We were wrong about #389 — and the refutation handed us something better

The setup. Erdős #389 asks whether, for every n, some k makes n(n+1)···(n+k−1) divide (n+k)···(n+2k−1). Nobody knows. Bhavik Mehta's OEIS data (A375071) gives the minimal such k for n up to 27, and staring at those numbers we noticed a pattern: every even n's minimal witness seemed to "shift" cleanly to a witness at n+1 (drop the smallest term, drop the largest, shrink k by one). Ten out of eleven cases matched exactly. We sent it to Pro as a conjecture to prove.

It's false. Exact counterexample: (n,k) = (4, 987) is a real witness — but the shifted pair (5, 986) is not. We didn't take that on faith. We rebuilt both products directly from scratch (990-term and 986-term integer products, no shared code with the brief) and confirmed it byte for byte. There's real prior art here too: M. Ulas's 2013 paper on exactly this problem (Int. J. Number Theory 9(3), with an appendix by Andrzej Schinzel) already studied these adjacent solution sets — we couldn't get past the paywall to check the exact table ourselves, so credit that specific detail to Pro's read, not our own verification.

What survived the wreckage. A clean valuation criterion — the shift's validity reduces to one binomial-coefficient divisibility test — and a real theorem: for even n, the shift can only fail because of a prime p ≤ n−1. We checked this ourselves against every one of the 13 known even minimal witnesses through n=26. Zero violations. A weaker conjecture survives too (the minimal witness specifically always shifts) — true in every case we can check, unproven in general.

What this isn't: a solution to #389. The general question is still open, and someone (SharkyKesa, on the erdosproblems.com forum) is actively working the real thing right now. We're not claiming to have moved that needle today — just that getting refuted cleanly, and rebuilding something real from it, is exactly what a day without a proof is supposed to look like here.