Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 186.07903
Autor: Erdös, Pál; Szemeredi, E.
Title: On a problem of P. Erdös and S. Stein (In English)
Source: Acta Arith. 15, 85-90 (1968).
Review: The system of congruences (1) ai (mod ni), n1 < ... < nk, is called a covering system if every integer satisfies at least one of the congruences (1). An old conjecture of Erdös states that for every integer c there is a covering system with n1 = c. This is still unproved. A system (1) is called disjoint if every integer satisfies at most one of the congruences (1). Denote by f(x) the largest value of k for which there is a disjoint system (1) satisfying nk \leq x. Erdös and Stein conjectured that f(x) = o(x). The authors prove a stronger result, namely Theorem 1 below. Theorem 1. For every \epsilon > 0 if x > x0(\epsilon) we have (c1 denotes a suitable positive constant) x/\exp ((log x) ½+\epsilon) < f(x) < x/(log x)c1.
Reviewer: P.Chowla
Classif.: * 11A07 Congruences, etc.
11B25 Arithmetic progressions
Index Words: number theory
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