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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. Erdos repeated both conjectures at the number theory conferences in Boulder,

PROOF. By our hypothesis there exist congruences #=a* (mod &<), l^i^n,such that if T is the set of integers satisfying none of the congruences, then x^T t lfg#^2n , yet TT £ 0. T contains negative integers; let x 0 be the greatest nonpositive element of T. Then the congruences x=a,i-Xo (mod ô») satisfy (A). Now we assume we have n congruences satisfying. (A). Suppose x^a (mod b) is one. Since (A) implies b\a, there exists a prime p such that p a \ b but p«\a. Suppose b =p a q. Then we could replace this congruence with x=a (mod p a ) without losing (A). Moreover, if a>l and p\a y our original congruence could be replaced with x^a (mod p), still without losing (A). Thus we may assume all our congruences are of the form #=a (mod p a ) (for various primes p), where a> 1 implies p\ a. This is a start toward (B).We illustrate our proof of (C) by taking the case p = 2. By the last paragraph we can assume our congruences are of three types: 1 Supported in part by NSF grants GP 6663 and GP 8075. 1326

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