224,4931691075,5425069447,13051463048,2213222 A076445 — The smaller of a pair of powerful numbers (A001694) that differ by 2.25,70225,130576327,189750625,512706121225,13837575261123,99612037019889,1385331749802025,3743165875258953025,10114032809

links

#365Do all pairs of consecutive powerful numbers

n

and

n + 1

come from solutions to Pell equations? In other words, must either

n

n + 1

be a square?Is the number of such

n \leq x

bounded by

(lo g x)^{O (1)}

?A060355 #366Are there any

2

-full

n

such that

n + 1

3

-full? That is, if

p ∣ n

then

p^{2} ∣ n

and if

p ∣ n + 1

then

p^{3} ∣ n + 1

.A060355

status