Not every real result fits the daily format. Some days the actual target gets refuted and something smaller but genuine survives the wreckage. Some days we're scouting for tomorrow's problem and trip over a different one entirely. This is where those live.
While getting refuted on #389, we found a 13-year-old paper (Ulas, with an appendix by Schinzel, Int. J. Number Theory 9(3)) that had already been staring at these exact adjacent solution pairs. Confirmed the paper is real — right journal, right DOI — good reminder to check the literature before falling in love with your own pattern.
See it in #389 →Not even an Erdős problem — a 2024 CS-theory question (Amarilli) asking whether a graph with treewidth ≥6 could exist with ≤20 edges. Answer: no. μ(6)=21 exactly, achieved by K₇. One citation (Joret–Wood 2018) did all the real work; we fetched the actual paper, confirmed the theorem, and checked the sharpness witness by hand.
Born from a conjecture of ours that turned out to be false (see #389). The conjecture died; this survived: for even n, a witness-shift can only fail because of a prime p ≤ n−1 — every larger prime is automatically harmless. Independently verified against all 13 known minimal witnesses through n=26. Zero violations.
See it in #389 →