The Additive Completion of kth Powers

THE ADDITIVE COMPLETION OF Kth-POWERS

Let k ≥ 2 be an integer. For fixed N , we consider a set A of non-negative integers such that for all integer n ≤ N , n can be written as n = a + b, a ∈ A , b a positive integer. We are interested in a lower bound for the number of elements of A . Improving a result of Balasubramanian [1], we prove the following theorem: Theorem 1. |AN | ≥ N1− 1 k { 1 Γ(2− 1 k )Γ(1 + 1 k ) + o(1) } . 1. STATMEN...

Values of the Euler function free of kth powers

We establish an asymptotic formula for the number of positive integers n x for which φ(n) is free of kth powers. © 2006 Elsevier Inc. All rights reserved. MSC: 11N37; 11A25

Sums of kth Powers in the Ring of Polynomials With Integer Coefficients

A Binary Additive Equation Involving Fractional Powers

with integers m1, m2; henceforth, [θ] denotes the integral part of θ. Subsequently, the range for c in this result was extended by Gritsenko [3] and Konyagin [5]. In particular, the latter author showed that (1) has solutions in integers m1, m2 for 1 < c < 3 2 and n sufficiently large. The analogous problem with prime variables is considerably more difficult, possibly at least as difficult as t...

ON THE NUMBER OF ELEMENTS THAT ARE NOT kth POWERS IN A GROUP

Let k be a positive integer, and suppose that the number of elements of a group G that are not k th powers in G is nonzero but finite. If G is finite, we obtain an upper bound on |G|, and we present some conditions sufficient to guarantee that G actually is finite.