AI 中文总结
研究最小模为7的独特覆盖系统的最小公倍数，通过连续过滤论证构造出最小公倍数为10080的系统，并证明其最小性。
AI 中文摘要
我们确定了最小模为7的独特覆盖系统的最小公倍数。此前Klein构造了一个最小公倍数为15120的此类系统，并猜想该值是最小的。我们给出了一个最小公倍数为10080的构造，并证明不存在更小的最小公倍数。证明通过连续过滤论证进行，从10080以下7的可能倍数开始，先应用倒数和过滤器，再用除数完备整数规划过滤器，最后用更强的部分和过滤器，剩余少数困难情况通过Gurobi完整计算验证。
英文摘要
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.
Comments12 pages, no figures. Includes complete Gurobi computations; code available at https://github.com/zhangshiliang502-droid/Distinct-Covering-System-With-m-7
