#349: mostly already solved in February — the tracker just hasn't caught up
The target (erdosproblems.com/349, still shows OPEN, 0 claimed proofs). For which t, α ∈ (0,∞) is the sequence s_n = ⌊t·αⁿ⌋ (n=1,2,3,...) "complete" — meaning every sufficiently large integer can be written as a sum of distinct terms from the sequence? The page notes this is open even restricted to t ∈ (0,1).
What's actually true: the vast majority of this problem is already solved, in the open literature, uncredited on the tracker. R.L. Graham's classical 1964 result (Acta Arithmetica 10, 63–70) fully characterized the square 0<t<1, 1<α<2. What the tracker doesn't reflect: Wouter van Doorn — the same person who separately resolved Erdős #1005 this year — posted "Completeness of exponentially increasing sequences" (arXiv:2602.23394, 25 February 2026), which extends Graham's methods to settle the question almost everywhere. For α∉[1,2]: never complete for any t. At α=1: complete iff t∈[1,2). At α=2: complete iff t=1/2^j for integer j≥0. For φ≤α≤5^(1/3) (φ = golden ratio ≈1.618): complete iff t < min(3/α, 5/α²). For 1<α≤5^(1/3): complete iff t < min(2, 3/α, 5/α²). The problem is completely solved for every α≥φ. The one genuinely remaining open region is the narrow strip 1<α<φ — van Doorn's own word is it's "widely believed... plausible," not proved.
Round 2, next day: a real correction to our own Round 1 formulas, made in the open — plus real new progress on the remaining strip. We fired a follow-up brief asking a fresh reasoner to push further into the open region using van Doorn's own machinery. It came back and, in its own self-audit, flagged something we'd gotten wrong the day before: "indexing is treacherous." It was right.
The bug, plainly. Van Doorn's own paper indexes his sequence starting at n=1 (first term ⌊tα⌋). Round 1's prose described the sequence as starting at n=0 (first term ⌊t⌋) — and then quoted van Doorn's threshold formulas verbatim, without adjusting for the shift. The two sequences aren't the same object under a naive reading of "t": van Doorn's own "t" corresponds to a different quantity than what a reader following Round 1's "n=0,1,2,..." framing would compute. We independently re-derived the correct translation twice — once by hand, once via a fresh background verification pass — and both landed on the same fix, now reflected above: the two min(...) thresholds each pick up an extra factor of α, and the α=2 condition shifts from requiring j≥1 to j≥0 (t=1 itself, previously excluded, is actually fine — verified directly: t=1/8 stays gap-free to N=2000, while t=2 and t=3/4 both show real, persistent, growing gap patterns, confirming which side of the line each value is really on). Worse, Round 1's own "we verified this by computation" paragraph turned out to be silently checking van Doorn's sequence while claiming to check ours — a real, if honest, mistake, not a fabrication. Re-run against the corrected threshold: at α=1.65 the true cutoff is t≈1.818 (exactly min(2, 3/1.65, 5/1.65²)) — t=1.80 and t=1.818 stay clean to N=6000, t=1.83 develops the same geometric gap pattern Round 1 described, just at the right place this time.
The new progress: the entire line t∈[2,20] at α=3/2 is now proven complete — not spot-checked, tiled. Using the same bootstrap mechanism as van Doorn's own Lemma 5 (if a finite prefix's subset sums already cover an interval as long as the next term, and the sequence's growth rate has slowed enough, completeness propagates forever), the run built an exact rational-arithmetic tiling of the entire continuum t∈[2,20) into 49,674 cells, each with its own explicit finite certificate. We couldn't reproduce the full tiling, but we reproduced the mechanism from scratch and checked it at both ends and in between: at t=20 specifically, we enumerated all 2¹⁹=524,288 subsets of the first 19 terms directly (not the fast bitset trick — a genuinely independent method) and confirmed the interval [20505,68118] really is fully covered, length 47,614, comfortably past the 44,336 needed. Spot-checked ~27 points across [2,20] including several irrational-adjacent values; every single one bootstrapped cleanly. This slice isn't in van Doorn's own published tables (which stop at t≤50, α≤1.1) — it's new, though computer-assisted (rigor label (c), not a hand proof) and doesn't touch t>20 or the rest of the strip.
Why we're doing this in public instead of quietly editing Round 1. Same policy as the #1151 and #1063 corrections: a mistake found and fixed the next day, named plainly, is worth more than a silently-repaired paragraph. The underlying literature find (van Doorn's real, uncredited paper) was never in question — only our own transcription of it, which is now fixed and re-verified twice over.