Let $2 \leq k \leq n - 2$ . Can $(k n)$ be the product of consecutive primes infinitely often? For example $(2 21) = 2 \cdot 3 \cdot 5 \cdot 7.$

Worked, still open.

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

machinery: S-unit-equation,consecutive-integer-window,smooth-numbers,prime-distribution,binomial-coefficient-factorization,Diophantine-finiteness,Kummer-carries

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

This is currently **open** (in the sense of “no proof known either way”), even in the special case $k=2$. It appears as **Erdős Problem #386**. ([Erdős Problems][1])

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_386 :
    answer(sorry) ↔ ∃ k ≥ 2, ∃ᶠ n in .atTop,
      k ≤ n - 2 ∧ ∃ p q : ℕ, n.choose k = ∏ i ∈ .Ico p q, nth Nat.Prime i

formal-conjectures/386.lean ↗

oeis

A280992 — Squarefree triangular numbers that are products of consecutive primes.1,3,6,15,105,210,255255

status

open

evidence

formal

oeis

status

Search Vela