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.
Update, same day. A follow-up brief came back with real progress — still not a proof, but the obstruction is now exact and the verified range moved. Pro derived an identity: for a dangerous endpoint prime p, the shifted valuation equals a p-adically-rescaled copy of the source's valuation, minus a fixed "parity debt" Zp. We didn't take that on faith — we stress-tested it against ~3,900 randomly generated (m, N, p) triples in its stated domain (even n) with a from-scratch Legendre's-formula implementation, zero exceptions, and it reproduced every worked example in the brief exactly, including the tight n=18 case where the shift survives with literally zero valuation to spare.
The verified-witness range grew too. n=28's minimal witness — k = 18,253,129,921,815 — is now certified, not just a valid upper bound. The trick: it piggybacks on a public exhaustive computational certificate for a different Erdős problem, #396 (Justin Dehorty's project), which already proved 18,253,129,921,842 is the smallest N with a valid source at this size. We went and read that certificate ourselves against the live repo — the exact partition boundary and the "15 false alarms, 0 real witnesses below it" report are really there, not just described to us. That extends the shift conjecture's verified range from 13 minimal even witnesses (n≤26) to 14 (n≤28), still zero violations. Two more candidates (n=30, n=32) are named as valid shifts but not yet proven minimal — we confirmed their arithmetic too, exactly as reported.
What's still missing is the actual proof mechanism. The obstruction formula shows precisely what a general argument would need — that minimality always forces the p-adic debt to get paid — and shows why the obvious approach ("minimal witnesses have spare slack") is false: the n=18 case pays its debt with nothing left over. Pro's own honest lean is mild skepticism a universal theorem exists at all.
Update: what n=30 actually needs. No public K=14 certificate exists yet to piggyback on the way n=28 did — Dehorty's certified results still stop at K=13, even though OEIS has now separately published the raw K=14 and K=15 witness values (we checked the actual OEIS data file ourselves; both numbers are really there). But the picture isn't "go compute for months" either. A real #389-specific shortcut: for large primes, the source condition collapses to checking only 15 numbers — N, N−1, ..., N−14 — against a simple carry-counting rule, not the full 29-term window. We verified that reduction exactly, over 3,000+ generated cases, zero exceptions — including a genuinely subtle exception the brief caught on its own: the prime 29 gets a one-unit "free pass" that a naive reuse of #396's screen would wrongly disallow. Combined with a proposed integer-only (no floating point) sieve with a proven no-false-negative margin, the honest compute estimate for actually certifying n=30 drops from roughly 100,000–300,000 core-hours (porting #396's exact validator as-is) to roughly 1,000–10,000 (a purpose-built #389 sieve) — a real, order-of-magnitude difference, not just optimism. Still not free, still not done. N=359,503,904,702,190 remains a verified witness, not yet a certified minimum.