Covering System with Restricted Moduli: Theory, Existence and Computational Structure

We study covering systems whose moduli are restricted to specific multiplicative families, including primes, prime powers, and smooth numbers. Combining analytic tools such as reciprocal-sum inequalities and sieve obstructions with computational methods using SAT solvers, ILP, and CRT-based search, we obtain explicit nonexistence results for prime-only covering systems, construct new examples for prime-power families, and provide extensive evidence for threshold behavior in smooth-moduli families. Our work unifies theoretical and algorithmic approaches, yielding new insights into covering systems under multiplicative constraints and suggesting several directions for future research.