Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 329.10005
Autor: Erdös, Paul; Simonovits, Miklos
Title: On a problem of Hirschhorn. (In English)
Source: Am. Math. Mon. 83, 23-26 (1976).
Review: In 1973 M. D. Hirschhorn gave the following problem [Amer. math. Monthly 80, 675-677 (1973; Zbl 266.10009)]: Let q1 > 1 be given and qn+1-qn = prodi \leq n (1-{1 \over qi} ) -1. Then does it necessarily follow that qn = (1+o(1))n log n? In this note, the authors settle the problem in the affirmative. The background of this problem is that, if pn denotes the n-th prime, then the well-known sieve method gives that the number of integers between a and b which are not divisible by any of p1, ... ,pn is approximately (b-a) prodi \leq n (1-{1 \over pi} ). The interval (pn,pn+1], contains exactly one prime, i.e., exactly one integer not divisible by any pi(i \leq n). This suggests that pn+1-pn = prodi \leq n (1-{1 \over pi} ) -1. We know in this special case by the prime number theorem that qn = (1+o(1))n log n.
Reviewer: D.Suryanarayana
Classif.: * 11A41 Elemementary prime number theory
11B37 Recurrences
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