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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.
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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.

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

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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