Deterministic Smooth Number Cover and Lemoine Obstructions

We derive a deterministic covering theorem: for any even N ≥ 8, the interval [1,N] is covered by Y -smooth numbers and their complements with respect to N + 1 provided Y ≥ ⌈N/3⌉. We further identify the unique obstruction type just below this threshold: in the window ⌊N/5⌋ < Y < ⌊N/3⌋, we prove the cover fails if and only if N +1 admits a Lemoine representation N +1 = p+2q with primes p, q > Y . Motivated by computation, we propose a strengthened (well-balanced) Lemoine-type conjecture asserting that such representations can be chosen with both primes located within a polylogarithmic distance of N/3.