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erdős #683

← #682 · #684 → (packet.json; erdosproblems.com)

Is it true that for every $1 \leq k \leq n$ the largest prime divisor of $(k n)$ , say $P ((k n))$ , satisfies $P ((k n)) \geq min (n - k + 1, k^{1 + c})$ for some constant $c > 0$ ?

Worked, still open.

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

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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

This is **not known** in general — it’s an **open problem of Erdős**.

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_683 : answer(sorry) ↔
    (∃ c > (0 : ℝ), ∀ n k : ℕ, 0 < k ∧ k < n → P n k > min (n - k + 1 : ℝ) (k ^ (1 + c)))

formal-conjectures/683.lean ↗

oeis

A006530 — Gpf(n): greatest prime dividing n, for n >= 2; a(1)=1.1,2,3,2,5,3,7,2,3,5,11,3,13,7,5,2,17,3,19,5,7,11,23,3,5,13,3,7,29,5,31,2,11,17,7,3,37,19,13,5,41,7,43,11,5,23,47,3,7,5,1 A074399 — a(n) is the largest prime divisor of n(n+1).2,3,3,5,5,7,7,3,5,11,11,13,13,7,5,17,17,19,19,7,11,23,23,5,13,13,7,29,29,31,31,11,17,17,7,37,37,19,13,41,41,43,43,11,23,A121359 — Greatest prime factor of pyramidal number A000292(n).2,5,5,7,7,7,5,11,11,13,13,13,7,17,17,19,19,19,11,23,23,23,13,13,13,29,29,31,31,31,17,17,17,37,37,37,19,41,41,43,43,43,23

links

#368How large is the largest prime factor of

n (n + 1)

?A074399 #928Let

α, β \in (0, 1)

P (n)

denote the largest prime divisor of

n

. Does the density of integers

n

P (n) < n^{α}

P (n + 1) < (n + 1)^{β}

status

open

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

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