Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-28T04:46:59.990Z Has data issue: false hasContentIssue false
  • English
  • Français

On the power free values of polynomials

Published online by Cambridge University Press:  26 February 2010

C. Hooley
C. Hooley
Affiliation:
Department of Mathematics, The University of Durham.
Get access

Extract

In 1953 Erdős† proved in a characteristically ingenious manner that an irreducible‡; integral polynomial f(n) of degree r ≥ 3 represents (r – 1)-free integers (that is to say integers not divisible by an (r – 1)th power other than 1) infinitely often, provided that the obvious necessary condition be given that f(n) have no fixed (r – 1)th power divisors other than 1. His method did not, however, give a means for determining an asymptotic formula for N(x), the number of positive integers n not exceeding x for which f(n) is (r – 1)-free, nor did it even show that the positive integers n for which f(n) is (r – 1)-free had a positive lower density.

Type
Research Article
Information
Mathematika , Volume 14 , Issue 1 , June 1967 , pp. 21 - 26
Copyright
Copyright © University College London 1967

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

page 21 note † “Arithmetical properties of polynomials”, Journal London Math. Soc, 28 (1953), 416–425.Google Scholar

page 21 note ‡ Erdós also considers reducible polynomials, but these present neither the same interest nor difficulty.

page 21 note § “On the square-free values of cubic polynomials”, Crelle 1966 (in the press). We shall refer to this subsequently as [I].

page 24 note † “Sur les courbes algébriques et les varietes qui s'en d'duisent”, Actuel Scientif. Industr., 1041 (Paris, 1948).Google Scholar

page 24 note ‡ This uniformity is perhaps most quickly appreciated in this case by appealing to Weil's theory of character sums.

page 25 note † Journal London Mart. Soc, 27 (1952), 7–15; Lemma 10.Google Scholar