2026/07/21 by Jiheng Zhang, Shiliang Zhang
#math.NT
We determine the minimum possible least common multiple of a distinct covering system whose minimum modulus is 7. Klein previously constructed such a system with least common multiple 15120 and conjectured that this value was minimal. We give a construction with least common multiple 10080, and we prove that no smaller least common multiple can occur. The proof is organized as a successive filtering argument. Starting from the possible multiples of 7 below 10080, we first apply a reciprocal-sum filter, then a divisor-completed integer-programming filter, then a stronger partial-sum filter. The few remaining hard cases are finally certified by complete Gurobi computations.