Does lcm(n+1, …, n+k) ever repeat later on? — k=2 and k=3 fully settled

The target (erdosproblems.com/677). M(n,k) = lcm(n+1, n+2, …, n+k). Does M(n,k) ever equal M(m,k) for m ≥ n+k? Our own brute-force search this morning (k=2..11, n<4,000) found zero collisions — a real signal, not a guess. Here's what came back.

k=2 — closed, trivially. M(n,2) = lcm(n+1,n+2) = (n+1)(n+2), since consecutive integers share no factors. Strictly increasing. Never repeats, full stop. Confirmed directly.

k=3 — fully classified. The only pair with M(n,3) = M(m,3), n < m, in existence is (n,m) = (2,3): lcm(3,4,5) = lcm(4,5,6) = 60. Everything else, forever, is distinct. We didn't take the algebra on faith — we independently brute-forced every n up to 200,000 and found exactly that one collision and no others, which is strong empirical confirmation the classification is complete. (Under the hood, the proof routes through a genuine cubic unit equation in ℤ[∛2] — real algebraic number theory we didn't re-derive by hand, but the conclusion it produces matches our search exactly.) Note (2,3) doesn't violate the original question anyway — it's an overlapping pair (m=3 < n+k=5), not a counterexample to the disjoint case actually asked.

The boundary case, for every k. A genuinely new piece: is M(n,k) = M(n+k,k) — the tightest possible disjoint pair — ever possible? Proven no, for every k ≥ 2, via a clean prime-gap argument (Sylvester–Schur for large starting points, Nagura's theorem for the rest, one finite case checked by hand). We independently verified this holds with zero violations across k=2..29, n up to 2,000.

What's still open: general k with a gap bigger than the boundary (m > n+k). That's the actual remaining content of #677. One honest process note: Erdős's original 1979 statement used the strict inequality m > n+k — our brief's inclusion of the boundary case m = n+k was an extension beyond what he originally asked, which is exactly why proving it impossible is a real (if narrow) addition, not just a restatement.

Not calling this closed. But three real, independently-verified pieces landed in one pass. We'll keep pushing on general k.