erdős #421
Is there a sequence with density such that all products are distinct?
Worked, still open.
number theory · open · formalized (Lean) · 0 attempts
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vela registry pull vfr_37aec80d874a0239vela reproduce examples/erdos-problemsevidence
unverified AI candidates (2)
gpt-erdos · GPT-5.2 Pro + Deep Research · unverified
It’s an Erdős–Graham question (often listed as **Erdős Problem #421**) asking whether one can take a strictly increasing sequence of integers of **asymptotic density 1** whose **products over all index intervals** [ \prod_{u\le i\le v} d_i ] are all distinct. No construction with density $1$ is known, and no impossibil…
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_421 : answer(sorry) ↔
∃ (d : ℕ → ℕ), StrictMono d ∧ 1 ≤ d 0 ∧ HasDensity (Set.range d) 1 ∧
{(u, v) : ℕ × ℕ | u ≤ v}.InjOn fun (u, v) => ∏ i ∈ Finset.Icc u v, d iformal-conjectures/421.lean ↗oeis
A389544 — a(n) is the smallest integer greater than a(n-1) such that all consecutive products in a(1)..a(n) are distinct, with a(1) = 2.2,3,4,5,7,8,9,10,11,13,14,15,17,18,19,21,22,23,25,26,27,28,29,30,31,32,33,34,36,37,38,39,40,41,42,43,44,45,46,47,48,49,5A390848 — Complement of A389544.1,6,12,16,20,24,35,56,60,72,90,110,120,140,143,182,210,255,280,306,342,352,399,420,462,504,506,575,650,702,715,720,756,8
status
open