Sums and differences of four kth powers

Equal sums of four seventh powers

In this paper, the method used to find the smallest, nontrivial, positive integer solution of a1 + a 7 2 + a 7 3 + a 7 4 = b 7 1 + b 7 2 + b 7 3 + b 7 4 is discussed. The solution is 149 + 123 + 14 + 10 = 146 + 129 + 90 + 15. Factors enabling this discovery are advances in computing power, available workstation memory, and the appropriate choice of optimized algorithms. Introduction Diophantine...

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

Sums and Differences of Three k-th Powers

If k ≥ 2 is a positive integer the number of representations of a positive integer N as either x1 + x k 2 = N or x k 1 − x2 = N , with integers x1 and x2, is finite. Moreover it is easily shown to be Oε(N), for any ε > 0. It is known that if k = 2 or 3 then the number of representations is unbounded as N varies, but it is conjectured that the number of representations is bounded for k ≥ 4. Inde...

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...

Sums of powers via integration

Sum of powers 1 + · · · + n, with n, p ∈ N and n ≥ 1, can be expressed as a polynomial function of n of degree p + 1. Such representations are often called Faulhaber formulae. A simple recursive algorithm for computing coefficients of Faulhaber formulae is presented. The correctness of the algorithm is proved by giving a recurrence relation on Faulhaber formulae.