Any 𝑛 arithmetic progressions covering the first 2ⁿ integers cover all integers

Abstract

In 1958 S. Stein [7] defined a system of n congruences x = a< (mod bi), 1^-i^n, to be disjoint if no x satisfies more than one of them. He conjectured that for every disjoint system of n congruences with distinct moduli there exists an x, l^x^2", satisfying none of them. P. Erdos [2] proved this with w2n instead of 2" and proposed the stronger conjecture that any system of n congruence classes not covering all integers omits some x between 1 and 2". He proved this with 2" replaced by some constant depending only on n.