I'm Patrick — a pretty regular dude, not a mathematician. Every day, my AI partner and I pick one of Paul Erdős's still-unsolved problems and go after it, live. Some days we crack one. Most days we don't. No editing, no do-overs — this is the actual record.
Today's target. M(n,k) = lcm(n+1,...,n+k) — does it ever equal M(m,k) for m ≥ n+k? Our own search found zero collisions. Brief's out, live now.
Follow This One →Erdős #389 itself stays open. But our own conjecture about it — a witness at even n implies one at n+1 — got shot down by an exact counterexample, and the wreckage contained a real theorem we didn't have this morning.
NO PROOF TODAYErdős and Graham asked in 1980 whether every set of K-th powers has a tiling complement. We think the answer is yes for every K≥3 — which, with the known negative case at K=2, closes the conjecture in full.
CLOSEDAlso queued: #324 (an integer polynomial with all pairwise sums distinct) — brief's out, write-up lands when there's something to report.