Power free values of polynomials

Abstract

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 ( x 2 ) where e 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.