THE ADDITIVE COMPLETION OF Kth-POWERS
نویسنده
چکیده
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. STATMENT OF RESULT AND PRELIMINARY LEMMAS. 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) } . Lemma 1. [1]. If f(n) ≥ 0, then a+bk≤N f(a + b) ≥ ∑N n=1 f(n). Proof. It is obvious. The equality doesn’t hold in general because an integer n may have more than one representation as n = a + b. 1 2 Lemma 2. If f(x) = g( x N ), g continuous and g differentiable except at a finite number of points, then
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تاریخ انتشار 2008