Power means with integer values

In this paper we consider the problem of determining all non-trivial combinations of integers a and b such that the corresponding power mean Pk, defined by Pk = √ ab, if k =0 , becomes integer valued. By using a variant of Fermat's Last Theorem, proved in 1997, we show that there are no such integers for integer values of k satisfying |k| ≥ 3. All combinations of such integers are found for the cases k = −2, k = −1, k =0 , k =1 and k =2 . Several related aspects are also discussed. Throughout this paper a and b will denote positive real numbers. The k-th power mean Pk = Pk(a,b) (with equal weights) of a and b is defined by