POWER FREE VALUES OF POLYNOMIALS III By M. N. HUXLEY and M. NAIR [Received 11 July 1978] 1. Introduction An integer which is not divisible by the fcth power of an integer greater than 1 is said to be k-free. Let/(a;) be an irreducible polynomial of degree g with integer coefficients. For k ^ 2, let A k (N) be the number of positive integers n ^ N for which f(n) is &-free, and let B k (N) be the number of primes p ^ N for which /( p) is fc-free. It is conjectured that A k (N) tends to infinity with JV, provided that for each prime p there is some residue class (modp k ) for which p k does not divide f(n), and that B k (N) also tends to infinity, provided now that for each prime p there is some reduced residue class (modp k ) for which p k does not divide f(N). It is further conjectured that there is some <p < 1 (depending on f(x) and k) with -A k {N)~a k h, (1.1) (1.2) when N* ^h^N, N -> oo, (1.3) where I denotes log N, and a k , b k are non-zero constants, given by certain infinite products. An historical discussion of these conjectures can be found in [2]. The conjectures are known when k ^ g and 9 = 1 , and for certain smaller ranges of k and <p. Nair [2, 3] has established them for k > \g, q> = 1, where A is a constant less than 1. More precisely he can show that f(n) represents (g — 2)-free numbers for g ^ 18, even if n be restricted to the sequence of primes. Our improvements in the present paper do not change the constant A, but we can now show that f(n) represents {g — 2)- free numbers for g ^ 15, and thsXf{p) with p prime represents {g — 2)-free numbers for g ^ 16; our argument just fails if g = 15 and n is restricted to the primes. We state our results as follows. THEOREM A. As N -> 00 and as for some integer tin 2 < t < k—lthe two conditions h/N««WM -» 00 (1.4) Proc. London Math. Soc. (3) 41 (1980) 66-82