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Power free values of polynomials

Published online by Cambridge University Press:  26 February 2010

M. Nair
M. Nair
Affiliation:
Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD.
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If an integer does not have a k-th power of a positive integer, other than 1, for a divisor, it is said to be k–free. Let f(n) be an irreducible polynomial, with rational integer coefficients, of degree g, having no fixed k-th power divisors other than 1. We define

i.e. Nk(x) is the number of positive integers n not exceeding x such that f(n) is k-free. One would expect that f(n) is square-free for infinitely many n and further that, given x sufficiently large, there is an n with x < n ≤ x + h, such that f(n) is square-free for h = 0(x2) where ε is any real number > 0. These conjectures, however, seem to be extraordinarily difficult to prove. We begin with a brief account of the best results that have been attained so far.

Type
Research Article
Information
Mathematika , Volume 23 , Issue 2 , December 1976 , pp. 159 - 183
Copyright
Copyright © University College London 1976

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